AN ADVANCED COURSE OF INSTRUCTION IN CHEMICAL PRINCIPLES THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO DALLAS ATLANTA SAN FRANCISCO MACMILLAN & CO.. LIMITED LONDON BOMBAY CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, LTD. TORONTO AN ADVANCED COURSE OF INSTRUCTION IN CHEMICAL PRINCIPLES , ARTHUR A. NOYES DIRECTOR OF THE GATES CHEMICAL LABORATORY (CALIFORNIA INSTITUTE OF TECHNOLOGY AND MILES S. SHERRILL ASSOCIATE PROFESSOR OF THEORETICAL CHEMISTRY MASSACHUSETTS INSTITUTE OF TECHNOLOGY NEW YORK THE MACMILLAN COMPANY 1922 PRINTED IN THE UNITED STATES OF AMERICA PRELIMINARY EDITIONS COPYRIGHT, 1017 AND iQ2o, BY ARTHUR A. NOYES COMPLETE REVISION COPYRIGHT, 1022, BY ARTHUR A. NOYES AND MILES S. SHERRILL Set up and electrotyped. Published May, 1922 Press of J. J. Little & Ives Co. New York PREFACE IN this book are presented the results of the authors' many years' ex- perience with their own classes in the development of a thorough course of instruction in the laws and theories of chemistry from quantitative stand- points. The course is intended for junior, senior, or graduate students of physical chemistry in colleges, scientific schools, and universities, who have completed the usual freshman and sophomore courses in chemistry, physics, and mathematics. The purpose of the course is to give such students an intimate knowledge of fundamental chemical principles and a training in logical, scientific thinking, such as will enable them to attack effectively the practical prob- lems arising in their subsequent educational or professional work in any of the branches of chemistry or related sciences. Descriptive text-books of physical chemistry afford a general survey of chemical laws and theories which may suffice for the purposes of students who are not preparing for a professional career on the educational, research, or industrial sides of chemistry, but they do not give that intensive training which is essential for pursuing successfully more specialized courses of scientific study or for applying chemical principles to industrial problems. No one regards a course of lectures on the principles of mathematics as a suitable method of giving beginners the ability to handle that science, and with scarcely more reason can descriptive courses on physical chemistry be expected to afford a working knowledge of chemical principles in their quantitative aspects. Only by constantly applying the principles to concrete problems will the student acquire such a knowledge and the power to use it in new cases. Accordingly, the course of instruction is so planned as to make the student think about the significance of the principles presented and work out for himself the method of treatment of special cases upon the basis of those principles. To this end, the text is interspersed with problems, which prevent the student from memorizing the principles or complacently believing that a formal knowledge constitutes a real understanding of them. These problems are for the most part not of the usual type, involving merely substitution in formulas and mathematical operations, but are such as require clear logical thinking in the application of the principles to the cases under consideration. They are not merely supplementary or inci- dental to the text, but they are the feature about which the whole presen- yi PREFACE tation centers. The aim striven for has been to make each problem serve a definite purpose, and to have it involve independent thought, yet in such measure as shall not be beyond the mental capacity of college students who have completed good general courses in mathematics, physics, and chemistry. The problems have been gradually developed as a result of many years' trial of the plan with large classes of such students. While the authors realize that the presentation is in its details still far from perfect, they have decided to defer publication of it no longer, in the hope that this educational method, novel in this subject, may be further developed through the experience of other teachers, from whom suggestions will be gratefully received. In order to attain the purpose in view and yet keep the book within the limits of an undergraduate course, the subjects treated have been carefully selected from the point of view of their practical importance to the chemist; many topics being omitted that are commonly included in descriptive courses on physical chemistry or in complete treatises on the subject. The book consists mainly, as the table of contents shows, in a development of the atomic, kinetic, and ionic theories through a consideration of the physical properties directly related to them, and in a treatment, with the aid of these theories, from mass-action, phase, and thermodynamic view-points, of the principles relating to the rate and equilibrium of chemical reactions. The newer theories of atomic structure and of radiation have been reluctantly omitted; because, in spite of their transcendent interest and their im- portance to the future of chemistry, in their present stage of development they are appropriately treated in an advanced course of the research type, rather than in a fundamental undergraduate course of a systematic character. Lack of tune has led also to the omission of those more specialized parts of the subject that treat of the physical properties of substances in the various states of aggregation, such as the metallic, crystalline, liquid, and dispersed states; for, important as are some of the generalizations already arrived at in these fields, their inclusion in this course would have involved sacrifice of an adequate and intensive treatment of the more fundamental principles. A systematic notation (summarized on page 297) has been employed throughout the book. The course of instruction, as carried out by the authors, consists of class-room exercises, mainly of the recitation type, given to sections of twenty to thirty students. At each exercise a number of problems are assigned which are to be solved and handed in at the beginning of the next exercise; the students being advised to use slide-rules so as to shorten the arithmetical work. The instructor then solves these problems on the blackboard, questioning members of the class as to how he shall proceed. He emphasizes the principles involved and the best way of looking at the problems under consideration*. In the assignments it is customary not to PREFACE vii give out all at once the whole group of problems following a section of the text, but to assign only the first one or two of such a group, together with the last problems of the preceding group. This enables the instructor to discuss the principles relating to a new group of problems after the students have given some thought to them, but before they have passed to another topic; thereby insuring a better understanding of the general view-point from which it is best to attack the problems and removing difficulties which individual students may experience. Frequent written tests are given, con- sisting of review problems different from those already solved by the class, but involving the same principles. In order that the student may better appreciate the principles he is studying, it is desirable that the class work be accompanied by a brief laboratory course, or if that is not practicable by lecture experiments, whose primary purpose should be to give concrete illustrations of the nature of the basic phenomena under consideration (such as vapor-pressure, ion-migration, reaction-rate); for the experimental methods by which the properties are determined are not described in this book, except in so far as these are es- sential to an understanding of the phenomena. Moreover, no attempt has been made to present the historical and research aspects of the topics, because of the impossibility of doing this at all adequately in a brief sys- tematic treatment of the well-established principles of the subject. Many teachers will doubtless desire to supplement this systematic presentation by mentioning these historical and research aspects, and by suggesting the collateral reading of reviews and original articles that have appeared in chemical periodicals, or of text-books of physical chemistry in which these aspects are more fully discussed. To cover satisfactorily the whole subject as here presented requires from 120 to 150 exercises; but, in order to make it readily adaptable to a one-year go-hour course, certain articles and problems which are less important or more difficult are indicated by asterisks. It is also suggested that, when the subject must be completed in a one-year course, no attempt be made to include the thermodynamic treatment presented in Chapters X-XIII, but that, after taking up the chapter on thermochemistry, the course be completed by considering (without reference to its thermodynamic derivation) the applications of the important van't Hoff equation, as pre- sented in Arts. 167-169. When the course extends through a year and a half, the second year work will naturally consist of Chapters IX-XIII, these constituting a systematic course on thermodynamic chemistry, which in- deed may be pursued as a graduate subject by those who have not studied the earlier chapters of the book. The authors desire to express their indebtedness to many of their col- leagues for valuable suggestions, and especially to Professors S. J. Bates and E. B. Millard, and to Mr. Roger Williams, who have furnished detailed lists viii PREFACE of corrections and general criticisms. The authors desire also to thank Drs. G. N. Lewis and Merle Randall for the privilege of including, in ad- vance of its publication in their book on Thermodynamics and the Free Energy of Chemical Substances, a table showing the values of the electrode- potent^als recently derived by them. Pasadena and Cambridge, March, IQ22. CONTENTS PART I THE ATOMIC, MOLECULAR, AND IONIC THEORIES AND THE PROPERTIES OF SUBSTANCES DIRECTLY RELATED TO THESE THEORIES CHAPTER I: THE COMPOSITION OF SUBSTANCES AND THE ATOMIC THEORY PAGE 1. The Field of Chemistry. Its General Principles the Subject of this Course . . . . ... >.,, , 3 2. Pure Substances and Mixtures, Elementary and Compound Sub- stances, the Elements, and the Law of Definite Proportions . 3 3. The Law of Combining Weights 4 4. The Atomic Theory 5 5. Determination of Combining and Atomic Weights .... 6 6. Values of the Atomic Weights 8 7. Chemical Formulas, Formula- Weights, and Equivalent- Weights 8 CHAPTER II: THE MOLAL PROPERTIES OF GASES AND THE MOLECULAR AND KINETIC THEORIES I. THE VOLUME OF GASES IN RELATION TO PRESSURE, TEMPERATURE, AND MOLAL COMPOSITION 8. The Volume of Perfect Gases in Relation to Pressure and Temperature 10 9. Law of Combining Volumes and the Principle of Avogadro . . 12 10. Empirical Definition of Molecular Weight and of Mol ... 12 11. General Expression of the Laws of Perfect Gases 13 12. Dalton's Law of Partial Pressures 14 13. Determination of Molecular Weights 16 14. Derivation of the Atomic Weights of Elements and of the Mo- lecular Composition of Compounds 17 15. Deviations from the Perfect-Gas Laws at Moderate Pressures . 19 16. Pressure-Volume Relations of Gases at High Pressures ... 19 x CONTENTS H. THE KINETIC THEORY PAGE 17. The Fundamental Kinetic Hypotheses 22 18. The Kinetic Equation for Perfect Gases 22 19. The Kinetic Energy of the Molecules and the Avogadro Number 24 20. The Kinetic Equation for Imperfect Gases . 25 21. The Characteristics of the Molecules 29 22. Distribution of the Velocities and Kinetic Energies of the Mole- cules of a Gas .... V . . 31 HI. THE ENERGY RELATIONS OF GASES 23. Energy in General and the Law of its Conservation .... 35 24. Work Attending Volume Changes in General . . . . . . 37 25. The Energy-Content of Systems in General in Relation to Tem- perature and Heat- Capacity . . . , 39 26. The Energy-Content and Heat-Capacity of Perfect Gases . . 40 27. The Heat- Capacity of Perfect Gases in Relation to their Mo- lecular Composition . ' . . . . . 41 28. The Energy-Content and Heat-Capacity of Perfect Gases in Re- lation to the Kinetic Theory 42 29. The Energy-Content of Imperfect Gases in Relation to Volume and Pressure : . 44 30. The Energy-Content of Imperfect Gases at Constant Temperature in Relation to the Kinetic Theory 45 CHAPTER III: THE MOLAL PROPERTIES OF SOLUTIONS AND THE MOLECULAR THEORY I. VAPOR-PRESSURE AND BOILING-POINT IN GENERAL 31. Vapor-Pressure 47 32. Relation of Boiling-Point to Vapor-Pressure 48 33. Change of Vapor-Pressure with Temperature. The Clapeyron Equation 48 34. Vaporization in Relation to the Kinetic Theory 51 II. SOLUTIONS IN GENERAL 35. The Nature and Composition of Solutions 55 m. VAPOR-PRESSURE AND BOILING-POINT OF PERI.ZCT SOLUTIONS WITH ONE VOLATILE COMPONENT 36. Raoult's Law of Vapor-Pressure Lowering 57 37. Relation of Boiling-Point Raising to Vapor-Pressure Lowering and Molal Composition 59 38. Determination of Molecular Weights 62 39. Partial Vapor-Pressure of Volatile Solutes. Henry's Law . . 63 CONTENTS xi IV. VAPOR-PRESSURE AND BOILING-POINT OF CONCENTRATED SOLUTIONS WITH TWO VOLATILE COMPONENTS PAGE 40. The Vapor-Pressure of Concentrated Solutions 67 41. Vapor-Pressure and Boiling-Point of Concentrated Perfect Solu- tions in Relation to their Molal Composition ..... 67 42. Vapor-Pressure of Concentrated Solutions in General in Relation to their Composition 69 43. Boiling-Point of Concentrated Solutions in General in Relation to their Composition 72 V. DISTRIBUTION BETWEEN PARTIALLY MISCIBLE SOLVENTS 44. Determination of Equilibrium-Conditions by the Perpetual- Motion Principle 76 45. Distribution of a Solute between Two Non-Miscible Solvents . 76 46. The Lowering by Solutes of the Solubility of One Solvent in Another Solvent 78 47. The Vapor-Pressure Relations of Partially Miscible Liquids . . 79 VI. THE FREEZING-POINT OF SOLUTIONS 48. Freezing-Point and Its Relation to Vapor-Pressure and Molal Composition 80 49. Determination of Molecular Weights ........ 83 VII. OSMOTIC PRESSURE OF SOLUTIONS 50. Osmotic Pressure 84 51. Relation of Osmotic Pressure to Vapor-Pressure 85 52. Relation between Osmotic Pressure and Molal Composition . . 87 Vin. REVIEW OF THE PRINCIPLES RELATING TO THE MOLAL PROPERTIES 53. Review Problems 89 CHAPTER IV: THE ATOMIC PROPERTIES OF SOLID SUBSTANCES I. HEAT-CAPACITIES OF SOLID SUBSTANCES 54. Properties in the Solid State . 91 55. The Heat-Capacity of Solid Elementary Substances .... 91 56. The Heat-Capacity of Solid Compound Substances .... 93 57. Determination of Atomic Weights 94 H. GENERALIZATIONS RELATING TO ATOMIC WEIGHTS 58. Methods of Atomic Weight Determination and the Periodic Law 95 xii CONTENTS CHAPTER V: THE ELECTROLYTIC BEHAVIOR OF SOLUTIONS AND THE IONIC THEORY I. THE MOLAL PROPERTIES OF SOLUTIONS OF IONIZED SUBSTANCES PAGE 59. Effects of Salts on the Molal Properties of Aqueous Solutions and their Explanation by the Ionic Theory 96 H. ELECTROLYSIS AND FARADAY'S LAW 60. Electrolytic Conduction 98 61. Chemical Changes at the Electrodes 98 62. Faraday's Law 99 63. The Mechanism of Conduction between Electrodes and Solutions 100 m. ELECTRICAL TRANSFERENCE 64. The Phenomenon of Electrical Transference 104 65. The Law of Transference 105 66. Transference-Numbers . - 105 67. The Mechanism of Conduction in Solutions 106 68. Transference in Relation to the Mobility of the Ions .... 107 69. The Moving-Boundary Method of Determining Transference . 108 70. Change of Transference-Numbers with the Concentration . . 109 71. The Composition of Ions Determined by Transference Experi- ments no IV. ELECTRICAL CONDUCTANCE 72. Conductance, Specific Conductance, and Equivalent Conductance 112 73. Conductance in Relation to the Mobility of the Ion-Constituents 113 74. Change of Equivalent Conductance with the Concentration . . 114 75. Conductance in Relation to the Concentration and Mobility of the Ions . . ... . . . . . u6 76. Conductance of the Separate Ion-Constituents - 119 77. Determination of the Concentration of Ions and of Largely Ionized Substances by Conductance Measurements . . . 121 V. THE IONIZATION OF SUBSTANCES 78. lonization of Substances Not Largely Ionized ... . . . 123 79. lonization of Largely Ionized Substances ....... 123 VI. APPLICATIONS OF THE LAWS OF CONDUCTANCE AND TRANSFERENCE 80. Review Problems . . 127 CONTENTS xiii PART II THE RATE AND EQUILIBRIUM OF CHEMICAL CHANGES FROM MASS-ACTION AND PHASE VIEW-POINTS CHAPTER VI: THE RATE OF CHEMICAL CHANGES I. THE EFFECT OF CONCENTRATION ON REACTIONS BETWEEN SOLUTES OR GASEOUS SUBSTANCES PAGE 81. Concept of Reaction-Rate 131 82. The Mass-Action Law of Reaction-Rate between Solutes . . 131 83. First-Order Reactions 132 84. Second-Order and Third-Order Reactions 133 85. The Mechanisn^ of Reactions and the General Mass-Action Law of Reaction-Rate 134 86. The Mass-Action Law of Reaction-Rate between Gaseous Sub- stances 135 87. Simultaneous Reactions 136 II. THE EFFECT OF CONCENTRATION AND SURFACE ON REACTIONS BETWEEN SOLUTES AND SOLID SUBSTANCES 88. Rate of Reaction between Solutes and Solid Substances . . . 139 89. Solid Substances Dissolving in their Own Solutions .... 139 in. THE EFFECT OF CATALYSTS 90. Catalysis ..... . . . . 140 91. Carriers . 140 92. Contact Agents . . . . . . . . ( . . . . . . 141 93. Hydrogen-Ion and Hydroxide-Ion as Catalysts . . . . . 142 94. Enzymes 143 95. Water as a Catalyst .............. 143 IV. THE EFFECT OF TEMPERATURE 96. Principles Relating to the Effect of Temperature 144 CHAPTER VII: THE EQUILIBRIUM OF CHEMICAL CHANGES AT CONSTANT TEMPERATURE I. THE LAW OF MASS-ACTION 97. The Equilibrium of Chemical Reactions 145 98. The Mass-Action Law of Chemical Equilibrium 146 xiv CONTENTS H. THE MASS-ACTION LAW FOR REACTIONS BETWEEN GASES PAGE 99. The Mass-Action Law in Terms of Partial Pressures . . . 148 100. Gaseous Dissociation 148 101. Metathetical Gas Reactions 150 m. THE MASS-ACTION LAW FOR REACTIONS BETWEEN SOLUTES 102. lonization of Slightly Ionized Univalent Acids and Bases . . 152 103. lonization of Largely Ionized Substances . 153 104. The lonization of Water . . . 154 105. The Hydrolysis of Salts 154 1 06. lonization of Dibasic Acids and their Acid Salts 157 107. Displacement of One Acid or Base from its Salt by Another . 159 108. Neutralization-Indicators . . ... . 161 IV. THE MASS-ACTION LAW FOR REACTIONS INVOLVING SOLID PHASES 109. Form of the Mass-Action Expression . ^ . ' . ' ;. * . . . 165 no. Reactions Involving Solid and Gaseous Substances .... 165 in. Solubility of Unionized and Slightly Ionized Substances . . 166 112. Solubility of Largely Ionized Substances 167 113. The Mass-Action of Imperfect Solutes. The Concept of Activity 169 CHAPTER VIII: EQUILIBRIUM OF CHEMICAL SYSTEMS IN RELATION TO THE PHASES PRESENT I. GENERAL CONSIDERATIONS 114. General Considerations 174 II. ONE-COMPONENT SYSTEMS 115. Representation of the Equilibrium-Conditions by Diagrams . 176 III. THE PHASE RULE 116. The Concept of Variance and Inductive Derivation of the Phase Rule r 8o 117. Discussion of the Concept of Components 181 1 1 8. Derivation of the Phase Rule from the Perpetual-Motion Principle 183 IV. TWO-COMPONENT SYSTEMS 119. Systems with Solid and Gaseous Phases ^7 120. Systems with Solid, Liquid, and Gaseous Phases. Pressure- Temperature Diagrams j88 121. Systems with Solid and Liquid Phases. Temperature-Compo- sition Diagrams I9 o CONTENTS xv PAGE 122. Systems with Liquid and Gaseous Phases 197 123. Systems Involving Solid Solutions 197 V. THREE-COMPONENT SYSTEMS 124. Systems with Liquid and Solid Phases. Temperature-Composi- tion Diagrams 201 125. Systems with Gaseous, Liquid, and Solid Phases in Relation to the Phase Rule and Mass-Action Law 203 PART III THE ENERGY EFFECTS ATTENDING CHEMICAL CHANGES, AND THE EQUILIBRIUM OF CHEMICAL CHANGES IN RELATION TO THESE EFFECTS CHAPTER IX: THE PRODUCTION OF HEAT BY CHEMICAL CHANGES I. THE FUNDAMENTAL PRINCIPLES OF THERMOCHEMISTRY 126. The First Law of Thermodynamics . . -. 207 127. Heat-Effects Attending Isothermal Changes in State . . . 208 128. Change in Energy-Content and in Heat-Content . . . . . 209 129. Expression of Changes in Heat-Content by Thermochemical Equations . 211 130. Indirect Determination of the Heat-Effects of Chemical Changes 214 131. Influence of Temperature on the Heat-Effects Attending Chem- ical Changes . '. ..,'... .......... 215 H. GENERAL RESULTS OF THERMOCHEMICAL INVESTIGATIONS 132. Heat-Effects Attending Changes in the State of Aggregation of Substances 217 133. Heats of Reaction in Aqueous Solution 218 134. Applications of Thermochemical Principles 221 CHAPTER X: THE PRODUCTION OF WORK BY ISOTHERMAL CHEMICAL CHANGES IN RELATION TO THEIR EQUILIBRIUM CONDITIONS I. THE SECOND LAW OF THERMODYNAMICS AND THE CONCEPT OF FREE-ENERGY 135. The Second Law of Thermodynamics and its Application to Iso- thermal Changes in State 223 136. The Concepts of Work-Content and Free-Energy .... 226 xvi CONTENTS II. FREE-ENERGY CHANGES ATTENDING PHYSICAL CHANGES PAGE 137. Free-Energy Changes Attending Changes in Volume and Pres- sure 230 138. Free-Energy Changes Attending the Transfer of Substances be- tween Solutions of Different Concentrations 231 139. Free-Energy Changes Attending the Transfer of Substances be- tween Different Phases 234 m. FREE-ENERGY CHANGES ATTENDING CHEMICAL CHANGES 140. The Free-Energy Equation for Chemical Changes between Perfect Gases *-* . 238 141. The Free-Energy Equation for Chemical Changes between Perfect Solutes 240 142. The Free-Energy Equations for Chemical Changes between Solid Substances and Perfect Gases or Perfect Solutes . . 242 CHAPTER XI: THE PRODUCTION OF WORK FROM ISOTHER- MAL CHANGES BY ELECTROCHEMICAL PROCESSES I. CHANGES IN STATE AND IN FREE ENERGY IN VOLTAIC CELLS 143. Introduction 243 144. Changes in State in Voltaic Cells 243 145. The Production of Work and the Corresponding Decrease of Free Energy in Voltaic Cells . . ... ... . . . 247 H. THE ELECTROMOTIVE FORCE OF CELLS UNDERGOING ONLY CHANGES IN CONCENTRATION "OR PRESSURE 146. Change of the Electromotive Force of Voltaic Cells with the Concentration of the Solutions 249 147. The Electromotive Force of Concentration-Cells . . . . . 250 HI. THE ELECTROMOTIVE FORCE OF CELLS UNDERGOING CHEMICAL CHANGES AND ITS RELATION TO THEIR EQUILIBRIUM CONDITIONS 148. Chemical Changes in Voltaic Cells 252 149. The Electromotive Force of Cells in Relation to the Equilibrium Conditions of the Chemical Reactions . . . ... . 252 IV. ELECTRODE-POTENTIALS AND LIQUID-POTENTIALS 150. The Nature of Electrode-Potentials '. . . 254 151. The Expression of Electrode-Potentials ....... 255 152. Change of Electrode-Potentials with the Ion-Concentrations, and the Concept of Molal Electrode-Potentials .... 256 CONTENTS xvii PAGE 153. Values of the Molal Electrode-Potentials 259 154. Derivation of Related Molal Electrode-Potentials from One Another 260 155. The Electromotive Force of Cells with Dilute Solutions in Re- lation to the Molal Electrode-Potentials 260 156. The Electromotive Force of Cells with Concentrated Solutions . 261 157. The Nature of Liquid-Potentials and Expressions for their Ap- proximate Evaluation . 262 158. Derivation of the Liquid-Potential Equations 264 159. Determination of Ion-Concentrations and of Equilibrium-Con- stants by Means of Electromotive-Force Measurements . . 266 V. THE EQUILIBRIUM OF OXIDATION REACTIONS IN RELATION TO THE ELECTRODE-POTENTIALS 160. Derivation of the Equilibrium-Constants of Oxidation Reac- tions from the Molal Electrode-Potentials 269 VI. VOLTAIC ACTION, ELECTROLYSIS, AND POLARIZATION 161. Concentration-Changes Attending Voltaic Action and the Re- sulting Polarization 271 162. Electrolysis in Relation to Minimum Decomposition-Potential 272 163. Electrolysis in Relation to Polarization . . . ... . . 273 CHAPTER XII: THE EFFECT OF TEMPERATURE ON THE WORK PRODUCIBLE BY ISOTHERMAL CHEMICAL CHANGES AND ON THEIR EQUILIBRIUM CONDITIONS I. THE FUNDAMENTAL SECOND-LAW EQUATION 164. The Quantity of Work Producible from a Quantity of Heat that Passes from One Temperature to Another . . . . . 277 H. THE EFFECT OF TEMPERATURE ON THE EQUILIBRIUM OF UNIVARIANT SYSTEMS 165. The Effect of Temperature on the Pressure at which the Phases of Univariant Systems are in Equilibrium 280 HI. THE EFFECT OF TEMPERATURE ON THE EQUILIBRIUM OF CHEMICAL REACTIONS IN GENERAL 1 66. The Effect of Temperature on the Free-Energy Decrease Attending Any Isothermal Change in State 284 167. The Effect of Temperature on the Equilibrium of Chemical Changes Involving Perfect Gases ........ 285 1 68. The Effect of Temperature on the Equilibrium of Chemical Changes Involving Perfect Solutes .... . /. . . . 288 xviii CONTENTS IV. THE EFFECT OF TEMPERATURE ON THE ELECTROMOTIVE FORCE OF VOLTAIC CELLS PAGE 169. The Effect of Temperature on the Electromotive Force of Voltaic Cells 290 CHAPTER XIII: SYSTEMIZATION OF FREE-ENERGY VALUES 170. Importance of Systemizing Free-Energy Values 292 171. Expression of Free-Energy Changes by Equations .... 292 172. Determination of the Free-Energies of Substances .... 294 173. The Equilibrium-Conditions of Chemical Reactions Derived from the Free-Energies of the Substances Involved . . . 295 NOTATION AND INDEX Notation . . . . . . . . . . ; 297 Index 299 PART I THE ATOMIC, MOLECULAR, AND IONIC THEORIES AND THE PROPERTIES OF SUBSTANCES DI- RECTLY RELATED TO THESE THEORIES CHAPTER I THE COMPOSITION OF SUBSTANCES AND THE ATOMIC THEORY 1. The Field of Chemistry. Its General Principles the Subject of this Course. Chemistry treats of the composition and molecular structure of substances, of their properties in relation to their composi- tion and molecular structure, of changes in their composition, and of the effects attending such changes. Physics treats of the properties of substances without relation to their composition, and of changes in state that do not involve changes in composition. This course on Chemical Principles is devoted to a consideration of the more important general principles which express the properties and reactions of chemical substances. The subject here considered is sometimes called general, theoretical, or physical chemistry. The last of these names, which is now most commonly employed, carries the unfortunate implications that the subject is a separate branch of chemistry coordinate with inorganic, organic, analytical, or applied chemistry, and that it deals mainly with the physical sides of chemical science; whereas, in reality, it deals with the principles common to all branches of chemistry and is concerned primarily with purely chemical phenomena, such as the properties of substances in relation to their molecular structure, and such as the rate and equilibrium of chemical reactions and the attendant effects. The general plan adopted for the presentation of the subject in this book may be seen by reference to the headings of the Parts and Chapters in the Table of Contents given on the preceding pages. 2. Pure Substances and Mixtures, Elementary and Compound Substances, the Elements, and the Law of Definite Proportions. Out of the materials occurring in nature there can be prepared sub- stances which, when subjected to suitable processes of fractionation (that is, to operations which resolve the materials into parts or frac- tions), always yield fractions whose properties are identical when measured at the same pressure and temperature. Such substances are called pure substances; other substances which can be resolved by such processes into fractions with different properties being called mixtures. For example, whether a solid material is a pure substance or mixture 3 COMPOSITION OF SUBSTANCES may be determined by partially melting or vaporizing it or by partially dissolving it in solvents, and by comparing the value of the density, melting-point, or some other sensitive property, of the unmelted, un- vaporized, or undissolved part with that of the original material. The fundamental idea involved in the preceding considerations is that there exists an order of substances, called pure substances, of rela- tively great stability toward resolving agencies, each one of which has a perfectly definite set of properties, sharply differentiated from those of other pure substances; so that there is not a continuous series of pure substances whose properties pass over into one another by insensi- ble gradations. Nearly all pure substances, called compound substances because of the behavior to be now mentioned, can be converted, by subjecting them to sufficiently powerful resolving conditions, into a small number of other pure substances, called elementary substances, which are not further decomposed by chemical or physical processes. In corre- spondence with this fact all substances are considered to be composed of a small number of kinds of matter, about ninety in all, called the dements; each elementary substance consisting of only one of these elements, and each compound substance of two or more of them. Although some elements, those contained in the so-called radio- active substances, have been found to be undergoing spontaneous disintegration, elementary substances obviously constitute a separate order of substances of very great, but not unlimited, stability toward decomposing agents. The general principle of the definiteness of the properties of pure substances applies also to the elementary composition of pure sub- stances. This fact is expressed by the law of definite proportions, which states that a pure substance, however it be prepared, always contains its elements in exactly the same proportions by weight. 3. The Law of Combining Weights. To the various elements definite numerical values can be assigned which when multiplied or divided by small whole numbers accurately express the weights of the elements which are combined with one another in all pure substances^ Such numerical values are called the combining weights of the elements. Adopting 1 6 as the combining weight of oxygen, the combining weight of any other element may be defined to be that weight of it which combines with 16 parts of oxygen, or with a quantity of oxygen (such as 8, 24, or 32 parts) which stands to 16 in the ratio of small whole THE ATOMIC THEORY 5 numbers. It will be seen from this definition that a combining weight is an arbitrary multiple of an exact quantity ^ The principle that the composition of all pure substances can be accurately expressed in terms of the so-defined combining weights of the elements is called the law of combining weights. Prob. i. The Concept and Law of Combining Weights. a. Two oxides of sulfur are found by analysis to contain 50.05% and 40.04% sulfur. Derive from these analytical results two values for the com- bining weight of sulfur, and show that these values stand to one another in the ratio of small whole numbers, b. Derive a combining weight for iron from the fact that one of its oxides contains 30.06% of oxygen, c. Derive another value for the combining* weight of iron from the fact that one of its sulfides contains 36.47% of sulfur, whose combining weight was found in a\ and show that the two values of the combining weight of iron stand to each other in the ratio of small whole numbers. 4. The Atomic Theory. The fact that elements combine with one another only in the proportions of their combining weights or multiples of them originally suggested the following fundamental atomic and molecular hypotheses: (1) The matter constituting each element is made up of extremely small particles, called atoms, which are alike in every respect, and which are not subdivided hi ordinary chemical or physical changes; there being as many different kinds of atoms as there are elements. (2) The atoms associate with one another, usually in small numbers, forming a new order of distinct particles, called molecules. Pure sub- stances usually consist of only one kind of molecule, while mixtures always contain two or more kinds; and the molecules of elementary substances consist of atoms of the same kind, while those of compound substances consist of atoms of different kinds. By supplementing these hypotheses with certain other hypotheses, developing them to their logical conclusions, and applying these con- clusions to physical and chemical phenomena, many of the physical properties and chemical reactions of substances have been explained. Indeed, the adequacy of the fundamental hypotheses has now been confirmed in so many ways that there is no longer any doubt that substances are really made up of atoms and molecules of substantially the assumed character. These hypotheses and the conclusions drawn from them constitute the atomic and molecular theories; the former term being commonly 6 THE COMPOSITION OF SUBSTANCES used when properties that are more directly related to the characteris- tics of the atoms or to the atomic structure of the molecule are under consideration; the latter term when properties that are determined primarily by the characteristics of the molecule as a whole are being studied. A substance which contains only a single kind of molecule is called a chemical substance (or by some authors a molecular species). Pure substances normally consist of single chemical substances; but they may consist of two or more chemical substances which are capable of conversion into one another and which are always present in definite proportions at any given temperature and pressure in consequence of the rapid establishment of equilibrium between them. Thus, the pure substance liquid water contains the two chemical substances whose molecules are H 2 O and H 4 O 2 , these being always present in definite proportions at any given temperature and pressure, since equilibrium is almost instantaneously established between them; but the pure substance water- vapor at small pressures is a single chemical substance, for it contains only molecules of the form H^O. Pure substances which, like water, water-vapor; and ice, are converted into one another by changes of pressure and temperature, are commonly spoken of as the same substance; but they may consist of different chemical substances, as has just been illustrated. The relative weights of the atoms of the various elements and of the molecules of the various substances are called their atomic weights and molecular weights, respectively; and as a standard of reference, the weight of the oxygen atom taken as 1 6 is adopted. The atomic theory evidently requires that the weights of elements that combine with one another be proportional to the weights of their atoms or to multiples of those weights; in other words, that the atomic weights be equal to the combining weights or to multiples of them. 5. Determination of Combining and Atomic Weights. The determination of the atomic weight of an element involves two distinct steps: (i) the determination of its combining weight; and (2) the determination of that multiple of its combining weight which is its atomic weight. The combining weight of an element is experimentally determined by preparing one of its compounds or the elementary substance itself in a state of the highest possible purity, and completely converting it into another pure substance, the other elements contained in the sub- THE ATOMIC THEORY 7 stances being of known combining weights. The quantities of the initial substance taken and of the final substance obtained are accu- rately weighed, and from the ratio of these weights the combining weight is calculated. For example, the combining weight of hydrogen has been determined by burning a weighed quantity of pure hydrogen gas with oxygen and weighing the water formed; also by finding the weight of the oxygen consumed. Values for the weight ratios H 2 : H 2 O and H 2 : O were thus established, and from each of them the combining weight of hydrogen was computed. The following prob- lem illustrates the methods of determining combining weights in general, and it shows some of the fundamental determinations on which our present system of atomic weights is based. Prob. 2. Determination of Combining Weights. Determine the exact combining weights of silver, potassium, and chlorine from the following data: In a series of eight experiments 801.48 g. of pure potassium chlorate were ignited or treated with hydrochloric acid, yielding 487.66 g. of potassium chloride. In another series of five experiments 24.452 g. of pure potassium chloride were dissolved in water and precipitated with silver nitrate, whereby 47.013 g. of silver chloride were obtained. In a third series of ten experiments 82.669 g. of pure silver were dissolved in nitric acid and precipitated with hydro- chloric acid, yielding 109.840 g. of silver chloride. Ans. Ag, 35.978 x; Cl, 11.825 y\ K, 13.038 z; where x, y, and z are ratios of whole numbers not determinable by gravimetric methods. The multiple of the analytically determined combining weight which is the most probable atomic weight of each element was origi- nally derived for most of the elements from theoretical principles relat- ing to certain physical properties, especially the pressure-volume relations of gaseous compounds, the heat-capacities of the elementary substances in the solid and gaseous states, and the isomorphism of corresponding compounds of different elements in the crystalline state. The principles relating to the first two of these properties and their applications to atomic weight determinations are considered in Chapters II and IV. The system of atomic weights so derived finds its chief justification, however, in the fact that it has led to molecular formulas of chemical substances which represent their chemical behavior in a remarkably satisfactory manner. It has also led to the formulation of the periodic system of the elements, which affords a qualitative representation of many of the physical and chemical properties of elementary and compound substances as a progressively changing and periodically recurring function of the atomic weights. 8 THE COMPOSITION OF SUBSTANCES 6. Values of the Atomic Weights. The existing data and all newly determined data relating to atomic weights are critically con- sidered at frequent intervals by an International Committee on Atomic Weights, and a table is published each year showing the most probable values. The following table contains the latest values adopted for the more important elements. The values are given to such a number of decimal places that the last one is probably in error by not more than one or two units. VALUES OF THE ATOMIC WEIGHTS Aluminum . Al 27.0 Lead . . . Pb 207.20 Antimony . Sb 120.2 Lithium . . Li 6.94 Argon . . A 39.9 Magnesium . Mg 24.32 Arsenic : . As 74.96 Manganese . Mn 54.93 Barium .' . Ba 137.37 Mercury . Hg 200.6 Beryllium . Be 9.1 Neon . Ne 20.2 Bismuth . > . Bi 209.0 Nickel . . Ni 58.68 Boron . . B 10.9 Nitrogen . N 14.008 Bromine . . Br 79.92 Oxygen . . O 16.00 Cadmium Cd 112.40 Phosphorus . P 31.04 Calcium . . Ca 40.07 Platinum . Pt 195.2 Carbon C 12.005 Potassium . K 39.10 Chlorine . . Cl 3546 Radium . Ra 226.0 Chromium . Cr 52.0 Silicon . . Si 28.1 Cobalt . . Co 58.97 Silver . Ag 107.88 Copper . Cu 63.57 Sodium . . Na 23.00 Fluorine . . F 19.0 Strontium . Sr 87.63 Gold . . . Au 197.2 Sulfur . . S 32.06 Helium . He 4.00 Thallium . Tl 204.0 Hydrogen . H 1.008 Tin . . . Sn 118.7 Iodine . I 126.92 Titanium . Ti 48.1 Iron . . . Fe 55-84 Zinc . . . Zn 65.37 7. Chemical Formulas, Formula- Weights, and Equivalent- Weights. In order to express the gravimetric composition of compounds, the symbols of the elements are considered to represent their atomic weights and are written in sequence with such integers as subscripts as will make the resulting formula express the proportions by weight of the elements in the compound. The formula is commonly so written as to represent also the number of atoms of each element present in the molecule. The formula represents, in addition, a definite weight of the substance, namely the weight in grams which is equal to the sum of the numbers represented by the symbols of the elements in the formulas. This weight is called the formula-weight of the substance. Thus the formula HNOs denotes 1.0084-14.01+ (3X16.00) or 63.02 grams of nitric acid. THE ATOMIC THEORY g Those weights of various substances which enter into chemical reactions with one another are called equivalent-weights. Adopting one formula- weight or i.ooS grams of the element hydrogen as the standard of reference, the equivalent-weight or one equivalent of any substance is denned to be that weight of it which reacts with this standard weight of hydrogen, or with that weight of any other sub- stance which itself reacts with this standard weight of hydrogen. Thus the equivalent-weight of each of the following substances is that fraction of its formula-weight which is indicated by the coefficient preceding the formula: jCfcjiOs; iAg;jZn; Jfii; iNaOH; jBa(OH) 2 ; JH 2 SO 4 ; iH 3 PO 4 ; f A1C1 3 ; iK 4 Fe(CN) 6 . The equivalent-weight of a substance may have different values depending on whether it is considered with reference to a reaction of metathesis or to one of oxidation and reduction. Thus, the metathetical equivalent of ferric chloride is jFed 3 , but its oxidation-equivalent (with respect to its conversion to ferrous chloride) is iFed 3 ; the metathetical equivalent of potassium chlorate is iKC!O 3 , but its oxidation-equivalent (with reference to its reduction to KC1) is KC10 3 . CHAPTER II THE MOLAL PROPERTIES OF GASES AND THE MOLEC- ULAR AND KINETIC THEORIES I. THE VOLUME OF GASES IN RELATION TO PRESSURE, TEMPERATURE, AND MOLAL COMPOSITION 8. The Volume of Perfect Gases in Relation to Pressure and Temperature. Properties whose magnitudes are determined pri- marily by the number, and not by the nature, of the molecules present are called mold properties. The properties of this kind exhibited by gas&Stre considered in this chapter. The properties of gases, in general, conform more and more closely to certain limiting laws, known as the laws of perfect gases, as then- pressure approaches- zero. The principles relating to the effects of pressure and temperature on the volume of perfect gases will be first considered. At any definite temperature the pressure p of a definite weight m of any perfect gas is inversely proportional to its volume v; or its pressure- volume product p v is independent of its pressure or volume. Since density (d) is defined to be the ratio (m/v) of the weight of a sub- stance to its volume, this law may als^be stated as follows: At any definite temperature the density of any definite perfect gas is propor- tional to its pressure. This principle is known as Boyle's law. r The pressure-volume product increases with rise in temperature by the same fractional amount for differei^ perfect gases when they are heated from the same initial to the same final temperature. For ex- ample, the pressure-volume product increases by 36.6i2percent of its value at o when the temperature becomes 100, whatever be the nature of the perfect gas. This law is known as the law of temperature-effect. It is mathematically expressed by the equation ff&piVi)/(piVi)~ f(/i, AJ), where f(/ t , t^) denotes the same function of the temperatures ti and AJ whatever be the nature of the perfect gas. The form of the functional relation between the pressure-volume product and the temperature evidently depends on the definition of temperature adopted; and this relation assumes the simplest form when the concept of absolute temperature is employed. For absolute 10 PRESSURE-VOLUME RELATIONS OF GASES n temperature is so defined that it is directly proportional to the pressure- volume product of a perfect gas; that is, it is defined by the equation T/To = (pv)/(poVo), where p v Q is the value of pv at any specified temperature TO, which for definiteness may here be taken as that of ice when melting under a pressure of one atmosphere. The value of To is dependent on the temperature-scale adopted. We may deter- mine its value on the centigrade scale from the following considera- tions. This scale is based on the convention that there shall be a difference of 100 degrees between the temperature of melting ice and the temperature of boiling water, when both are at a pressure of one atmosphere. Now by writing the precedinJgjnuation in the form (T-TQ)/T O = (PV-POVQ)/PQVQ, and substifc^ Lj'o the value 100 and for the second member the quantil > givoyi^he preceding paragraph, we find the value 273.14 for To, the temperature of melting ice at a pressure of i,.ajm. on the absolute^gp|?ade gfe. Since this temperature is calleet'p? .<|Byj|^|^^^cfentigrade scale, and since this scale, like the absolute scale, is 1 Hn the convention of proportionality between increase of temperature and the increase in the pressure- volume product of a perfect gas^he absolute centigrade temperature T is always equal to the ordinary centigrade temperature / increased by 273.14, or approximately 273; black-face type being used here and throughout the book to show approximate values, given with an accuracy of o.i 0.2 percent, of fundamental con- stants which it is well to rem(|nber. The laws above stated, the definition of absolute temperature, and the obviojp, proportionality between the value of the pressure- volume product and the weight m of the gas are all expressed by the equation pv=mRT, in which R is a cogitant, evidently representing the value otpv/T for qj^gram of the gas, which has different values for different $P Prob i. Derivation from Experimental Data of pv/T for One Gram of Carbon Dioxty* a. From the fact that the density of carbon dioxide is 0.0019765 g. per ccm. at o and i atm., calculate the exact value of p v/T under these conditions for one gram of the gas. b. Taking into account the fact that at this temperature and pressure the density has been found, by extrapolating from a series of measurements made at pressures between o.i and i atm., to be 0.68% greater than that re- quired by the perfect -gas law, calculate for this gas the value of the constant R, the pressure being expressed in atmospheres and the volume in cubic centimeters. Ans. b, 1.8649. 12 THE MOLAL PROPERTIES OF GASES 9. Law of Combining Volumes and the Principle of Avogadro. The foregoing physical laws acquire an important chemical significance by reason of the law of combining volumes, which may be stated as follows: Those quantities of perfect gases that are involved in chem- ical reactions with one another have, at the same temperature and pressure volumes, which are equal or small multiples of one another. Thus the quantities of hydrogen and of oxygen which unite with each other to form water have volumes whose ratio approaches the exact value 2 : i as the pressure of the gases approaches zero. In other words, those quantities of different substances which as perfect gases have the same value of the product pv/T are equal to, or are small multiples of, the quantities which are involved in chemical reactions with one another. Now, since according to the molecular theory substances react by molecules, these quantities of different perfect gases which have the same value of p v/ T must contain either an equal number of molecules or small multiples of an equal number. These considerations suggest a simpler principle, known as the principle of Avogadro, which may be stated as follows: Those^ quanti- ties of different perfect gases which have the same volume at the same temperature and pressure, and therefore the same value oipv/T at any temperature and pressure, contain an equal number of molecules. The densities of different perfect gases at the same temperature and pressure are therefore proportional to the weights of their molecules. This principle, originally hypothetical, has now been so fully verified that it has become one of the fundamental laws of chemistry. 10. Empirical Definition of Molecular Weight and of Mol. The principle of Avogadro evidently enables the relative weights of the molecules of different gaseous substances to be determined. To ex- press these relative weights more definitely, the ratio of the weight of the molecule of any substance to the weight of the molecule of oxygen taken as 32 is commonly considered, this ratio being called the molecu- lar weight M of the substance. The number 32 is adopted as the refer- ence quantity of oxygen, since (as shown in Prob. 15) it corresponds to the adoption of 1 6 as the atomic weight of oxygen. The so-defined molecular weight of a substance may* evidently be experimentally determined by finding the number of grams of it which have that value oi pv/T which 32 grams of oxygen have, when both substances are in the state of perfect gases. This number of grams is called one mol or one molal weight of the substance. PRESSURE-VOLUME RELATIONS OF GASES 13 The mol is a unit of great importance in chemical considerations, both because it is directly related to molecular weight and because it makes possible a general expression of the laws of perfect gases. 11. General Expression of the Laws of Perfect Gases. Repre- senting by R the constant value of p v/ T for one mol and by ^V the number of mols of the gas present, the laws of perfect gases may be expressed in a general form by the following equation, hereafter called the perfect-gas equation: The numerical value of the gas-constant R depends on the units in which the pressure, volume, and temperature are expressed. In scientific work, temperature is always expressed in centigrade degrees (here on the absolute scale); volume is ordinarily expressed in cubic centimeters or liters; and pressure in dynes per square centimeter, millimeters of mercury, or atmospheres. One dyne is a force of such magnitude that when it acts on a freely moving mass of one gram it increases its velocity each second by one centimeter per second; and the pressure of one dyne per square centimeter is called one bar, io 6 bars being called one megabar. One atmosphere is a pressure equal to that exerted by a column of mercury 76 cm. in height at o at the sea-level in a latitude of 45, The value of the gas-constant R when the pressure is in atmospheres and the volume is in cubic centimeters is 82.07 (approximately 82) Its value when the pressure is in dynes per square centimeter and the volume is in cubic centimeters is 8.3i6Xio 7 ergs per degree. Its value will later be shown to be 1.985 calories per degree. Prob* 2. Value of the Atmosphere. Calculate the value of the atmosphere in dynes per sqcm. and in megabars. The density of mercury at o is 13.60. The force of gravity acting on any freely moving body increases its velocity each second by g centimeters per second. . The value of g at the sea-level in a latitude of 45 is 980.6 (980) centimeters per second. Ans. 1.013 Xio 6 dynes per sqcm. Prob. 3. Calculation of the Gas-Constant. a. Calculate precisely the value of the gas-constant R when the pressure is in atmospheres and the volume in cubic centimeters, from the fact that one gram of oxygen has a volume of 7005 ccm. at o and o.i atm., under which conditions oxygen does not deviate appreciably from the perfect- gas law. b. Calculate also the value of R when the pressure is in dynes per square centimeter, and the volume is in cubic centimeters. c. Calculate the volume of one mol of a perfect gas at 20 and i atm. Ans. c, 24060 ccm. (approximately 24 1.) 14 TEE MOLAL PROPERTIES OF GASES Applications of the Perfect-Gas Laws. Note. In the following problems, and in general throughout the book, assume, unless otherwise indicated, as an approximation sufficient for many purposes, that gases at pressures not exceeding i atm. con- form to the perfect-gas equation. v, Prob. 4. Seven grams of a certain gas have a volume of 6.35 1. at ^ 20 and 720 mm. How many grams make one mol? y Prob. 5. a. What is the volume occupied by 12 g. of ether vapor / XC 4 Hi O) at 80 and 600 mm.? b. What is the density? c. What is the ratio of the density to that of oxygen at the same temperature and pressure? ^ Prob. 6. How many grams of iron must be taken to produce by its action on sulfuric acid one liter of hydrogen (H 2 ) at 27 and i atm.? It will be seen in Art. 12 that,, though air is a mixture, it has in different localities a fairly constant composition and that the pres- sures of its separate constituents are determined by the perfect-gas laws. The pressure of the air as a whole, and its pressure-volume- temperature relations in general, can therefore be simply computed with the aid of the perfect-gas equation by regarding air as a single substance having a molal weight equal to the number of grams of air which have the same value of p v/T as 32 grams of oxygen. The so- defined molal weight of air is computed from the most accurate density determinations to be 28.98 (approximately 29). Prob. 7. Calculation of the Molal Weight of Air. Calculate the molal weight of pure dry air from the fact that its density is 0.0012930 at o under a pressure of 760 mm. of mercury at sea-level in latitude 45. 12. Dalton's Law of Partial Pressures. The (total) pressure p of a mixture of substances in the gaseous state is evidently the sum of the pressures p\ t p 2 . . . of the separate substances. That is, p=p\-\- p2 + . The pressures of the separate substances are called the partial pressures. Partial pressures in the case of perfect gases may be calculated as described below. They cannot be directly measured, except in cases where a semipermeable wall, that is, a wall permeable for only one of the substances present in the gas mixture, can be found. Thus, when a platinum vessel containing a mixture of hydrogen and nitrogen at a high temperature is immersed in an atmosphere of hydrogen, hydro- gen passes through the platinum walls until its pressures within and without the vessel become equal; the difference then observed between the total pressure of the mixture within the vessel and the pressure of the hydrogen outside is due to the nitrogen, which does not pass through the wall; and this difference is equal to its partial pressure. PRESSURE-VOLUME RELATIONS OF GASES 15 The pressure of a mixture of substances in the state of a perfect gas is found to be equal to the sum of the partial pressures calculated by the principle that each chemical substance (defined as in Art. 4 to be a substance consisting of molecules of one particular kind) has the same pressure as it would if it were alone present in the volume occu- pied by the mixture. This principle, which is of great importance in chemical considerations, is known as Dalton's law of partial pressures. /~SLn accordance with this law and the perfect-gas law, the partial / pressures of the separate substances are evidently given by the expres- /sions, pi = NiRT/v, p 2 = NiRT/v, ... where Ni, N 2 ... represent the number of mols of the respective substances; and the partial 'pressure of any of the substances is evidently equal to the total pres- j sure of the gas multiplied by the ratio of the number of mols of that i substance to the total number of mols in the mixture. This ratio is ^commonly called the mol-fraction (x) of that substance. That is, V=f *i, where xi= N,/( #! + #,+ .. .). Air affords an important example of these considerations. Pure dry air has at different times and places a fairly constant composition; namely, it contains approximately 21.0 mol-percent of oxygen (O 2 ), 78.0 mol-percent of nitrogen (N2), 0.94 mol-percent of argon (A), and 0.03 mol-percent of carbon dioxide (CO 2 ), and, in accordance with Dalton's law, the partial pressures of these four substances in air are 21.0, 78.0, 0.94, and 0.03 percent, respectively, of the total pressure of the air. As will be shown later, it is these partial pressures, not the total pressure of the air, which determine the quantities of the respec- tive substances which dissolve in water when it is shaken with air, and it is the partial pressure of the oxygen which determines the equilibria of oxidation reactions taking place in air. Applications of the Law of Partial Pressures. Prob. 8. A Bessemer converter is charged with 10000 kg. of iron containing 3% of carbon, a. How many cubic meters of air at 27 and i atm. are needed for the combustion of all the carbon, assuming one third to burn to COj and two thirds to CO? b. What are the partial pressures of the gases evolved? Ans. a, 1950 cubic meters. Prob. 9. In making sulfur dioxide for use in the contact process of manufacturing sulfur trioxide, 30 kg. of sulfur are charged into each burner per hour, and such an excess of air is introduced (to insure complete combustion) as to produce a gas mixture containing 10 mol- percent of oxygen. How many cubic meters of air at 20 and i atm. must be introduced per hour? Ans. 205 cubic meters. 1 6 THE MOLAL PROPERTIES OF GASES Determination of the Dissociation of Gases by Density Measurements. Prob. 10. A certain fraction 7 (or a certain percentage 100 7) of gaseous nitrogen tetroxide is dissociated according to the reaction N 2 O 4 = 2NO 2 at any definite pressure p and temperature T. a. Find an expression for the total number of mols i (of N 2 O 4 and NO 2 ) that result under these conditions from one formula-weight M (92 g.) of nitrogen tetroxide taken, b. Find an expression for the volume v that would be occupied by one formula-weight of the nitrogen tetroxide at the pres- sure p and temperature T. c. Find a corresponding expression for the density d of the vapor. Ans. d=Mp/(iRT). Prob. ii. Calculate the dissociation of nitrogen tetroxide at 50 and 500 mm. from the following data: a balloon filled with it at this tempera- ture and pressure weighs 71.981 g.;/the same balloon when evacuated weighs 71.217 g., and when filled at 25 with water (whose density at 25 is 0.997) weighs 555.9 g. Ans. 45%. ^ 13. Determination 'of Molecular Weights. The (experimental determination of the molecular weight of a gaseous substance consists in measuring the volume o!^, weighed quantity of the substance at an observed pressure and temperature. From these quantities (v, m, p, and T) the molecular weight of the substance is calculated with the aid of the perfect-gas equation. Prob. 12. Experimental Determination of Vapor-Density by Hof- mann's Method. 0.1035 g- of a volatile liquid is introduced into the vacuum above a mercury column in a graduated tube standing in an open vessel of mercury. The tube is entirely surrounded by a jacket through which steam at 100 is passed. The mercury column falls till it stands 260 mm. above the mercury-level in the vessel below, and the volume of the completely vaporized liquid is observed to be 63.0 ccm. The barometric pressure at the time is 752 mm. At 100 the density of mercury is 13.35, and its vapor-pressure is negligible. Make a sketch showing the apparatus; and calculate the density and molecu- lar weight of the vapor. Ans. d =0.001645; M = 76.8. Prob. 13. Experimental Determination of Molecular Weight by the Air-Displacement Method of Victor Meyer. A cylindrical bulb pro- vided with a long stem is filled with dry air; and it is heated, while still open to the atmosphere, to a constant temperature (of about 218) within a jacket filled with vapor of boiling naphthalene. The stem, which protrudes above the vapor jacket, has a side-arm which delivers into a graduated tube filled with water and inverted over water, so that the air to be later expelled from the bulb can be collected and measured. A tiny glass bottle containing 0.2210 g. of a volatile liquid substance is dropped into the heated bulb, and at the same moment the bulb is cut off from communication with the atmosphere by closing the top of the stem. The liquid rapidly vaporizes and drives out of the bulb a volume of air equal to the volume occupied by the vapor PRESSURE-VOLUME RELATIONS OF GASES 17 of the substance, the pressure being always that of the atmosphere. This air collected in the measuring apparatus is found to have a volume of 24.65 ccm. at 22 under the barometric pressure of 752 mm.; but of this pressure 20 mm. is due to the pressure of water vapor. Make a sketch showing the apparatus; and calculate, a, the number of mols of air expelled, and b, the molecular weight of the vaporized substance. Ans. a, 0.000980; b, 225. Since gases at atmospheric pressure conform to the perfect-gas laws only approximately, and since gaseous densities are not commonly determined with so great accuracy as the composition of substances, the exact value of the molecular weight of a compound is usually derived from the gravimetric composition, the density being employed only to determine what multiple or submultiple of the value so derived is in accordance with Avog;adro's principle. Prob. 14. Exact Evaluation of Molecular Weights. a. A certain oxide contains exactly 72.73% of oxygen. What does this show in regard to its molecular weight? b. At o and i atm., one liter of this (gaseous) oxide is found to weigh exactly 1.977 g- What is the mo- lecular weight of the oxide corresponding to this datum? c. What is the exact molecular weight derived by considering the data relating both to the composition and density? d. What do these two values of the molecular weight show as to the percentage deviation of the density from that which is required by the perfect-gas equation? Ans. d, 0.68%. 14. Derivation of the Atomic Weights of Elements and of the Molecular Composition of Compounds. The exact values of atomic weights are based on analytical determinations of the combining weights, as described in Art. 5. The multiple of the combining weight adopted as the atomic weight of any element is derived by finding the smallest weight of the element contained hi one molecular weight of any of its gaseous compounds. This is the true atomic weight only in case some one of the compounds studied contains in its molecule a single atom of the element; and the adopted atomic weight is therefore strictly only a maximum value, of which the true atomic weight may be a submultiple. The probability that the true atomic 'weight has been found evidently increases with the number of the gaseous com- pounds whose molecular weights have been determined. The multiples of the combining weights adopted as the atomic weights have, however, not been derived solely from molecular-weight determinations. From the laws relating to certain other properties, mentioned in Art. 5, independent values of the atomic weights have 1 8 THE MOLAL PROPERTIES OF GASES been obtained, which confirm and extend those derived from molecular weights. The molecular composition of a gaseous substance (that is, the number and nature of the atoms in its molecule) can evidently be de- rived from its molecular weight, its composition by weight, and the atomic weights of the elements contained in it. The chemical formu- las of substances whose molecular weights in the gaseous state are known are ordinarily so written as to express this molecular com- position. Such formulas are called molecular formulas. Derivation of Atomic Weights and Molecular Formulas. Prob. 15. Certain compounds have molecular weights M (referred to that of oxygen as 32) and percentages of oxygen x as follows: Sulfur tri- oxide, M =80.07, x = 59-95 5 water, M = 18.02, x = 88.80; carbon dioxide, M =44.00, x = 72.73. a. Find the weight of oxygen contained in one molecular weight of each of these oxides, b. What conclusion as to the atomic weight of the element oxygen can be drawn from these results (which are based on the convention that the molecular weight of oxygen gas is 32)? c. What does this show as to the number of atoms in the molecule of oxygen gas? Prob. 16. a. Calculate the combining weight of the element con- tained in the oxide whose composition was given in Prob. 140. b. What conclusion as to its atomic weight can be drawn from this combining weight and the density given in Prob. 146? c. What con- clusion can be drawn as to the molecular formula of the oxide? Pr*. b. 17. a. The chloride of a certain element is found by analysis to contain 52.50% of chlorine, whose atomic weight is 35.46; and it is found to have at 150 a vapor-density 4.71 as great as that of air at the same temperature and pressure. What conclusion can be drawn from these facts as to the exact atomic weight of the element? b. The hydride of the same element is a gas which contains 5.91% of hydrogen; and it is found to be produced without change in volume when hydrogen (H 2 ), whose atomic weight is 1.008, is passed over the solid elementary substance. What conclusion can now be drawn as to the atomic weight of the element? c. What are the simplest molecular formulas of the hydride and of the chloride consistent with these conclusions? Prob. 18. A certain hydrocarbon is composed of 92.25% of carbon and 7.75% of hydrogen, whose atomic weights are 12.00 and 1.008, respectively. Its density when in the form of vapor at 100 and i atm. is 2.47 times as great as that of oxygen under the same conditions. Calculate its exact molecular weight, and derive its molecular formula. The molecules of elementary substances may consist of single atoms or of two or more atoms. Thus the molecular formulas of some of those whose density in the gaseous state has been determined are : H 2 , N 2 , O 2 , F 2 , C1 2 , Br 2 , 1 2 , P 4 , As 4 , He, Ne, A, Hg, Cd, Zn, Na, K. PRESSURE-VOLUME RELATIONS OF GASES 19 At very high temperatures some of these have been shown to dissociate into simpler molecules; thus above 1500 the molecule of iodine con- sists of a single atom. A knowledge of the molecular formulas of substances is of impor- tance principally because the chemical relationships of different sub- stances are far more clearly brought out by such formulas than by the simpler ones expressing merely composition by weight. The structure theory, which underlies the science of organic chemistry, is based upon a knowledge of the molecular weights of substances. 15. Deviations from the Perfect-Gas Laws at Moderate Pres- sures. The deviations from the perfect-gas law, so long as they do not exceed a few percent, are almost exactly proportional to the pressure. This fact is expressed by the following equation, which will be called the gas equation at moderate pressures: In this equation the coefficient o^eyidently represents the fractional deviation at unit-pressure of the actual value of p v from that required by the perfect-gas law. The value of a varies with the nature of the gas, and for a given gas with the temperature. The following table shows the values of the percentage deviation (100 a) of pv from NRT at one atmosphere for various gases at o, together with their condensation- temperatures at one atmosphere. DEVIATIONS AT MODERATE PRESSURES Formula of gas He H 2 N 2 NO CO 2 NH 3 SO 2 Percentage deviation -f 0.06 +0.05 0.04 0.12 0.68 1.52 2.38 Condensation-temp. 269 253 196 151 78 34 10 Prob. 19. Calculation of the Deviation-Coefficient. a. Calculate the value of the deviation coefficient a for nitrous oxide (N 2 O) at o from the fact that its density at o and i atm. is 0.001978 g. per can. b. State what percentage error would be made in calculating by the perfect- gas equation the volume of this gas at o and 3 atm. Ans. a, 0.0073. 16. Pressure-Volume Relations of Gases at High Pressures. In the figure are plotted the values of pv/T in atmospheres and cubic centimeters as ordinates against the values of the pressure in atmos- pheres as abscissas for one mol of hydrogen at o, of nitrogen at o, of carbon dioxide at 60 and at 40, and of a perfect gas (marked P. G. in the figure). At temperatures between o and 60, helium (He) and neon (Ne) have curves similar to that of hydrogen (H 2 ); oxygen (O 2 ), carbon monoxide (CO), and nitric oxide (NO) have curves similar 20 THE MOLAL PROPERTIES OF GASES to that of nitrogen; and nitrous oxide (N 2 O), ammonia (NH 3 ), and ethylene (CzH.^) have curves similar to those of carbon dioxide (CO 2 ). 20 100 200 Pressure in Atmospheres 300 Pressure-Volume Relations of Gases at High Pressures. Prob. 20. Estimate with the aid of the figure the percentage amount by which at 60 atm. the volume of H 2 at o, of N 2 at o, and of CO 2 at 40 and at 60 Deviates from that of a perfect gas. Prob. 21. Summarize the general conclusions in regard to the pressure-volume relations of gases that can be drawn from the curves of the figure, from the statements in the preceding text, and from the condensation-temperatures given in Art. 15. Prob. 22. Determine from the figure how many kilograms of CO 2 can be charged into a 1 5-liter steel tank which will safely wit 1 stand a pressure, a, of 50 atm.; 6, of 100 atm.; c, of 150 atm. Assume ixiat the tank might be exposed (under a summer sun) to a temperature a - high as 40. Ans. a, 1.7; 6, 9.6; c, 11.5. PRESSURE-VOLUME RELATIONS OF GASES 21 The pressure- volume- temperature relations of gases are closely represented up to moderately high pressures by the following expres- sions, known as the van der Wads equation, in which a and b are constants (with respect to changes of volume, pressure, and tempera- ture) varying with the nature of the gas: (p + ^r) (o-Nb) = NRT; or for one mol, (p+Q (v-b) = RT. With reference to the volume occurring in the second of these ex- pressions, it may be noted that, in general, quantities referred to one mol are called molal quantities, and those referred to one gram specific quantities; and that in this book the former are designated by a wavy line, and the latter by a straight line, above the symbol of the quantity. Thus, v denotes the molal volume, the volume of one mol, and v denotes the specific -volume, the volume of one gram, of substance. The term specific is also applied to quantities which are referred to unit-values of other determining factors such as the volume, length, or cross-section; thus specific resistance denotes the electrical resist- ance of a mass of the substance one centimeter long and one square centimeter in cross-section. The applicability of this equation and its theoretical significance are discussed in Art. 20. Any equation which, like the van der Waals equation, the perfect- gas equation, or the gas equation at moderate pressures, expresses the relation between the volume, pressure, and temperature of a sub- stance within a certain range of conditions is called the equation of state of the substance under those conditions. 22 THE MOLAL PROPERTIES OF GASES II. THE KINETIC THEORY 17. The Fundamental Kinetic Hypotheses. The molecular theory has, with the aid of certain hypotheses in regard to the motion of molecules, been developed into a body of principles, collectively known as the kinetic theory. ^The two most fundamental of these hypotheses, which express from a molecular viewpoint the nature of pressure and of temperature, respectively, are as follows: The molecules of gaseous or liquid substances are continually in motion in all directions, the motions taking place in accordance with the ordinary laws of mechanics. The molecules are constantly collid- ing with one another and with the walls of the containing vessel; but no loss of kinetic energy results from these collisions. The pressure of the substance on the walls is the integrated effect of the impacts of its molecules. This may be called the kinetic pressure hypothesis. At the same temperature the molecules of different fluid (that is, gaseous or liquid) substances have the same kinetic energies, whatever be the masses of their molecules or the pressure or volume of the substance; that is, for two different substances at the same tempera- ture, \rn\u? = J/W2M2 2 , where m\ and nh represent the masses of the two kinds of molecules, and u\ and u^ their velocities. The kinetic energy of the molecules increases with the temperature, and it is determined solely by it. Mathematically expressed, Jww 2 =f(D, where i(T) is a function which is identical for all fluid substances. This may be called the kinetic temperature hypothesis. 18. The Kinetic Equation for Perfect Gases. From the kinetic pressure hypothesis may readily be derived an expression in terms of the molecular quantities for the pressure-volume product of a perfect gas, as shown in the following problem. In this derivation two simplifying assumptions, admissible in the case of a perfect gas, are made; namely, that the volume of the molecules is negligible in com- parison with the volume of the gas, and that any force of attraction between the molecules is negligible because of the relatively large mean distances between them. This expression is: In this equation n denotes the number of molecules in the volume v of the gas at the pressure />, m the mass of a single molecule, and u the velocity of the molecules. It may be called the kinetic equation for perfect gases. THE KINETIC THEORY 23 Prob. 23. Derivation of the Kinetic Equation. A volume v of a perfect gas is contained in a cubical vessel whose edges have the length /. The gas consists of n molecules, each of mass m. Consider that all the molecules are moving with the same velocity, u, and that the motions of these molecules, which in reality take place in all directions, are resolved in the three directions perpendicular to the faces of the cube, which is equivalent to assuming that one third of all the molecules is moving in each of these directions, a. Find the number of impacts which any one molecule makes on the opposite faces in a unit of time. b. Find the total number of impacts which all the molecules make on all the faces in a unit of time. c. Show that the number of impacts which the molecules make on a unit of surface in a unit of time is j^nu/v. d. Find the change of momentum which attends each impact, taking account of the fact that the molecule rebounds from the wall with a velocity equal to that with which it strikes it. e. Find the change of momentum of all the molecules that strike a unit of surface in a unit of time. /. Noting that this, in accordance with the general equation of mechanics f=m(du/di), is equal to the force (/) exerted on the unit of surface or to the pressure, show that the pressure-volume product is expressed by the equation pv = The derivation given in the preceding problem is not conclusive, since it involves an assumption as to the directions of motion of the molecules; but a rigorous analysis leads to the same kinetic equation. It is important to note, however, on account of its bearing on certain surface phenomena, that the rigorous treatment, which takes into account the fact that molecules approach the surface not only normally but obliquely in all directions, leads to the conclusion that the actual number of impacts made by all the molecules on unit-surface in unit- time is larger than that derived above; this actual number being given by the expression 0.23 nu/v, instead of by ^nu/v. From the kinetic equation Boyle's law may be immediately de- duced. For, since a change only in the volume of the gas will not cause a variation in the number of molecules n (unless they are thereby caused to associate or dissociate), and since the kinetic energies %mu 2 of the separate molecules will remain constant, in accordance with the kinetic temperature hypothesis, so long as the temperature remains unchanged, it follows that also the product p v will undergo no change when the volume is varied at a constant temperature. The law of temperature-effect is also a consequence of the kinetic equation; for any definite cnange in temperature must evidently cause the same change in the pressure- volume product of different gases, since it causes the same change in the kinetic energy of their molecules. 24 THE MOLAL PROPERTIES OF GASES The principle of Avogadro also is readily derived from the kinetic equation for perfect gases and from the kinetic temperature hypothesis. Thus, for two perfect gases with equal pressure-volume products, piVi=pzV2, and therefore \n\m\u^ =\n^m^\ and for the case that they have also equal temperatures, \rn\u? =^m^u^; whence follows HI =2. That is, those quantities of different perfect gases which at any definite temperature have equal pressure- volume products consist of an equal number of molecules. Certain principles relating to the velocity of the molecules, and even the exact absolute values of their velocities, can be derived from the kinetic equation in the ways illustrated by the following problems. Velocity of the Molecules. Prob. 24. a. Show how the velocities of the molecules of different gases at the same temperature are related to the molecular weights of the gases; illustrating by hydrogen and oxygen. b. Show how the velocities of the molecules of the same gas at different temperatures are related to those temperatures; illustrating by oxygen at o and 100. Prob. 25. Calculate the velocity of the molecules of oxygen at 20 in kilometers and in miles per second. Ans. 0.478 km.; 0.30 mile. 19. The Kinetic Energy of the Molecules and the Avogadro Num- ber. An expression may be readily derived for the kinetic energy K due to the translatory motion of the n molecules present in any mass of a perfect gas. Namely, for this kinetic energy, which is obviously equal to \nmu^^ the following expression is obtained at once by combining the kinetic equation for perfect gases, pv =lnmu 2 , with the perfect-gas equation, pv = NRT: From this equation the increase dK/dT of the kinetic energy of the molecules in one mol of gas per degree of temperature is seen to be f R, which is readily shown to have the value 12.474 ergs or 2.978 (ap- proximately 2.98) calories. Prob. 26. Kinetic Energy of the Molecules. a. Calculate the translatory kinetic energy in ergs and calories of the molecules in one mol of a perfect gas at 20. b. Show that the increase in this kinetic energy per degree is 2.98 cal. Note that i cal. equals 4.182 Xio 7 ergs. The corresponding expression for the kinetic energy of a single molecule evidently is: 2 3 RT 3 RT J m u 2 = f /-ft = f - 2 n/ N * n THE KINETIC THEORY 25 The quantity n/ N, which represents the number of molecules in one mol of any substance (for example, in 2.015 grams of hydrogen or 32.00 grams of oxygen), is a fundamental constant known as the Awgadro number (n). Its value has been recently determined by a variety of independent methods. The most probable value is 6.06 X10 23 . The above derived expression for %mu 2 is the general form, holding true for all fluid substances, of the functional relation between tem- perature and kinetic energy of the separate molecules, which is postu- lated in the kinetic temperature hypothesis. For, since according to that hypothesis this kinetic energy is the same function of the tempera- ture for all fluid substances, and since it has been shown to be pro- portional to the absolute temperature for a perfect gas, it follows that it is proportional to the absolute temperature for all fluid substances. In other words, the mathematical expression of the kinetic tempera* ture hypothesis, which was given in the indefinite form Jww 2 =f(T), may now be written in the definite form %mu 2 =%kT; where k is a constant (equal to R/n) with respect to any variation whatsoever of the nature or state of the fluid substance. The value of the fundamental constant k is readily calculated from the previously given values of R and n, and is thus found to be i.372Xio~ 16 ergs per degree. Prob. 27. The Number and the Mass of the Molecules. a. Show that the number of molecules in i ccm. of a perfect gas at 20 and i atm. is 2.51 Xio 19 . b. Calculate the mass of one molecule of hydrogen, and that of one atom of chlorine. 20. The Kinetic Equation for Imperfect Gases. As the pressure of a gas increases, the two effects that were neglected in the deriva- tion of the kinetic equation for perfect gases (Art. 18) must be taken into account; namely, the effect of the volume occupied by the mole- cules themselves, and the effect of forces of attraction between the molecules. These influences affect respectively the two factors in- volved in the derivation of the equation the number of impacts on the unit of surface in the unit of time, and the force corresponding to each impact. For the number of impacts made by a row of molecules moving between the opposite faces of the cube will evidently be increased when the distance which the molecules have to traverse becomes appreciably shortened by collisions between them. And the force produced by the impact of each molecule will evidently be 26 THE MOLAL PROPERTIES OF GASES diminished if, as it approaches the wall, it is appreciably attracted by the molecules behind it so that it strikes the wall with a smaller momentum. The correction to be applied to the kinetic equation for perfect gases for the volume of the molecules may be derived by considering the number and character of the collisions (whether head-on, glancing, or tangential) and the shortening of the path by each type of collision, and by integrating the separate effects. It is evident that the resultant effect will be dependent on the size of the molecules; and under certain simplifying assumptions it can be shown, if the volume of one mole- cule be represented by , that the distance between the opposite faces of the cube actually traversed by the molecules is reduced in conse- quence of the collisions from / to / (v 40)/. The number of im- pacts by all the molecules on unit-surface in unit-time is increased in the same proportion, namely, nu . j.-n. Nj nu from (as found in Prob. 230) to , r- 6v 6(v-4wt>) *The effect of the attraction of the molecules on the momentum with which a molecule strikes the wall can be derived from the follow- ing considerations. The decrease in its momentum mu must be pro- portional to the force acting on it and to the time during which that force acts. The force of attraction must, however, be proportional to the number of molecules n/v in the unit of volume; and the time during which the molecule is subjected to this attraction, which is the time required for it to trasverse the outer layers of the gas, must be inversely proportional to its velocity, which is uv/(v 4/fc/>); this velocity being greater than u, the velocity between collisions, because of the just considered effect of the volume of the molecules. The change in momentum attending the impact of the molecule on the wall will therefore be reduced from 2mu to 2 (mu 2 (v 4^0) ) , where a is a proportionality-constant dependent on the nature of the gas under consideration. The change of momentum of all the mole- cules striking the unit of surface in the unit of time, which is equal to * Articles, paragraphs, and problems to which an asterisk is preSxed may be omitted in briefer courses. THE KINETIC THEORY 27 the pressure p, is found by multiplying this quantity by the quantity derived in the preceding paragraphs. The product so obtained is nmu z an 2 3(fl-40) 3*> 2 *It is evident from this equation that the externally manifested pressure p is the resultant of an outward-directed pressure (repre- sented by the first term of the equation) caused by the motion of the molecules, and an inward-directed pressure (represented by the second term) caused by the attraction of the molecules within the gas for those in the outer or surface layers. By considering the corrections for the volume and attraction of the molecules there may be derived (in the way just shown) the following expression, which may be called the kinetic equation for imperfect gases: In this equation denotes the volume of a single molecule of the substance, and a represents a constant characteristic of the sub- stance which expresses the effect of the attraction of its molecules. Now, since it has been shown (in Art. 18) that for any gaseous or liquid substance \nmu z = NRT, and since the number of mole- cules n may evidently be replaced in the second and fourth terms by the number of mols N of the substance with new proportionality- constants, this expression may be transformed into the following one, known as the van der Waals equation, already mentioned in Art. 16: By considering one mol of the substance, N becomes equal to unity, and the equation assumes the simpler form which is often conveniently employed. It is evident from the derivation that the quantity b in the van der Waals equation denotes four times the volume of the molecules in one mol of the gas; and that the term aN 2 /v z or aft 2 denotes the inward- directed pressure due to the attraction of the molecules. This pres- sure is called the cohesion pressure; and the constant a may be called the cohesion constant. 28 THE MOLAL PROPERTIES OF GASES Since the cohesion-constant a and the volume-constant b vary with the nature of the molecule, it is evident that the van der Waals equa- tion involves the assumption that the molecules of the substance do not undergo a change through association as the pressure increases, or through dissociation as the temperature rises. These effects are, in fact, of not infrequent occurrence; and the limits of applicability of the equa- tion must therefore be determined experimentally for each substance. With many substances, probably with all of those whose molecules are all of one kind, the van der Waals equation gives values of the pressure or volume which agree with the observed values within one or two percent, so long as the pressure does not much exceed 20 atmospheres, or so long as the molal volume is not less than 1000 ccm. But it may lead to very erroneous results when applied, as it some- times is, at much higher pressures (for example, to the computation of the critical constants). At these higher pressures a satisfactory correspondence with the observed values can be attained by expressing the quantities a and b as definite functions of the molal volume, in- stead of regarding them as constants. A comparison of the observed pressures with the calculated pressures given in the 'following table illustrates the validity of the van der Waals equation for a few simple gases at molal volumes of 2000, 1000, and 500 ccm., or at pressures between 12 and 40 atmospheres. The constants used in calculating the pressures are given in the last two columns. In evaluating them the pressure is expressed in atmospheres and the volume in cubic centimeters, and they are referred to one mol of the gas. They are so chosen as to reproduce the observed pressure- volume relations accurately at pressures up to 10 or 12 atmospheres. VALIDITY OF THE VAN DER WAALS EQUATION Sub- stance Temper- ature Molal Volume Observed Pressure Calculated Pressure Constants io" a b N 2 IOOO 22.2 22.2 1.26 47 500 44.2 44.4 C0 2 40 2000 IOOO 500 12. 1 22.8 40.6 12.2 23.2 44.1 6.23 128 NH 3 IOOO 500 I6. 5 23.1 I6. 7 26.4 IO.O4 163 H 2 200 IOOO 500 29.9 46.8 30.1 57-8 10.02 210 THE KINETIC THEORY 29 *Prob. 28. Applications of the van der Waals Equation. The constants a and b in the van der Waals equation have for carbon dioxide the values 6.23Xio 6 and 128, respectively, when the pressure is ex- pressed in atmospheres and the volume in cubic centimeters, a. Cal- culate the pressure of carbon dioxide at 40 when the volume of one mol of the gas is 2000 ccm.; b, when it is 1000 ccm. c. Find the ratio of these computed pressures to the actual pressures, which are 12.11 and 22.84 atm., respectively, d. Find the ratio of the cohesion pres- sures to these two external pressures, e. Find the ratio of the correc- tion for the volume of the molecules to the volume of the gas for these two cases. /. Find the ratio of the cohesion pressure to the total pressure, and the ratio of the volume correction to the total volume, for carbon dioxide at o and i atm. (where the molal volume is not greatly different from 22,400 ccm., that for a perfect gas), g. From these ratios derive the percentage deviation of pi) from RT for carbon dioxide at o and i atm. ; and compare the result with that given in the table in Art. 15. Ans. a, 12.16 atm.; b, 23.24 atm. *Prob. 2g. Determination of the Constants of the van der Waals Equa- tion. When one mol of NH 3 has a volume of 2000 ccm. at 30 its pressure is 11.03 atm -> an d when the gas is heated at this (constant) volume to 60, its pressure becomes 12.37 at m. Calculate the values in atmospheres and cubic centimeters of the constants a and b for NHs in the van der Waals equation. Ans. a = io.o3Xio 6 ; 6 = 163. *Prob. 30. Principles Relating to the Internal Pressures. -*- a. From the results of the preceding problem find the values of the two oppositely directed internal pressures of which the external pressure of 11.03 a tm. is the resultant, for NH 3 at a molal volume of 2000 ccm. at 30. b. Find the values of the two internal pressures at 60; and state the principles which determine the effect of temperature upon each of them when the gas is kept at constant volume, c. Find the value of the cohesion pressure at 30 when the molal volume is 20,000 ccm. Ans. a, 2.51 and 13.54 atm.; c, 0.0251 atm. 21. The Characteristics of the Molecules. From the van der Waals volume-constant and the Avogadro number the volume and diameter of a single molecule can be readily calculated, as illustrated by Prob. 31. From the diameter of the molecule the number of its collisions per second and the mean distance traversed between successive collisions can be derived as shown in the text below. Prob. ji. Determination of the Dimensions of the Molecules. a. From the values, 128 and 163 ccm., of the volume-constants for CO 2 and for NH 3 calculate the actual volume of the molecules in one mol of each of these gases, and in 1000 ccm. of each of the gases at 20 and i atm. b. Assuming the molecules are spherical, calculate with the aid of the Avogadro number the diameters of the molecules of CO 2 and NH 3 in centimeters. 30 THE MOLAL PROPERTIES OF GASES The results of determinations of these molecular magnitudes for two substances, hydrogen and carbon dioxide, at 20 and one atmosphere are given in the following table. The number of molecules per cubic centimeter and the velocity of the molecules are for comparison also included. MOLECULAR MAGNITUDES Hydrogen Carbon dioxide Number of molecules per cubic centimeter . 2.5Xio 19 2.5Xio 19 Velocity in centimeters per second . . . i.pXio 5 0.4 Xio 5 Diameter of the molecule in centimeters . 2.4Xio~ 8 3.3Xio~ 8 Collisions by any one molecule per second . io.2Xio 9 6.2Xio 9 Mean distance between collisions in centimeters i7.2Xio~ 6 6.iXio~ 6 These figures show that at atmospheric pressure the number of collisions per second is enormous and that the distance which on the average a molecule traverses between collisions is extremely small. They show, however, that this distance is several hundred times greater than the diameter of the molecule. *From the diameter of the molecule, as stated above, certain other quantities characteristic of the molecular condition of gases can be derived. The most important of these are the number of collisions between the molecules per second, and the average distance traversed by a molecule between the collisions, a quantity commonly called the mean free path. An approximate relation between the diameter of the molecules and these quantities and a fairly close estimate of their magnitudes can be derived as illustrated by Probs. 32 and 33, with the aid of the approximate assumptions that the molecules are spherical, that any one molecule under consideration is moving with a uniform velocity equal to that required by the kinetic equation, and that the other molecules among which it is moving are stationary. The number of collisions C per second which one molecule of radius r moving with velocity u makes with the other molecules when n of these are present in volume t>, and the mean free path / of the molecules, are thus found to be given by the expressions: *It is seen that, as would be expected, the number of collisions per second is proportional to the number of molecules (n/v) per unit volume, to the cross-section (irr 2 ) of the molecules, and to their velocity (u); and that the mean distance traversed between collisions is in- THE KINETIC THEORY jx versely proportional to the number of molecules and to their cross- section. By a more rigorous analysis, not involving the above stated simplifying assumptions that the molecule under consideration has a uniform velocity equal to that required by the kinetic equation and that the other molecules are stationary, it can be shown that the mean free path (for spherical molecules) is only 0.71 times as great as that computed by the above equation. Relations- between the Diameter of the Molecules, the Number of Col- lisions, and the Mean Free Path. *Prob. J2. a. Considering, as illustrated in the figure, that a molecule of radius r is moving with a velocity u among stationary molecules, find an expression for the volume of the space within which the centers of the stationary molecules must be located in order that the ""TTV ^ moving molecule may collide with ^-^ them during one second, b. Find an expression for the number of stationary molecules with which the moving molecule collides per second, in case a volume v of the gas contains n molecules uniformly distributed, c. Derive from this equation the expression given in the preceding text for the mean free path of the molecules, stating the reasons involved. *Prob. jj. From the diameters, of the C0 2 and NH 3 molecules calcu- lated in Prob. 31, find for the case that each gas is at 20 and i atm. the number of collisions made by one molecule per second, and the mean free path. *The mean free path is a factor that determines the magnitudes of certain properties of gases, especially of their viscosity and their elec- trical and thermal conductivities; and conversely, its own magnitude can be computed from the results of measurements of these properties. Since the number of collisions and the diameter of the molecules can be calculated from the mean free path by means of the equations given above, there are available a number of independent methods of deter- mining all these molecular quantities. 22. Distribution of the Velocities and Kinetic Energies of the Molecules of a Gas. It has been assumed thus far that all the molecules of a gas have the same velocity and kinetic energy. Evi- dently this cannot be the case, since as a result of the collisions there is a constant interchange of the kinetic energies, with consequent variation of their values from zero up to values much greater than the mean kinetic energy. It can be shown that, in view of the very 3 2 THE MOLAL PROPERTIES OF GASES large number of molecules and the frequency of their collisions, their kinetic energies and velocities must be distributed in accordance with the laws of probability, of which the distribution of errors of observa- tion around the mean value affords a familiar illustration. This law of the distribution of kinetic energies and velocities is known as the Maxwell distribution law. 0.7 0.6 D.6I I 0.3 0.21 0.1 E- K .05 dK 1.0 1.5 Kinetic Energy K 2.0 2.5 3.0 The figure shows graphically the distribution of kinetic energies required by this law. The two graphs OABC and ODEF refer to o and 136.5, corresponding to absolute temperatures which stand to each other in the ratio 2:3. As abscissas are plotted the kinetic energies K (equal to %tmi 2 ) of the molecules expressed in terms of the mean value of the kinetic energy at o as unity; and as ordinates are plotted the values P of such a function of K that the product P dK, represented by the area beneath any element of the graph, is equal to the fraction dn/n of the whole number of molecules n (in one mol) that have kinetic energies between K and K-\-dK\ the area beneath the whole graph being equal to unity and corresponding to the total number of the molecules. Otherwise interpreted, the product P dK is the fraction of the time that any given molecule has kinetic energies between K and K-\-dK. Correspondingly, the relative values of the ordinates P, for different points of the curve represent the relative fractions of all the molecules that have kinetic energies within any THE KINETIC THEORY 33 definite infinitesimal amount of those represented by the correspond- ing abscissas K; or they represent the relative fractions of the time that any one molecule has these kinetic energies. In the case of the graph OA BC, where the kinetic energies are expressed in terms of the mean kinetic energy at the same temperature, the quantity P, com- monly called the probability function, has the form Prob. 34. Fraction of the Molecules having Kinetic Energies between Definite Limits. Determine from the figure the percentages of the total number of molecules which at o have kinetic energies lying be- tween limits which are multiples of the mean kinetic energy at o, as follows: a, o.o and 0.2; b, 0.2 and 0.6; c, 0.6 and i.o; d, i.o and 2.0. Prob. 55. Effect of Temperature on the Fraction of the Molecules having Kinetic Energies Greater than any Definite Absolute Value. Determine from the figure the percentages of all the molecules that, a, at o, and b, at 136.5, have kinetic energies greater than twice the mean kinetic energy at o. The fact, illustrated by the preceding problem, that the fraction of the molecules having kinetic energies exceeding any definite value increases rapidly with rising temperature is of much importance in relation to the effect of temperature on the vapor-pressure of liquid substances and on the rate of chemical reactions. The graphs in the figure represent also the distribution of the squares of the velocities of the molecules in terms of the mean velocity- square at o. Similar graphs can be drawn to show the distribution of the velocities themselves. When the velocity u is expressed in terms of the square-root of the mean value of the velocity-square at the same temperature, the probability function has the form: The percentages of the total number of molecules which possess values of the velocity that lie between limits which are certain multi- ples of the square-root of the mean velocity-square are as follows: Velocity multiples . . 0.0-0.4 0.4-1.4 1.4-2.5 2.5-00 Percentage of molecules . 8.2 76.9 12.7 2.2 *Differential equations expressing the distribution of kinetic ener- gies and velocities can readily be formulated as in Prob. 36. These equations cannot be directly integrated; but by methods of approxi- mation the integrals have been evaluated between various limits. 34 THE MOLAL PROPERTIES OF GASES *Prob. 36. Expressions for the Fraction of the Molecules having Definite Kinetic Energies or Definite Velocities. With the aid of the probability-functions given in the text, formulate differential equations for the fraction dn/n of the total number of molecules n which possess, a, kinetic energies (expressed in terms of the mean kinetic energy) between any value K and an infinitesimally greater value K+dK; and b, velocities (expressed in terms of the square-root of the mean velocity- square) between any value u and an infinitesimally greater value u+du. *Expressions for the fractions dn/n and n /n of the molecules that at any temperature T have absolute kinetic energies (i) between E and E-\-dE, and (2) greater than any definite value E Q , are given below. Equation (i) is obtained from that of Prob. 360, by replacing the energy-ratio K by the quantity JS/f k T ( J k T being the mean kinetic energy of the molecules); and equation (2) is obtained by integrating equation (i) between the limits EQ and oo : *The mean kinetic energy at any definite temperature is evident- ly directly related to the square-root of the mean velocity-square (\/Sw 2 /w) at the same temperature; this square-root being equal to the velocity derived from the kinetic equation %nmu 2 = NRT. In many kinetic relations the most probable kinetic energy and the most probable "velocity are also important quantities. These are denned to be the kinetic energy and velocity, respectively, which the greatest number of molecules possess. Their values can be shown, as in Prob. 37, to stand in a definite numerical ratio to the mean values just mentioned. Thus the square of the most probable velocity is just two-thirds of the mean velocity-square. Another important quan- tity is the mean velocity (2u/ri), which can be shown by a less simple process to be 0.921 of the square-root of the mean velocity-square. *Prob. 37. Relations between the Most Probable and Mean Values of the Kinetic Energy and of the Velocity. a. Determine from the figure the ratio between the mean kinetic energy and the most probable kinetic energy, b. Derive this relation also from the probability- function given in the text, noting that the most probable kinetic energy corresponds to the maximum value of that function, c. Derive similarly the relation between the square-root of the mean velocity- square and the most probable velocity. THE ENERGY RELATIONS OF GASES 35 III. THE ENERGY RELATIONS OF GASES 23. Energy in General and the Law of its Conservation. The essential idea underlying the concept of energy is the constancy of a quantity which is involved in all the changes taking place in the universe; and this is often explicitly expressed by the statement that energy is neither created nor destroyed in any process whatever. This statement is called the law of the conservation of energy, or the first law of thermodynamics. The law may be stated more concretely as follows: When a quan- tity of energy disappears at any place, a precisely equal quantity of energy simultaneously appears at some other place or places; and when a quantity of energy disappears in any form, a precisely equal quantity of energy simultaneously appears in some other form or forms; equal quantities of energy of different forms being understood to be such quantities as produce the same effect (for example, in modifying motion or raising temperature) when converted into the same form. The exactness of this law has been established by many careful quantitative investigations made for the purpose. The law is also confirmed by the correspondence of the conclusions drawn from it with well-established facts and principles. Among these may be mentioned the following principle, which is a conclusion based upon the failure of many attempts to produce a contrary result: The production of an unlimited amount of work by a machine or arrangement of matter which receives no energy from the surroundings is an impossibility. An ideal process like that here stated to be impossible is sometimes called perpetual motion of the first kind (to distinguish it from another kind of perpetual motion which will be later described). Of especial importance in chemical considerations are the energy effects attending changes in the state of systems. By the system is understood the definite kinds and quantities of matter under con- sideration. A system is said to be in a definite stale when the tem- perature, pressure, state of aggregation, quantity, and chemical composition of each of its parts is fixed; and a change in any of these conditions is called a change in the state of the system. Thus the system under consideration might consist of 16 g. of the element oxygen and 2.015 g- of the element hydrogen. This system might exist initially in the state of a mixture of 0.5 mol of oxygen gas (O 2 ), and i mol of hydrogen gas (H 2 ) at a temperature of 20 and a total pressure of ten 36 THE MOLAL PROPERTIES OF GASES atmospheres. It might now undergo a change in state, and exist finally in the state of one formula-weight of liquid water at 20 and one atmosphere. It follows from the law of the conservation of energy that, when the state of a system is fixed, its energy-content U is also fixed; and therefore also that any change in the state of a system is attended by a definite change AU in. its energy-content (equal to the difference U z Ui between its energy-content in the two states), whatever be the process by which the change takes place. This corollary from the law of the conservation of energy, stating that the change in the energy-content of a system is determined solely by its initial and final states, is of so much importance in chemical considerations that it has received a special name the law of initial and final states. Corresponding to the change in the energy-content of the system, there must, of course, be an energy effect in the surroundings; and, in general, the only method of determining the change in the energy- content of a system is to measure the energy produced in the sur- roundings. This is ordinarily done by causing the change in state to take place within a calorimeter and measuring the heat which is imparted to or withdrawn from the calorimeter and its contents; account being taken of any other form of energy which may at the same time be produced or destroyed. The energy lost or gained by the system may appear or disappear in the surroundings in any of its various forms; but in energy considera- tions it is primarily important only to differentiate the production of heat from that of the other forms of energy. Such other forms of energy (that may be associated with matter) are collectively desig- nated work. Under this term are included, for example, production of motion in a body, displacement of a force through a distance, change of volume under pressure, development of electrical energy, and production of chemical changes. All these forms of work are quanti- tatively transformable into one another; but the transformation of heat into work is subject to certain limitations, which will be considered later. It is for this reason that heat and work are differentiated from each other. It will be noted that the term work is here used in a broader sense than that in which it is used in the science of mechanics. With the aid of the concepts of energy-content, heat, and work, the law of the conservation of energy may now be expressed by the statement that the decrease in the energy-content of the system THE ENERGY RELATIONS OF GASES 37 when it undergoes a change in state is equal to the sum of the quanti- ties of heat and work produced in the surroundings. Of these three quantities the change in energy-content is most fundamental, since it has a definite value for any definite change in state; while the quantities of heat and of work produced may vary with the process by which the change in state takes place. Three units of energy are commonly used in scientific work the erg, the joule, and the calorie. The erg is the energy expended when a force of one dyne is displaced through one centimeter, or it is twice the energy possessed by a mass of one gram when moving with a velocity of one centimeter per second. The joule is a decimal multiple of the erg; namely, one joule equals 10 7 ergs. The mean calorie is one- hundredth part of the heat required to raise one gram of water from o to 100. This is identical within 0.02% with the ordinary calorie, which is the heat required to raise one gram of water from 15 to 16. One calorie (i cal.) is equal to 4.182 (approximately 4.18) joules, or 4.i82Xio 7 ergs, this value being the so-called mechanical equivalent of heat. Prob. 38. Evaluation of the Change in Energy-Content. When i g. of liquid water is vaporized at 100 against. a pressure of i atm., the heat withdrawn from the surroundings is 537 cal., and the work pro- duced by the expansion against the constant pressure is 168 joules. a. In which state does the water have the greater energy-content, and what is the difference between the two energy-contents in calories? b. How much heat would be drawn from the surroundings in case i g. of liquid water at 100 and i atm. was vaporized within a closed vessel (no work being produced in the surroundings) so as to form water-vapor at 100 and i atm? 24. Work Attending Volume Changes in General. The work produced when a system changes its volume is of especial importance in chemical considerations. It is 'most readily derived for the case that the volume undergoes a change in dimensions in one direction only. Suppose that a liquid or gaseous substance contained in a cylinder is enclosed by a movable piston of cross-section a, and that a force / is exerted upon this piston, for example, by a weight placed upon it. Suppose now that the piston rises through a distance dl. The increase of volume dv is then adl, and the force acting upon the unit of surface, which force is the pressure p, is f/a. The work dW pro- duced by the expansion is therefore given by the equations: 38 THE MOLAL PROPERTIES OF GASES That is, the work is equal to the product of the pressure into the infinitesimal increase of volume that takes place. It can be easily demonstrated that this equation also holds true in the general case in which the volume increases in dimensions in any number of directions. The general expression for the work produced when a body under- goes a change of volume from vi to %, is therefore: W = I pdo. */ vi In order to carry out the integration it is evidently necessary to know how the pressure varies with the volume of the system as the change in state under consideration takes place. The most common case in energy discussions is that in which the external pressure is considered to be substantially equal to the pressure exerted by the substance or substances of which the system is com- posed, these two pressures differing by only the infinitesimal amount which in the absence of friction suffices to cause the change in state to take place. In this case, in order to carry out the integration, it is necessary to know the functional relation between the pressure and the volume of the substance or substances involved, in other words, to know the equation of state of the substance or mixture of sub- stances of which the system consists. Prob. 39. Work Produced by the Change in Volume of a System at Constant Pressure. a. Calculate the work in ergs and calories that is produced when one formula- weight of water (iH 2 O) vaporizes at 100 against the pressure of i atm. The specific volume of liquid water at 100 is 1.043, a d that of saturated water vapor is 1650. b. Find the percentage errors that would be made by neglecting the volume of the liquid, and by regarding the vapor as a perfect gas. c. Formulate a general expression for the work produced when a system undergoes a change of volume under a constant pressure; and show that under these approximate assumptions the work W produced by the vaporiza- tion at a constant temperature T of that quantity of a liquid which forms N mols of saturated vapor is given by the expression W =NRT. d. Show that the value of the gas-constant R is 1.985 cal. per degree. Prob. 40. Work Producible by the Change in Volume and Pressure of a Perfect Gas at Constant Temperature. a. Calculate the work in calories produced when i mol of a perfect gas at 2 atm. expands at 20 till its pressure becomes i atm., the opposing external pressure being kept always substantially equal to that of the gas. b. Calculate the work in calories produced when under these conditions iCO 2 at 2 atm. expands at 20 till its pressure becomes i atm. The molal volume of CO 2 at 20 and 2 atm. is 11,890 ccm. (or 1.16% less than that of a perfect gas). Ans. b, 408 cal. THE ENERGY RELATIONS OF GASES 39 25. The Energy-Content of Systems in General in Relation to Temperature and Heat-Capacity. The change in the energy-content which attends a change in the temperature of any system is experi- mentally determined by measuring the quantities of heat and work withdrawn from or imparted to the surroundings. The ratio of the quantity of heat dQ absorbed when the temperature of the system rises from T to T+dT to the rise of temperature dT is called its heat- capacity C at T\ that is, C =dQ/dT. The heat-capacity is substan- tially equal to the quantity of heat absorbed when the temperature rises one degree. When the heating takes place without change of pressure, the heat absorbed per degree is called the heat-capacity at constant pressure Cp. When the heating takes place without change of volume, the heat absorbed per degree is called the heat-capacity at constant volume C v . The heat-capacity of any homogeneous system is obviously the product of its weight by the heat-capacity of one gram of it, which is called its specific heat-capacity C (or often its specific heat). The heat-capacity of one mol or one formula- weight of a pure substance is called its molal or formal heat-capacity C. The heat- capacity of one atomic weight is called the atomic heat-capacity. Prob. 41. Measurement of the Heat-Capacity of Gases. Through a coil within a water calorimeter at 20.000 having a total heat-capacity of 1 100 cal. per degree n.o g. of carbon dioxide preheated to various temperatures were passed at a pressure of i atm. in a series of experi- ments, and the rise in temperature in the calorimeter was measured, with the following results: Initial gas temperature . . 200 400 600 800 Final calorimeter temperature 20.389 20.861 21.378 21.930 a. Calculate the quantity of heat given off to the calorimeter in each experiment, b. Plot on a large scale these heat quantities Q as ordi- nates against the initial gas temperatures t as abscissas, c. Determine with aid of the plot the molal heat-capacity of carbon dioxide at 200 and that at 600. When a system is heated without change of volume, no work is produced, and the change in its energy-content per degree is therefore simply equal to the heat absorbed from the surroundings or to the heat-capacity of the system at constant volume. When a system is heated at constant pressure, not only is heat absorbed from the surroundings, but also work is produced in them. The increase in energy-content per degree is therefore equal to the heat absorbed diminished by the work produced. 40 THE MOLAL PROPERTIES OF GASES *Prob. 42. Mathematical Expressions for the Heat-Capacities. Formulate partial-derivative expressions for the definitions of the heat- capacity at constant volume and that at constant pressure, and. for the relation of each of these heat-capacities to the corresponding change in energy-content per degree. Ans. C' = \'T\ ' e ^ c " 26. The Energy-Content and Heat-Capacity of Perfect Gases. Experiments have shown that when a perfect gas expands at a constant temperature without producing any work (for example, when it expands within a calorimeter from one vessel into another vessel previously evacuated), there is no heat-effect in the surroundings (no change of temperature in the calorimeter). Such experiments have established the important law that the energy-content of a definite quantity of a perfect gas at any definite temperature has the same value, whatever be its volume and pressure. In other words, for the change in state which occurs when a perfect gas changes its volume and pressure at a constant temperature, the change At/ in its energy- content is zero. From this law of perfect gases the following principles can be derived with aid of the law of initial and final states, as in Probs. 43 and 44 : (1) At any definite temperature the increase per degree dU/dT of the energy-content of any definite quantity of a perfect gas has the same value whatever be the initial and final pressures and volumes; and therefore the heat-capacity at constant volume of any perfect gas is independent of the pressure and volume. (2) The molal heat-capacity of a perfect gas at constant pressure is greater than that at constant volume by an amount equal in value to the gas-constant R, whatever be the gas and whatever be the temperature or pressure; that is, Cp C v = R = i .985 cal. per degree. Derivation of the Heat-Capacity Relations of Perfect Gases. Prob. 43. Derive the first principle stated in the preceding text by considering that a perfect gas changes by two different processes from a volume Vi at pressure pi and temperature T\ to a volume % at pressure fa and temperature T z . Prob. 44. Derive the second principle stated in the preceding text by a consideration similar to that employed in the last problem. This second principle obviously enables the heat-capacity at con- stant volume to be calculated from the heat-capacity at constant THE ENERGY RELATIONS OF GASES 41 pressure. This is of importance, since the former quantity is one which it is difficult to determine accurately by direct experiment, owing to the fact that any rigid container used for holding the gas and preventing its expansion has unavoidably a much larger heat- capacity than the gas itself; while the latter quantity can be readily and accurately measured (as illustrated by Prob. 41). In view of this difficulty of determining directly the heat-capacity at constant volume, it is important to note also that the heat-capacity ratio Cp/C v can be derived from measurements of certain properties of gases; especially, from measurements of the change of pressure which results when the gas expands adiabatically (that is, without imparting heat to the surroundings or withdrawing heat from them), and from measurements of the velocity of sound through the gas. These methods need not, however, be here described. 27. The Heat-Capacity of Perfect Gases in Relation to their Molecular Composition. The heat-capacity of perfect gases depends primarily on the complexity of their molecules, as is shown by the following principles. The molal heat-capacity at constant volume C v has the smallest value for gases with monatomic molecules, such as mercury, helium, and argon; and it has the same value, namely f R or 2.98 cal. for all such gases at all temperatures. The corresponding value of Cp is f R or 4.96 cal. It will be shown in Art. 28 that the kinetic theory affords a striking explanation of this simple behavior. The behavior of gases with diatomic molecules is less simple. Many of these gases, for example those with the formulas N2, O 2 , CO, NO, HC1, have substantially the same value of C v , and one which varies appreciably, but not very greatly, with the temperature, between o and 1200; thus for all these gases the value of C v is 4.9 cal. at 20, 5.6 cal. at 1000, and 6.3 cal. at 2000. Between the temperatures o and -i 80 the values for these gases change irregularly, but not very greatly; thus at 180, C v is 4.73 for nitrogen, 4.91 for oxygen, and 4.76 for carbon monoxide. The values for hydrogen (H 2 ) between 20 and 2000 are slightly (about 0.15 cal.) smaller than those for the gases just mentioned, and they become very much smaller at very low temperatures, attaining at 180 the value 4.3 cal., and at 210 the value 3.0 cal. shown by monatomic gases, this value then remaining constant at any rate down to -240 (33 A.). A few diatomic gases, for example those with the formulas C1 2 , Br 2 , I 2 , IC1, have at room 42 THE MOLAL PROPERTIES OF GASES temperature larger values of C v than do the other diatomic gases ; thus the value for chlorine is 5.9 cal. at 20. The only general statements that can be made in regard to the heat- capacities of triatomic and other polyatomic gases are that the values are much larger than those for the diatomic gases, that they increase more rapidly with the temperature, and that they increase with the complexity of the molecule; thus the values of C v at 200 are 6.5 for water vapor (H 2 O), 8.5 for carbon dioxide (CO 2 ) and sulfur dioxide (SO 2 ), 7.4 for ammonia (NH 3 ), and 41.6 for ether (C4HioO). The following table shows the functions which express, in general within one or two percent, the observed values of the molal heat-capacities at constant volume at atmospheric pressure, of some of the common gases at various temperatures. MOLAL HEAT-CAPACITIES AT CONSTANT VOLUME Substances Range of temperature Value of C v Hg, Na, K} Complete 2.98 +0.00 T 4.70+0.0007 r 4. 55+0. 0007 T C1 2 02000 5.65+0.00077 NH 3 0-500 5-83 + 0.0026 T + 0.000001 2 T 2 These principles in regard to heat-capacity, especially that relating to monatomic gases, have made it possible to determine (as illustrated in Prob. 45) the atomic weight of the inert elements, helium, neon, argon, etc., to which, because of their failure to form compounds, the usual method described in Art. 14 is not applicable. Prob. 45. Determination of Atomic Weight from the Heat-Capacity Ratio. From the experimentally determined velocity of sound in argon gas the heat-capacity ratio Cp/C v has been found to be 1.67. Find the atomic weight of argon by combining this result with its density, which at 20 and i atm. has been found to be 1.246 times as great as that of oxygen; stating the principles involved. 28. The Energy-Content and Heat-Capacity of Perfect Gases in Relation to the Kinetic Theory. In the discussion of the kinetic theory in Art. 18 only the translatory kinetic energy of the molecules (that due to their linear motion) was considered. It is evident, however, that the molecules may be set in rotation as a result of their collisions, and consequently that they may possess rotational kinetic THE ENERGY RELATIONS OF GASES 43 energy. It is evident also that the atoms within the molecules may possess kinetic energy due to vibration and potential energy arising from forces acting between the atoms and dependent upon their position with reference to one another; these constituting the atomic energy of the molecules. The proportion which each of these three kinds of energy contributes to the total energy of the molecules may be expected to vary with the temperature and with the nature of the molecules. In the case of monatomic molecules the conditions are simpler. For in the first place, since the molecule is identical with the atom, the atomic energy falls away. In the second place, it may well be true that the atoms, and hence also the molecules, are so small or so sym- metrical that their kinetic energy of rotation is negligible in comparison with their kinetic energy of translation. In this case the energy- content of the gas would consist solely in the translatory kinetic energy K of its molecules. And the heat-capacity C v at constant volume, which was shown in Art. 25 to be equal in general to the increase dU/dT in the energy-content per degree, would be equal to the increase dK/dT in this kinetic energy per degree; which increase dK/dT has been shown in Art. 18 to be equal to %R or 2.98 cal. per degree for one mol of any gas. Now, experimental determinations have in fact shown, as stated in Art. 27, that all monatomic gases have a molal heat-capacity at constant volume substantially equal to 2.98 cal. per degree. The conformity of this prediction of the kinetic theory with the facts is one of the striking successes of the theory. The fact stated in Art. 27 that most of the diatomic gases (nitrogen, oxygen, carbon monoxide, etc.) have values of C v which are nearly the same (about 5.0 cal. at 20) for the different gases and which do not vary much with the temperature indicates that the atomic energies and rotational energy of their molecules increase in nearly the same proportion as the translatory kinetic energy. The fact, however, that the heat-capacity of hydrogen gas, though not differing greatly from that of the other gases at room temperature, attains the value for a monatomic gas at very low temperatures, suggests that its molecules are then no longer taking up an appreciable proportion of atomic or rotational energy. Diatomic gases like chlorine which have larger values of C v util- ize this greater amount of absorbed heat to set their atoms into more rapid vibration and to separate them further from each other. This 44 THE MOLAL PROPERTIES OF GASES is in correspondence with the lower temperatures at which the mole- cules of these latter gases dissociate into their atoms. The molecules of gases with more than two atoms may be expected to take up more energy of rotation than the more nearly two-dimen- sional diatomic molecules and to convert more of the absorbed heat into atomic energy. This is in accordance with the fact that in general the heat-capacity increases with the complexity of the molecule. 29. The Energy-Content of Imperfect Gases in Relation to Volume and Pressure. Actual gases show deviations from the perfect-gas law that the energy-content of the gas at any definite temperature is independent of its volume and pressure. These deviations always lie in the direction that the energy-content of the gas increases with decrease in its pressure. They are small at atmospheric pressure, increase rapidly with the pressure, and become large at 30-1 oo atmo- spheres. The increase in energy-content attending an isothermal expansion is most accurately determined by porous-plug experiments, in which the gas is caused to flow continuously through a well-insulated tube containing a plug of porous material, as illustrated in the figure. K///////////////////////// ///// ////////////////////////////////////////A | T lPl ^ :$$ x-r-'X T*P**i -> Under these conditions the gas expands without taking up heat from the surroundings and without producing work, except that equivalent to the change in its pressure-volume product. It undergoes thereby a decrease in temperature, which is often called from its discoverers the Joule-Thomson effect. This decrease in temperature is accurately measured; and from its magnitude, from the heat-capacity of the gas, and from the change in its pressure-volume product, the increase in the energy-content that would have attended an isothermal expansion is calculated, in the way illustrated by the following problem. *Prob. 46. Determination of the Change in Energy-Content by Porous- Plug Experiments. Carbon dioxide at pressure p\ (e.g., 2 atm.) and temperature 7\ (e.g., 20.00) is caused to flow continuously through a well-insulated hardwood tube containing a porous plug of cotton. On passing through the plug its pressure falls to p z (e.g., i atm.), and it emerges from the tube at this pressure. After the gas has flowed so ENERGY RELATIONS OF GASES 45 long that every part of the apparatus has assumed the temperature of the gas in contact with it, the expansion of the gas takes place without ex- change of heat with the surroundings. Its temperature after passing through the plug is found to be T 2 (e.g., 18.86). These conditions are illustrated by the figure, a. What other process must be combined with this adiabatic process in order that the net result of the two pro- cesses may be the isothermal expansion of one mol of the gas from pressure pi to pressure p z at a temperature 7\? b. Formulate expres- sions for the work produced W, the heat absorbed Q, and the change in energy-content A 7, for each of these two processes. (Note that in the first process a volume Vi of the gas disappears on one side of the plug under a constant pressure pi, and that a certain volume %' of the gas is produced on the other side of the plug under a constant pressure pz.) c. Combine these results so as to give an expression for the change in energy-content that attends a change in pressure from pi to pz of one mol of the gas at a constant temperature I\. d. Calculate in calories the change in energy-content attending the expansion of i mol of carbon dioxide from a pressure of 2 atm. to a pressure of i atm. at 20, using the following data in addition to those given above: the molal volume of carbon dioxide at 20 and 2 atm. is 11,890 can., and at 20 and i atm. is 23,920 ccm.; its molal heat-capacity at 20 and at a constant pressure of i atm. is 8.92 cal. per degree, e. Calculate the heat that would be absorbed from the surroundings if the change in state given in d took place at 20 against an opposing pressure kept substantially equal to that of the gas, using the result obtained in Prob. 406. Ans. d, 6.78 cal. At high pressures and at low temperatures the Joule-Thomson cooling effect and the corresponding increase in energy-content attend- ing isothermal expansion are large even in the case of difficultly con- densible gases. Thus, when air at 2 atmospheres and at 20 expands to a pressure of one atmosphere under the conditions of a porous-plug experiment, there is a fall of temperature of 0.25; but if the initial pressure is 102 atmospheres there is a fall of 25 when the initial temperature is o, and a fall of temperature of 74 when the initial temperature is 100. Certain technical processes for liquefying air (for example, the Linde process) utilize only the Joule-Thomson effect, but certain other processes (such as the Claude process) provide for production of work during the expansion, whereby a greater cooling effect is obtained (as indicated by Prob. 46 e). *30. The Energy-Content of Imperfect Gases at Constant Tem- perature in Relation to the Kinetic Theory. The increase in energy- content attending the isothermal expansion of a gas, as computed from porous-plug experiments like those described in Art. 29, is evidently 46 THE MOLAL PROPERTIES OF GASES equivalent to the energy that must be imparted to the gas in order to separate its molecules from one another. In other words, it is equal to the internal work that has to be done in displacing the cohesion pressure through the increment of volume. It follows therefore that the increase dU in energy-content attending an increase in volume dv, and the increase A 7 in energy-content attending an increase in volume from Vi to %, will be given, if the cohesion pressure has the value required by the van der Waals equation, by the following expressions: --- ; or (for one mol fc %/ \Vi Prob. 47. Calculation of the Change in Energy-Content from the Cohesion-Constant. Calculate the increase in energy-content in calories attending the isothermal expansion of iC0 2 from 2 atm. to i atm. at 20, from the value of the cohesion-constant for carbon dioxide given in Prob. 28, and tabulate this value beside that derived in Prob. 46 from the data of porous-plug experiments. Conversely, the cohesion-constant can be derived from the results of porous-plug experiments. And in general the so-obtained values of this constant are approximately identical with those calculated from the pressure- volume relations by the van der Waals equation. Thus the cohesion-constant for carbon dioxide may be calculated from the data of Prob. 46 to be 6.62 Xio 6 , while that derived from the pressure-volume relations is 6.23Xio 6 . This agreement shows that the term aN 2 /v 2 in the van der Waals equation does in reality cor- respond to a cohesion pressure arising from an attraction between the molecules. CHAPTER III THE MOLAL PROPERTIES OF SOLUTIONS AND THE MOLECULAR THEORY I. VAPOR-PRESSURE AND BOILING-POINT IN GENERAL 31. Vapor-Pressure. A liquid in contact with a vacuous space vaporizes until the pressure of its vapor in that space attains a per- fectly definite value which is determined by the nature of the liquid and by the temperature. If, on the other hand, vapor having a pressure greater than this definite value is brought into contact with the liquid, condensation occurs until the pressure of the vapor falls to that value. In other words, for a given liquid at a given tempera- ture there is only one pressure which its vapor can have and exist in equilibrium with that liquid. This pressure is called the vapor- pressure of the liquid. This is to be distinguished from the pressure of the vapor, which when not in contact with the liquid may have any value from zero up to one somewhat exceeding the vapor-pressure. Solids likewise have definite vapor-pressures, which with certain substances (like iodine) are appreciable even at room temperature. The vapor-pressure of a liquid or solid substance increases rapidly with increasing temperature, as illustrated by the data of Prob. 3. When a liquid or solid is in contact with a space containing a gas (for example, when water or iodine is in contact with an air space), approximately the same quantity of the liquid or solid vaporizes as if the gas were not present, provided the gas is only slightly soluble in the liquid, and provided its pressure is not much greater than one atmosphere. When the gas is readily soluble in the liquid, or when its pressure is large, considerable deviations from this principle may result. * Prob. i. Pressure-Volume Relations of Wet and Dry Gases. 25 ccm. of dry air at 28 and i atm. are collected over water, whose vapor- pressure at 28 is 28 mm. a. What is the pressure if the volume is still 25 ccm,? b. What is the volume if the pressure is i atm.? Ans. b, 25.95. Prob. 2. Air-Bubbling Method of Determining Vapor-Pressure. 2000 ccm. of dry air at 15 and 760 mm. are bubbled through bulbs con- taining a known weight of carbon bisulfide (CS 2 ) at 15, and the mix- ture of air and bisulfide vapor is allowed to escape into the air at a 47 48 THE MOLAL PROPERTIES OF SOLUTIONS pressure of 760 mm. By reweighing the bulbs, 3.01 1 g. of the bisulfide are found to have vaporized. Find the vapor-pressure of carbon bisulfide at 15. Ans. 242 mm. Steam-Distillation of Liquids Insoluble in Water. Prob. j. Steam is bubbled through chlorbenzene (C 6 H 5 (?1) in a dis- tilling flask; and the vapors, which escape under a barometric pressure of i atm., are condensed as a distillate. The steam partially condenses in the distilling flask, and brings the mixture of water and chlorbenzene (which are not appreciably soluble in one another) to that tempera- ture where equilibrium prevails between each liquid and its vapor. Determine this temperature and the molal composition of the distillate with the aid of a plot of the following data, which represent the vapor- pressures of the pure substances at various temperatures: 70 80 90 100 Water ..... 234 355 526 760 mm. Chlorbenzene ... 98 145 208 292 mm. Prob.4. A current of steam is passed at atmospheric pressure through a -mixture of water and nitrobenzene (C 6 H 6 NO 2 ). Calculate, a, the temperature of the distilling mixture, and, b, the percentage by weight of nitrobenzene in the distillate, from the following data: the vapor- pressure of water at 100 is 760 mm. and changes by 3.58% per degree; that of nitrobenzene at 100 is 20.9 mm. and changes by 5.0% per degree. Ans. b, 15.5%. 32. Relation of Boiling-Point to Vapor-Pressure. The point of a liquid is the temperature at which it is in equilibrium with its vapor when both are subjected to any definite external pressure. In other words, it is the temperature at which the vapor-pressure, which increases as the temperature rises, becomes equal to the external pressure. When this temperature is exceeded by an infinitesimal amount, assuming that there is no superheating, the vapor forms throughout the mass of the liquid (not merely at its free surface), giving rise to the familiar phenomenon of boiling. Prob. 5. Change of Boiling-Point with Barometric Pressure. The vapor-pressure of water at 100 increases 27.2 mm. per degree. What variation of its boiling-point corresponds to a variation of the baro- metric pressure from 730 to 790 mm.? 33. Change of Vapor-Pressure with Temperature. The Clapeyron Equation. The vapor-pressure of liquids or solids increases very rapidly with the temperature, and roughly by equal multiples of its value for equal increments of temperature. Prob. 6. Approximate Relation b'effryn Vapor-Pressure and Tem- perature. From the data of Prob. 3 find and tabulate the ratios of the VAPOR-PRESSURE AND BOILING-POINT 49 vapor-pressure at 80 and 70, 90 and 80, and 100 and 90, a, for water; and b, for chlorbenzene. From the laws of thermodynamics, there can be derived, as shown in Art. 165, an exact differential expression for the increase of the vapor-pressure p of a liquid or solid at the absolute temperature T with the temperature. This expression, which is known as the Clapeyron equation, is as follows: i = A# dT ~ TAv In this equation Av represents the increase in volume when any definite quantity of the substance changes at the pressure p and temperature T from the state of liquid or solid to the state of saturated vapor, and AH denotes the heat absorbed from the surroundings when this quantity of substance undergoes the same change in state. This quantity of heat is commonly called the heat of vaporization, or when referred to one mol of the substance, its molal heat of vaporization. Since at the boiling-point of a liquid its vapor-pressure is equal to the external pressure upon the liquid and vapor, the Clapeyron equa- tion also expresses (more clearly in the inverted form) the change of boiling-point with the external pressure. In numerical applications of this equation, the energy quantities AH and Av X dp must be expressed in corresponding units. The latter quantity will be in ergs when the volumes are in cubic centimeters and the pressure is in dynes per square centimeter. The units of energy commonly employed and the relations between them were described in Art. 23. Prob. 7. Application of the Clapeyron Equation. a. Calculate with the aid of the Clapeyron equation the volume of one mol of satu- rated water- vapor at 100 from the following data: At 100 the vapor- pressure of water increases 27.2 mm. per degree, the heat of vaporiza- tion of one gram of it is 537 cal., and the specific volume of liquid water is 1.043. b. Calculate by the perfect-gas equation the volume of one mol of saturated water-vapor at 100. c. By comparing these two values of the molal volume, determine the percentage error that would be made in assuming that the saturated vapor conforms to the perfect- gas law. Ans. a, 29920 can.; c, 2.3%. A simpler, but less exact form of the Clapeyron equation can be derived (as in Prob. 8) by making the assumptions that the volume of the liquid or solid is negligible in comparison with that of the 50 THE MOLAL PROPERTIES OF SOLUTIONS saturated vapor and that the vapor conforms to the perfect-gas laws. This approximate Clapeyron equation is dlogp = AT dT ~ RT 2 ' In this equation p denotes the vapor-pressure of the liquid or solid substance at the absolute temperature T, AH its molal heat of vapori- zation at that temperature, and the symbol log the natural logarithm (to the base e). (This symbol is always used in this sense in this book; logarithms to the base 10 being denoted by the symbol logio. It may be noted that log x = 2.3026 logio x = approximately 2.3 logio x.) The approximate Clapeyron equation evidently yields, at moderate pressures where the volume of the liquid may usually be neglected, values of dp/dT or of d log p/dT which are less than those derived from the exact equation by nearly the same percentage amount as the actual volume of the saturated vapor is less than that calculated by the perfect-gas equation, or by the same percentage amount as the actual value of p v for one mol of the saturated vapor is less than that of RT. This amount, for example, is 2.3% for water at 100 (as found in Prob. 7), and 3.6% for ethyl alcohol at 78.3 (its boiling- point at one atmosphere). These quantities are evidently the values of the deviation-coefficient a for these vapors in the equation of state for gases at moderate pressures, given in Art. 15. The approximate Clapeyron equation can be integrated, usually without much error, between any two temperatures not greatly different from each other under the assumption that the heat of vaporization is constant between those temperatures, as illustrated in Prob. 9. Prob. 8. Derivation of the Approximate Clapeyron Equation. a. From the exact Clapeyron equation derive the approximate form given in the text with the aid of the simplifying assumptions there mentioned. b. From the results of Prob. 7 find the percentage error in dp/dT that would result from neglecting the volume of the liquid in the case of water at 100. c. Show that, aside from the correction for the volume of the liquid, the percentage error in dp/dT as calculated by the approxi- mate equation is equal to the percentage difference between RT and the actual value of pv for one mol of the saturated vapor. Prob. g. Integration of the Approximate Clapeyron Equation. a. Integrate the approximate Clapeyron equation so as to obtain a rela- tion between the vapor-pressures pi and p 2 at two different temperatures T\ and T 2 , assuming that the heat of vaporization does not vary between VAPOR-PRESSURE AND BOILING-POINT 51 the two temperatures, b. Calculate by the equation so obtained the boiling-point of water at 92.0 mm., and its vapor-pressure at 75. c. The actual boiling-point at 92.0 mm. is 49.88, and the actual vapor- pressure at 75 is 289.3 mm - State what inexact assumptions are involved in the equation which would account for the divergence. 34. Vaporization in Relation to the Kinetic Theory. When a liquid is placed in communication with an empty space, its molecules, which according to the kinetic theory are continually in motion, will escape from its surface into the space above it till the vapor attains such a pressure that the number of molecules entering the liquid surface from the vapor side becomes equal to the number of molecules leaving the surface from the liquid side. When this occurs a stationary condition of equilibrium is evidently established, and the then pre- vailing pressure of the vapor represents the vapor-pressure of the liquid. The kinetic principles underlying vaporization may be more fully developed by considering what determines the number of molecules that pass through the surface in the two directions. In the figure the line AB represents an imaginary plane above which the vapor V may be regarded as homogeneous, and the line CD represents another imaginary plane below which the liquid L may be A regarded as homogeneous. In the intervening region c A BCD, constituting the surface layer of the liquid, which in reality is of molecular dimensions, there is a progressive increase in the number of molecules per unit- volume from the number present in the vapor to the number present in the liquid. Now the molecules in the vapor, assuming it to be a perfect gas, exert no attraction upon one another; but between the molecules of the liquid, as between those of a gas at high pressure (Art. 20), there is a large attraction. Within the main body of the liquid (below CD) there is, to be sure, no resultant attraction on any given molecule, since it is attracted equally in all directions by the uniformly distrib- uted molecules surrounding it; but any molecule in the surface layer will evidently be attracted downwards (into the liquid) in consequence of the greater density of the molecules in the layers below it. All the molecules that strike the plane AB from above (that is, from the vapor side) will therefore have their downward velocities increased 52 THE MOLAL PROPERTIES OF SOLUTIONS by the attraction and will pass into the liquid. Of the molecules that strike the plane CD from below (that is, from the liquid side) only those will traverse the surface layer and pass through the plane AB into the vapor which have an upward velocity great enough to overcome the downward attraction exerted by the molecules within the surface layer. In other words, to escape from the liquid a mole- cule reaching the plane CD must have a velocity component x in the upward direction greater than a certain limiting value x\, which value is equal to the decrease in upward velocity which any molecule would experience, in consequence of the molecular attraction, in passing from CD to AB. The fraction of the molecules that have an upward velocity component greater than this limiting value is given by the Maxwell distribution law (Art. 22); and from this fraction the number of such molecules reaching unit-area of the plane CD in unit time may be calculated, as shown in Prob. 10. The number of mole- cules that must be present in unit-volume of the vapor in order that there may impinge on unit-area of the plane AB in unit time a number of the molecules equal to that which escapes into the vapor is readily derived from the fundamental kinetic pressure hypothesis, as has already been indicated hi Art. 18. It is evidently this number that determines the vapor-pressure of the liquid at any temperature. The great effect of temperature on vapor-pressure is readily seen to be a consequence of these kinetic considerations. For it was shown in Art. 22 that the number of molecules which possess kinetic energies or velocities greater than any definite (relatively large) kinetic energy or velocity increases very rapidly with the temperature; thus this number at the two temperatures is represented in the figure of Art. 22 by the area lying beneath the two graphs, respectively, and to the right of an ordinate drawn at the definite value of the kinetic energy (for example, at the value K = 2.o). The heat absorbed by the vaporization of a liquid at constant temperature is also accounted for by these considerations. For, when any molecule (of mass m) passes from the liquid to the vapor, it evidently loses a quantity of kinetic energy equal to \ m x?\ and cor- respondingly, when one mol of liquid (containing n molecules) vapor- izes, the system loses a quantity of kinetic energy equal to^nm xi 2 , consequently falls in temperature, and must absorb from the sur- roundings an equal quantity of heat (or other energy) if it is to return to its initial temperature. That is, the increase in energy-content VAPOR-PRESSURE AND BOILING-POINT 53 A U attending the vaporization of one mol of liquid at constant temperature, is given by the expression A7 = \nrnx?. A striking result of these kinetic considerations is that they lead (as shown at the end of Prob. 10) to the Clapeyron equation, which was derived, entirely independently, from the laws of thermodynamics. *Prob. 10. Kinetic Expression for the Vapor-Pressure of a Liquid. a. It can be shown with the aid of Maxwell's distribution law that, of all the molecules n contained in one mol, the fraction dn/n which have velocity-components along any one of the three rectangular axes with values between x and x + dx is given by the expression dn 0.69 _& , y-'Tr* *-* in which u denotes the square-root of the mean velocity-square. For- mulate an expression for the number (dn/v^ of such molecules in unit-volume of the liquid, representing its molal volume by V L . b. Derive the following expression for the number dn L of such molecules that strike unit-area of the plane CD in unit-time, noting that this number is equal to the number of them present in unit-volume multiplied by their upward velocity-component x and by a factor which represents the increase in the number of impacts on the plane resulting (as described in Art. 20) from the effect of the volume of the molecules, this factor being only a function of the molal volume of the liquid: n x ~ 9 dn L = 0.69 77^ - e * dx. c. By integrating this expression between the limits Xi and oo , find an expression for the total number of molecules n L striking the plane CD with a velocity-component greater than the critical one Xi (which barely permits their passage through the surface layer), d. By com- bining this result with the expression n G =o.2$ un/v G (given in Art. 18) for the number of molecules n G that strike in unit-time a unit-area of a plane bounding a perfect gas having one mol (and therefore n molecules) in the volume V G , derive the following expression: e. Derive for the vapor-pressure p the following expression, by re- placing V G by the expression for it given by the perfect-gas equation: RT - /.By multiplying both numerator and denominator of the exponent of e by nm, and noting that, as shown in the above text, At/ = \nrnx,?, and that, by Art. 19, \nmu z =RT, derive the expression: 54 THE MOLAL PROPERTIES OF SOLUTIONS g. Derive the approximate Clapeyron equation by differentiating the preceding equation with respect to T (regarding V L as constant), dividing the resulting differential equation by the original one, and transforming; noting that the heat A# absorbed by the vaporization of one mol of liquid at constant pressure and temperature is equal to the increase ACT" in the energy-content of the system plus the work RT produced in the surroundings, that is, that AH=AU+RT. SOLUTIONS IN GENERAL 55 II. SOLUTIONS IN GENERAL 35. The Nature and Composition of Solutions. A solution is a physically homogeneous mixture of two or more chemical substances; that is, one which has no larger aggregates than the molecules them- selves. Solutions thus denned may be gaseous, liquid, or solid; but only liquid solutions will be here considered. When one substance is present in large proportion it is called the solvent, and any substance present in small proportion is called a solute. In considering the equilibrium of solutions with the vapor or with the solid solvent, the term phase is conveniently employed. The phases of a system are its physically homogeneous parts, separated from one another by physical boundaries. Thus any gaseous mixture or any solution or any solid substance forms a single phase. A system may consist of any number of such phases. Thus a solution in contact with its vapor, or with the solid solvent, or with the solid solute, is an example of a two-phase system. A solution in contact both with the vapor and the solid solvent is a three-phase system. The composition of solutions is often expressed in terms of the mol- fractions of the substances defined as in Art. 12. Thus, representing, as will be done throughout this chapter, quantities referring to the solvent by letters with the subscript zero, and those referring to the solute or solution by letters without subscripts, the mol-fr actions x and XQ of the solute and solvent in a solution of two substances are defined by the equations x = N/(N +N) and x = N /(N + A 7 ), respectively. The proportion of solute may also be expressed in terms of the mol-ratio N/ No. Composition is also expressed in terms of the number of formula-weights, of mols, or of equivalents, of solute in 1000 grams of solvent. Composition so expressed will be desig- nated the formality f, the molality c, or the weight-normality c, respec- tively, of the solute in the solution. The term formality will be re- ferred to the solute as a whole; but the terms molality and weight- normality will be used with reference to some definite chemical sub- stance. Thus, a solution containing o.i formula- weight of H 2 SO 3 in 1000 g. of water is o.i formal (o.i f.) in H 2 SO 3 (considered without reference to the chemical substances which it forms in solution) ; but, owing to partial ionization, this solution is 0.066 molal (0.066 m.) or 0.132 weight-normal (0.132 wn.) in H 2 SO 3 , and 0.034 molal or 0.034 weight-normal in H + or in HSO 3 ~. Composition will in this book be expressed in the terms of mol-fraction, mol-ratio, formality, molality, 56 THE MOLAL PROPERTIES OF SOLUTIONS or weight-normality, unless otherwise stated. Composition expressed in these ways has the advantage of being independent of temperature. It is sometimes convenient, however, to employ true concentrations, which term strictly denotes the quantity of substance per unit-volume. The number of mols of solute per liter of solvent ( N/VQ) is called the mold concentration; and the solution is said to be x volume-molal in the solute. Thus a solution containing o.i mol of NH 3 in one liter of chloroform at 25 is o.i volume-molal (o.i vm.) in NH 3 at 25; while it is 0.0675 (weight) molal (0.0675 m -) m NH 3 , since 1000 g. of chloro- form have a volume of 675 ccm. at 25. In the case of aqueous solu- tions the molality and molal concentration evidently differ from each other in the same proportion that the density of water differs from unity; for example, by 0.3 percent at 25 and 4.3 percent at 100. In connection with certain properties, such as the electrical conductance of solutions, which are directly related to the volume of the solution (not of the solvent), there will be employed the form of concentration familiar in volumetric analysis, called normal concentration, which is denned to be the number of equivalents of solute per liter of solution (N/v). Thus an aqueous solution containing i equivalent of sulfuric acid in one liter of solution at 25 is i normal (i n.) hi H 2 SC>4 at 25; whereas this solution may be calculated with the aid of density data to be 1.020 weight-normal in H 2 SO 4 at any temperature. With reference to the proportions in which the substances are present, two groups of solutions may be distinguished: dilute solutions, those in which the mol-fraction of the solute is small (not greater than c.oi or 0.02); and concentrated solutions, those in which each sub- stance is present in considerable proportion. There is, of course, no sharp line of demarcation between these two groups of solutions. Some types of concentrated solutions and all dilute solutions con- form approximately more closely as the mol-fraction of the solute approaches zero to certain laws, which, in analogy with the laws of perfect gases, may be called the laws of perfect solutions. The funda- mental laws relating to the physical properties of perfect solutions are the vapor-pressure laws of Raoult and Henry, the corresponding laws of distribution between liquid phases, and tb| ( laws of the osmotic pressure of solutions. To consideration of thei& laws this chapter is mainly devoted. VAPOR-PRESSURE AND BOILING-POINT 57 III. VAPOR-PRESSURE AND BOILING-POINT OF PERFECT SOLUTIONS WITH ONE VOLATILE COMPONENT 36. Raoult's Law of Vapor-Pressure Lowering. Raoult's law, referred to above as one of the fundamental laws of perfect solutions, states that the addition o a A solvent causes at an^lemperature a fractional lowering of the vapor-pressure of the Solvent equal to the mol-fraction (x) of the solute. That is: __ _ p Q ~~N Q +N~ where p Q is the vapor-pressure of the pure solvent and p is its vapor- pressure in a solution consisting of N mols of solute and No mols of solvent. In this expression N Q is equal to the weight m Q of the solvent divided by its molecular weight Mo in the vapor, and N is equal to the weight m of the solute divided by its molecular weight M in the solution. The value of M Q is ordinarily that corresponding to the- molecular formula of the solvent; for, as stated in Art. 14, the molecular formula is commonly so written as to represent the molecular weight of the substance in the state of a perfect gas. In the case of very dilute solutions the mol-fraction N/(No+N) is evidently equal to the mol-ratio N/N Q ; ami even in the case of moderately dilute solutions the mol-fraction may be replaced by the mol-ratio without causing appreciable error. Although Raoult's law is exact only in the case of perfect solutions, it holds true approximately for all dilute solutions (usually within one or two percent up to a mol-fraction of 0.02), and for some con- centrated solutions, as will be described in Art. 41. The application of Raoult's law to solutions of moderate concentra- tion affords an important method of determining the approximate molecular weight and the molecular composition of dissolved sub- stances, as illustrated by the following problems. Determination of Molecular Weights and Molecular Composition. Prob. ii. At 30 the vapor-pressure of ethyl alcohol (C 2 H 5 OH) is 78.0 mm., and that of an alcohol solution containing 5% of a non- volatile substance is 75.0 mm. What is the molecular weight of the substance? Prob. 12. The experiment described in Prob. 2, ^Lrt. 31, was repeated, using in place of pure carbon bisulfide atf 8.00% solution of sulfur in carbon bisulfide. 2.902 g. of carbon bisulfide were found to have vaporized. Calculate the molecular weigtit oi^the sulfur, and find its molecular formula. 58 THE MOLAL PROPERTIES OF SOLUTIONS Raoult's law may also be stated in the following simple form, which indicates more clearly its real significance: the vapor-pressure (p) of the solvent in a perfect solution is proportional to its mol-fraction (#o) ; that is, representing by p Q the vapor-pressure of the pure solvent, Prob. 13. Mathematical Equivalence of the Two Forms of Raoult's Law. a. Show that the two statements of Raoult's law are mathe- matically equivalent, b. Show that the second statement of the law requires that the proportionality-factor be the vapor-pressure of the pure solvent, as is assumed in the mathematical expression of it. Raoult's law relates fundamentally to the distribution between the liquid phase and vapor phase of the chemical substance whose partial pressure in the vapor is under consideration. In other words, from the molecular standpoint, it relates to the distribution of the kind of molecules which give rise to this partial pressure. It shows that the number of these molecules which are present in unit-volume of the vapor when equilibrium has been reached is proportional to the ratio in the liquid of the number of this kind of molecule to the total number of molecules of all kinds. Moreover, since this molecule-ratio is unity in the case of a solvent which consists solely of the kind of molecule under consideration, the proportionality-factor in this case is obvi- ously the vapor-pressure of the pure solvent. It can be shown with the aid of the mass-action law that this is true also in the case that the solvent consists mainly of associated molecules (such as H^) and contains only a small proportion of the simple molecules (such as H 2 O) of which the vapor consists. When the vapor contains two or more kinds of molecules, owing to partial association of the simple solvent molecules or to the fact that the solute has an appreciable vapor-pressure, it is evident from these considerations that p and p Q in the Raoult equation denote, not the total vapor-pressures of the solution and solvent respectively, but the partial vapor-pressures due to the simple molecules. *The above-given forms of the Raoult equation also presuppose that the vapor conforms to the perfect-gas law. Fundamentally ? Raoult's law expresses the tendency of the solvent to escape from the solution, and it should therefore be independent of the equation of state of the vapor. It can be shown by thermodynamic considera- tions that the simple Raoult equation can be corrected for the devia- VAPOR-PRESSURE AND BOILING-POINT 59 tion of the vapor from the perfect-gas law when this is expressed (as in Art. 15) by the equation p v = NRT (i+ap) by modifying it by the addition of another factor so that it assumes the following form, which will be called the corrected Raoult equation: 37. Relation of Boiling-Point Raising to Vapor-Pressure Lowering and Molal Composition. A relation between the increase in the boiling-point and the decrease in the vapor-pressure of a solvent produced by a non-volatile solute can be derived from the effect of temperature on the vapor-pressure of the solvent in the way shown in the following problems. Calculation of Boiling-Point Raising from Vapor-Pressure Lowering. Prob. 14. a. What is the vapor-pressure in mm. of a solution at 100 containing 5 g. of glucose (C 6 Hi 2 O 6 ) in 100 g. of water? b. What is its boiling-point? Its vapor-pressure at 100, like that of water, increases 3.58% per degree. Prob. 15. The vapor-pressure of ethyl alcohol is 721.5 mm. at 77, 751.0 at 78, 781.5 at 79, and 813.0 at 80. a. Plot on a large scale these vapor-pressures as ordinates and the temperatures as abscissas. Calculate the vapor-pressures at 78, 79, and 80 of a solution consist- ing of 2 mols of a non-volatile solute and 98 mols of alcohol, and of one consisting of 4 mols of solute and 96 mols of alcohol; and plot these values on the diagram, b. With the aid of the diagram find the boiling-points at i atm. of pure alcohol and of the two solutions, c. Show from the geometrical relations of the diagram that the raising of the boiling- point is proportional to the lowering of the vapor-pressure at the boil- ing-point of the solvent, in the case of dilute solutions (for which the graphs may be considered to be parallel straight lines), d. Show from the geometrical relations that the ratio of the lowering of the vapor- pressure to the raising of the boiling-point, caused by adding the solute, is equal to the ratio of the increase in the vapor-pressure of the solvent caused by increasing the temperature to the increase in temperature, in the case of dilute solutions (for which the graphs may be considered straight lines). Representing by T T Q the raising of the boiling-point, and by pQ-p the lowering of the vapor-pressure, produced by increasing the mol-fraction of the solute from o to x, and representing by A/> /A7Ythe rate of change of the vapor-pressure of the solvent with the tempera- ture, the relations derived in the preceding problem for dilute solu- tions may be expressed by the equation: A7Y 60 THE MOLAL PROPERTIES OF SOLUTIONS A T* ^7^7^ The quantity . A , evidently corresponds to the quantity -jn * bpo/po dp/p in the approximate Clapeyron equation, and it is therefore equal to 7?T" ^ ~ =r- , where A# denotes the molal heat of vaporization of the solvent A# at its boiling-point TV The relation between boiling-point raising and fractional vapor-pressure lowering is therefore also given approxi- mately by the expression: By combining the expressions just derived with Raoult's equation for vapor-pressure lowering there may evidently be obtained the following relations between the raising of the boiling-point (TT ) and the mol-fraction x of the solute: / dT \ (RTf\ () T ~ T < \dpjpj*- \-JnJ * *The preceding equations become exact only when infinitesimal changes hi temperature, vapor-pressure, and mol-fraction are con- sidered; for only in that case are the graphs in the diagram of Prob. 15 from which these equations were derived strictly parallel straight lines. It will be seen from the diagram that the following differential equation, corresponding to equation (i) above, holds true for any solution with a non- volatile solute having any mol-fraction x: dT\ (dT\ (dp\ . /,~ "V^/.Vte/r *From this equation there can be derived, as shown in Prob. 16, by substituting for the last two partial derivatives the expressions for them given by the Clapeyron equation (Art. 33) and the Raoult equa- tion (Art. 36), the following equation: RT* dx M dT= ~^s (^r (5) In this equation dT represents the raising in the boiling-point T caused by increasing the mol-fraction of the solute from x to x+dx, and A# represents the heat absorbed in vaporizing one mol of solvent at the temperature T out of an infinite quantity of the solution (since it can be shown to have this significance when the Clapeyron equation is applied to vaporization out of a solution). VAPOR-PRESSURE AND BOILING-POINT 61 *Prob. 16. Boiling-Point of Solutions Conforming to RaouWs Law at Any Concentration. a. State in words what each of the partial derivatives in equation (4) signifies, b. Find from the exact Clapeyron equation (neglecting the volume of the liquid) and from the corrected Raoult equation (given at the end of Art. 36) expressions for the last two of these partial derivatives, respectively, c. By substituting these expressions in the partial derivative equation, and simplifying it with the aid of the equation of state pv = RT (i-f ap), derive the expression for the boiling-point raising formulated as equation (5). *Equation (5) holds true even when the saturated vapor of the solvent does not conform to the perfect-gas law, as shown by its derivation. It can be integrated (as in Prob. 17), usually with negligible error, under the assumption that A# does not vary with the temperature or with the mol-fraction of the solute, and that therefore it has the same value as the molal heat of vaporization A& of the pure solvent at its boiling-point. *Prob. 17. Integration of the Boiling-Point Equation. a. Inte- grate equation (5) given in the text, assuming that the heat of vaporiza- tion does not vary with the temperature or the mol-fraction of the solute, so as to obtain a relation between the boiling-point of the solu- tion, the boiling-point of the pure solvent, and the mol-fraction of the solute, b. Calculate the boiling-point of a solution consisting of 10 mols of a non-volatile solute and 90 mols of benzene. The heat of vaporization of one gram of benzene at its boiling-point 80.3 is 93.0 cal. *For small values of x and for the correspondingly small changes hi the boiling-point T equation (5) may be integrated under the assumption that i x is equal to unity and that T is constant and equal to To, and there then results equation (3), previously derived for dilute solutions, namely: T T RT \ 1 1 o = x. jrrt 7PT 1 2 The quantity - or =? occurring in equation (3) is evidently a constant characteristic of the solvent. Representing this quantity, which may be called the boiling-point constant, by a single letter B , and noting that the mol-fraction x approaches the mol-ratio N/ N as its value approaches zero, the law of boiling-point raising for perfect solutions may be expressed by the equation: T-T^B^. (6) No 62 THE MOLAL PROPERTIES OF SOLUTIONS It is evident from this equation that the boiling-point constant BQ is the ratio of the boiling-point raising to the number of mols of solute which are associated with one mol of solvent; or, briefly, it is the boiling-point raising per mol of solute in one mol of solvent. It can be experimentally determined by measuring the rise in boiling- point produced by dissolving a definite weight of a solute of known molecular weight in a definite quantity of solvent. It can also, of course, be calculated from the change of vapor-pressure of the solvent with the temperature or from the molal heat of vaporization of the solvent by the above-given expressions. There are, accordingly, three kinds of data with the aid of which the boiling-point constant can be evaluated, as illustrated in Prob. 18. In chemical literature is commonly recorded, not this boiling-point constant, but another constant, called the mold boiling-point raising, which is the boiling-point raising per mol of solute in 1000 grams of solvent. The value of this constant can obviously be readily calcu- lated from that of the boiling-point constant (as in Prob. 19). Prob. 18. Methods of Determining the Boiling-Point Constant. Calculate the boiling-point constant for ethyl alcohol from the following data. a. The heat of vaporization of one gram is 206 cal. at the boiling-point 78.3. b. Its vapor-pressure has the values given in Prob. 15. c. The boiling-point of a solution of i g. of naphthalene (CioH 8 ) in 50 g. of alcohol is 0.185 higher than that of pure alcohol. Prob. 19. Relation between the Boiling-Point Constant and the Molal Boiling-Point Raising. a. Calculate the molal boiling-point raising for water from its boiling-point constant, b. Formulate the algebraic relation between the two constants. The following table contains the values of the constants for a few important solvents. BOILING-POINT CONSTANTS Water Ethyl ether Ethyl alcohol Benzene mo CiHioO amon c&Ht Boiling-point constant . . 28.6 28.5 25.8 34.0 Molal boiling-point raising . 0.515 2.11 1.19 2.65 38. Determination of Molecular Weights. From the value of either of the boiling-point constants for a given solvent there may be calculated by direct proportion the number of mols 4 of solute corre- sponding to any observed raising of the boiling-point. The law of boiling-point raising, like Raoult's law from which it has been derived, therefore makes it possible to determine the molecular weights of substances in solution. VAPOR-PRESSURE AND BOILING-POINT 63 Determination of Molecular Weight and Molecular Composition. Prob. 20. a. When 10.6 g. of a substance are dissolved in 740 g. of ether (C 4 Hi O), its boiling-point is raised 0.284. What is the molec- ular weight of the substance? b. The substance is a hydrocarbon containing 90.50% of carbon. What is its molecular formula? Prob. 21. A solution of 3.04 g. of benzoic acid in 100 g. of ethyl alcohol boils 0.288 higher than pure alcohol. A solution of 6.34 g. of benzoic acid in 100 g. of benzene boils 0.696 higher than pure benzene. Calculate the molecular weight of benzoic acid in each of these solvents, and state what the results show in regard to its molecular formula in each solvent. Its composition by weight is expressed by C6H 5 CO 2 H. The molecular weights of substances are ordinarily found to be the same in the dissolved state as in the gaseous state; but hydroxyl com- pounds (such as the alcohols and organic acids) form in non-oxygen- ated solvents (such as benzene or chloroform) double or even more highly associated molecules. This indicates that the molecules of hy- droxyl compounds are associated also in the state of pure liquids. Oxygenated solvents (such as water, alcohols, acetic acid, etker, and acetone) have the power of breaking down these associated molecules into the simple ones; and in such solvents nearly all organic substances have the same molecular weight as in the gaseous state. 39. Partial Vapor-Pressure of Volatile Solutes. Henry's Law. In addition to Raoult's law, which relates to the vapor-pressure of the solvent in a perfect solution, there is another fundamental law which relates to the vapor-pressure of the solute in such a solution. This law, known as Henry's Law, may be stated as follows. The partial vapor-pressure of any chemical substance present in small proportion in a solution is proportional to its mol-fraction. That is, p =kx, where k is a proportionality-constant dependent on the nature of the solute and of the solvent and on the temperature. Prob. 22. Application of Henry's Law and Raoult's Law. The total vapor-pressure of a solution containing 3.00% by weight of ethyl alcohol (C 2 H 5 OH) in water is 760 mm. at 97.11, and the vapor-pressure of pure water at this temperature is 685 mm. Calculate with the help of Raoult's law and of Henry's law the vapor-pressures at 97.11 of ethyl alcohol and water in a solution containing 2.00 mol-percent tf ethyl alcohol. Ans. 139 mm. for C 2 H 8 OH. Henry's law may also be expressed in the following form, which makes it more obvious that it is a law of equilibrium which determines the distribution of the solute between the solution and a gaseous 64 THE MOLAL PROPERTIES OF SOLUTIONS phase, and which incidentally expresses the composition of the solution in terms either of molality or of molal concentration (denned as in Art. 35), instead of as mol-fraction. Any chemical substance present as a perfect gas in a vapor phase and as a solute in a perfect solution in equilibrium with it has at any definite temperature a molality or molal concentration c in the solution which is proportional to its (partial) pressure p in the gaseous phase. That is, c/p=K, where K is an equilibrium-constant which is determined by the nature of the chemical substance and of the solvent and by the temperature. The equilibrium concentration c is the solubility of the substance when its partial pressure in the gas phase is p, and the equilibrium- constant K may be called the solubility-constant of the gaseous sub- stance in the solvent. Henry's law is therefore a law of the solubility of gases. This second form of Henry's law is for perfect solutions equivalent to the first form. For, on the one hand, the mol-fraction N/ NQ+ N), as it approaches zero, becomes equal to the mol-ratio N/ No, and this is evidently proportional to the molality (ioooN/ m ) or to the molal concentration ( N/v Q ) ; and, on the other hand, the vapor-pressure of a substance in a solution is equal to its partial pressure iri a gaseous phase that is in equilibrium with the solution. / Henry's law in either of its two forms is conformed to more closely as the pressure of the gas and the concentration of the solute approach zero. Like the other laws of perfect solutions, it usually holds true within 2 to 3 percent, even when the pressure is one atmosphere and the concentration i molal. It is to be noted that Henry's law expresses conditions of equi- librium, and that these conditions are often attained between a gaseous and liquid phase only by long-continued intimate contact. From a molecular standpoint, Henry's law, like Raoult's law, relates to the distribution of some definite kind of molecule between the gas phase and the liquid phase. Hence in applications of it the same chemical substance in the two phases must be considered. Thus, when the chemical substance SO* dissolves in water it is largely converted into H 2 SO 3 and its ions H + and HSO 3 ~; and Henry's law therefore requires, not that the total concentration of solute in the solution, but that the concentration of the SO 2 itself, be proportional to the partial J pressure of the S0 2 in the vapor. When, however, the only change in / VAPOR-PRESSURE AND BOILING-POINT 65 the substance is that it partially combines with the solvent forming a solvate (a hydrate in the case of water), then the total concentration may be employed; for the fraction solvated is in dilute solution in- dependent of the concentration of the substance, as may be shown by the mass-action law. Thus, though the substance CO 2 on dissolv- ing in water is partly converted into the hydrate H 2 CO 3 (which is sub- stantially unionized, except at very small concentrations), yet the solubility of carbon dioxide gas, as found by determining the total quantity dissolved, changes with the pressure in accordance with Henry's law. These considerations also show that in applying Henry's law the partial pressure of the chemical substance in the gas phase, not the total pressure of the gas, must be considered. Thus the quantity of carbon dioxide dissolved by water in contact with air is not determined by the pressure of the air, but by the partial pressure of CO-2 in the air. Prob. 23. Determination of the Vapor-Pressure of Solutes. A mixture of air and ammonia containing i mol-percent of NH 3 is passed at 25 and i atm. through water. The saturated solution is found by titration to be 0.553 formal in NH 4 OH. Calculate the partial vapor- pressure of NH 3 in a i formal solution at 25. The vapor-pressure of water at 25 is 23.8 mm. Solubility of Gases in Water. Prob. 24. In a gas buret over mercury 60 ccm. of dry carbon dioxide at 25 and i atm. are placed, 40 ccm. of water are introduced, and the gas and water are shaken together at 25 till equilibrium is reached, keeping the pressure on the gas i atm. The volume of the (moist) gas is then found to be 28.9 ccm. Calculate the molal solubility of carbon dioxide in water at 25 when its partial pressure is i atm., neglecting effects that influence the result less than 0.5%. Ans. 0.0338. Prob. 25. At 20 100 g. of water dissolve 3.4 ccm. of oxygen, 1.7 ccm. of nitrogen, and 3.8 ccm. of argon when the pressure of each gas is i atm. a. Calculate the corresponding molal solubility of each gas. b. Calcu- late the mol-fraction of each constituent in the gas-mixture obtained by shaking water at 20 with air (free from CO 2 ) , expelling the dissolved gas by boiling and drying it. Tabulate the molal composition of this gas with that of air (given in Art. 12). Ans. b, 0.34 for O 2 . Prob. 26. Determination of the State of Substances in Solution. The partial vapor-pressure of NH 3 in an aqueous solution 0.3 formal in NH 3 and o.i formal in AgNO 3 is at 25 equal to that in a o.i formal solution of NH 3 in water. State and explain the conclusion that can be drawn as to the formula of the complex salt formed, considering all the silver nitrate to be combined with ammonia, and assuming that the solubility-constant of the NH 3 is not affected by the silver salt. 66 THE MOLAL PROPERTIES OF SOLUTIONS Aside from any chemical action, the addition to the water of a salt commonly decreases the solubility of a volatile substance at a given pressure in the gas phase; or conversely, it increases its vapor-pressure at a given concentration in the solution. This phenomenon, which is known as the salting-out effect, is subject to the following principles: (1) The decrease of the solubility-constant is approximately pro- portional to the concentration of the added salt, up to concentrations not much exceeding i normal. (2) The fractional decrease per equivalent of salt per 1000 grams of water is roughly the same for a definite salt, whatever be the nature of the solute. (3) This fractional decrease varies greatly with the nature of the salt; thus the decrease caused by o.i equivalent of salt per 1000 grams of water varies from about zero in the case of barium nitrate to about 4 percent in the case of potassium and sodium sulfates. Prob. 27. The Salting- out Effect. The solubility (that is, the concentration of the saturated solution) of carbon dioxide at 25 and i atm. is 0.0338 molal in pure water and 0.0331 in a i normal NaCl solution. The vapor-pressure of ammonia from a 0.5 molal solution of it in water at 25 is 6.65 mm. Predict from the principles of the salting- out effect the ammonia vapor-pressure for a solution 0.5 molal in NH 3 and 0.5 normal in NaCl. VAPOR-PRESSURE AND BOILING-POINT 67 IV. VAPOR-PRESSURE AND BOILING-POINT OF CONCENTRATED SOLUTIONS WITH TWO VOLATILE COMPONENTS 40. The Vapor-Pressure of Concentrated Solutions. Concentrated solutions may be divided for purposes of consideration into two groups as follows. One group consists of those solutions whose formation out of their pure components (when these are liquid) is not attended by any considerable change of temperature or volume, and whose 'properties in general are approximately the sum or average of those of the pure components. The characteristic of such solutions is that neither component exerts a specific influence on the properties of the other component. Such solutions conform approximately more closely as the condition characterizing them is more nearly fulfilled to the laws of perfect solutions. Concentrated solutions which are considered to conform to these laws may be called concentrated perfect solutions. The other group of concentrated solutions consists of those whose components exert a marked influence upon one another. For these solutions no general laws are known. The method commonly employed for experimentally determining at any definite temperature the partial vapor-pressures of solutions consisting of two or more volatile components is to distil off a small fraction from a large volume of the solution, adjusting the pressure on the liquid so that it boils at this temperature. The composition of the distillate is then determined by chemical analysis or by the measurement of some physical property, such as density; and from this composition and the pressure under which the distillation took place the partial vapor-pressures are calculated by Dalton's law. Prob. 28. Experimental Determination of Partial Vapor-Pressures. A solution of two substances AandB containing N\ mols of A and NB mols of B boils at the temperature T when a pressure p is exerted upon it. The first portion of distillate consists of N A f mols of A and N* mols of B. Derive an algebraic expression for the partial vapor- pressures p A and pv of the two substances in the solution, explaining the principles involved. 41. Vapor-Pressure and Boiling-Point of Concentrated Perfect Solutions in Relation to their Molal Composition. In concentrated perfect solutions the vapor-pressure of each component conforms approximately to Raoult's law. In other words, the partial vapor- pressure of each component is approximately equal to the product of its mol-fraction in the solution by its vapor-pressure in the pure state, whatever be the proportion in which the components are present; that is, p A = p QA X A , P* 68 THE MOLAL PROPERTIES OF SOLUTIONS Prob. 2Q. Vapor-Pressure-Composition Diagram for Perfect Solu- tions. At 50 the partial vapor-pressures of benzene and of ethylene chloride in solutions of these two substances have been found experi- mentally to have the following values: Mol-fraction Vapor-pressure Vapor-pressure ofCtHe o/CtHs of CzH t Ck i.ooo 268 mm. o mm" 0.707 190 68 0.478 128 123 0.246 66 178 o.ooo o 236 a. Plot on a large scale these partial vapor-pressures, and the corre- sponding total vapor-pressures, as ordinates against the mol-fractions as abscissas. Show that the three graphs are in almost complete accord with Raoult's law. b. Calculate the mol-fraction of benzene in the vapor which at 50 is in equilibrium with each of the three solutions for which the data are given in the above table, c. On a new larger- scale vapor-pressure-composition diagram (with ordinates covering only the interval of 230-270 mm.) draw a line showing the variation of the total vapor-pressure with the mol-fraction of the liquid mixture. Plot also on this diagram the compositions of the vapor calculated in b against the total vapor-pressures, considering that the abscissas now represent the mol-fraction of benzene in the vapor; Prob. 30. Distillation at Constant Temperature of Perfect Solutions. At 50 a small fraction is distilled off from a large volume of a solu- tion containing equimolal quantities of benzene and ethylene chloride, and this distillate is redistilled at 50. Derive from the diagram of Prob. 29 the mol-fraction of benzene in the first part of the second distillate. Ans. 0.56. *The boiling-point of a solution that contains two volatile components whose partial vapor-pressures both conform to Raoult's law can best be derived graphically from the vapor-pressures of the pure substances, as illustrated by the following problem. *Prob. 31. Boiling-Point-Composition Diagram for Perfect Solutions. The vapor pressure at .... 80, 83, 86, 89, 92 of a pure liquid A is . . . . 560, 610, 665, 725, 790 mm. and of a pure liquid B is . . 400, 435, 475, 520, 570 mm. a. With the aid of these data and Raoult's law, draw on a large-scale vapor-pressure-composition diagram for each temperature two lines one representing the total vapor-pressure of any solution of A and B ; and the other representing the partial vapor-pressure of A above any solution, b. Determine from the plot the composition of the liquid which at 570 mm. boils at each of these temperatures; also the composi- tion of the vapor which is in equilibrium with each of these solutions at J VAPOR-PRESSURE AND BOILING-POINT 69 its boiling-point. Determine also the boiling-point of the pure liquid A at 570 mm., and tabulate all of these results, c. On another large- scale diagram plot against these liquid-compositions the boiling-points as ordinates. Plot also the vapor-compositions against the correspond- ing boiling-points. Distillation at Constant Pressure of Perfect Solutions. *Prob. 32. A solution of 100 mols of each of the liquids of Prob. 31 is distilled at 570 mm. until its boiling-point rises 0.5. a. Find from the diagram of Prob. 31^ the molal compositions of the first and last por- tions of the distillate, b. Regarding the composition of the whole distillate as the mean of that of its first and last portions (which is approximately true when only a small fraction of the liquid distils over), calculate the number of mols of A and of B in the distillate and in the residue, c. The distillation of the residue is continued till its boiling- point rises 0.5 more. Calculate as in b the number of mols of A and B in this second distillate and in the residue, d. Tabulate the number of mols of A and B in the original liquid, the first distillate, the second distillate, and the final residue; the mol-fraction of A in each of these liquids; and the boiling-point of each of them. Ans. b, 52 mols of A, and 63 mols of B in the residue. *Prob. 33. a. The first distillate obtained in Prob. 320 is redistilled until the residue attains the composition of the second distillate ob- tained in Prob. 32^. Find the mol-fraction of A in the new distillate and its boiling-point, b.. The residue is now mixed with the second distil- late obtained in Prob. 32^, and the distillation is continued till the residue has the composition of the residue obtained in Prob. 32^. Find the mol-fraction of A in the distillate thus obtained and its boiling- point, c. Tabulate the composition and the boiling-point of the original equimolal solution and of the three fractions into which it has now been resolved. Ans. b, mol-fraction, 0.55; boiling-point, 85.5. The foregoing considerations show that any perfect solution sub- mitted to distillation resolves itself into a distillate containing a larger proportion, and into a residue containing a smaller proportion, of the more volatile component. It is evident that, in consequence of this behavior, the two components can be completely separated from each other by repeated fractional distillation, carried out as illustrated by Probs. 32 and 33. 42. Vapor-Pressure of Concentrated Solutions in General in Rela- tion to their Composition. There are comparatively few actual solutions which fulfil strictly the criterion stated in Art. 40 of being formed out of their components without any change of temperature or volume; and correspondingly, comparatively few concentrated 70 THE MOLAL PROPERTIES OF SOLUTIONS solutions conform completely to Raoult's law of perfect solutions. The law is therefore to be regarded as a limiting law, from which actual solutions deviate to an extent which is as a rule roughly indicated by the magnitude of the changes of temperature and volume attending the mixing of the components. The data given below illustrate the magnitude of the deviations from the law for a variety of solutions. The first two columns of figures show the change of temperature and the percentage change of volume which result when equimolal quantities of the two sub- stances at the same temperature are mixed. The last three columns show the percentage difference between the observed values of the partial (or total) vapor-pressures of the equimolal solution and those calculated by Raoult's law; namely, the values of the vapor-pressure ratio I00(obs. calc.)//>calc. DEVIATIONS FROM RAOULT'S LAW Mixture Components Change of Change of Percentage Deviation of Pressure No. A B Temperature Volume A B A+B 1 C 6 H 6 C1 C 6 H 6 Br 0.0 0.0% - o.i - o.i 2 C 6 H 6 C 2 H 4 C1 2 0.3 +0.34 +0.1 0.2 3 CHC1 3 (CH 3 ) 2 CO +12.4 -0.23 -23. -18. 4 CS 2 CH 2 (OCH 8 ) 2 - 6.5 +1.22 +10. + 7. 5 CSa (CH 3 ) 2 CO - 9.8 +i.2i +40. +58. 6 C 6 H 6 C 2 H 6 OH - 4.2 o.o +50 The character of the deviations from Raoult's law can be best appreciated by considering their causes from molecular and kinetic view-points. It has already been seen (in Art. 34) that the vapor- pressure of a solvent is determined primarily by the proportion of its molecules which at any temperature acquire an outward velocity sufficient to overcome the inward attraction of the other molecules and thus escape through the surface layer into the gas phase. Con- sider now the different conditions that may result when another sub- stance B is added to the solvent A. (i) In case the B molecules exert upon the A molecules the same attraction as the A molecules exert on one another, the attraction upon the A molecules in the mixture, and hence their escaping velocity, will be the same as in the pure solvent, the proportion of them that acquire this escaping velocity will be the same, and the absolute number of them which escape will be diminished only in the proportion in which the proportion of A molecules in the liquid is reduced. In this case, therefore, the vapor-pressure of A will be directly proper- VAPOR-PRESSURE AND BOILING-POINT 71 tional to its mol-fraction, as Raoult's law requires; and little if any heat-effect or volume change will take place on mixing the components. (2) In case the B molecules exert upon the A molecules a smaller attraction than they exert upon one another, the escaping velocity will be smaller and the proportion of the A molecules that acquire this velocity will be greater in the solution than in the pure solvent; and correspondingly the vapor-pressure of A will be greater than Raoult's law requires. In this case the mixing of the two components will be attended by an increase in volume and a fall in temperature (due to the expansion and attendant separation of the molecules in opposition to their attraction). (3) In case the B molecules exert upon the A molecules a greater attraction than the latter exert on one another, the opposite effects to those just considered will evidently result. (4) In case the B molecules form with the A molecules a chemical compound (such as AB, A2B, or AB 2 ), the fraction of A molecules in the liquid is decreased, not only as always by the presence of the new molecules, but in this case also by the conversion of some of the A molecules into those of the compound. This will evidently result in a smaller vapor-pressure than Raoult's law requires. In this case an evolution of heat and diminution of volume may be expected, since the synthetic formation of chemical compounds is usually attended by these effects. (5) In case the simple molecules A of the solvent are partially associated, to form double or more complex molecules, such as Aa or A 3 , some of the latter will be dissociated, in accordance with the mass-action law, when the solvent is diluted by the addition of the substance B, and the partial pressure of the simple A molecules in the vapor will be correspondingly increased. In this case, a fall in tempera- ture may be expected, since dissociation is commonly attended by absorption of heat. The foregoing considerations are not of a quantitative character; and on the chemical effects described in (4) and (5) may be super- posed the physical effects mentioned in (2) and (3). These molecular considerations, however, often assist in interpreting the vapor-pressure of mixtures. Prob. 34. Molecular Explanations of the Deviations from Raoult's Law. Suggest the most probable molecular explanation of the be- havioj of each of the mixtures for which data are given in the above table, taking into account all the foregoing considerations. 72 THE MOLAL PROPERTIES OF SOLUTIONS Prob. 55. Compound Formation Derived from the Vapor-Pressure Relations. At 35.2 the vapor-pressure of acetone, (CH 3 ) 2 CO, is 345 mm. and that of chloroform, CHC1 3 , is 293 mm. In a 50 mol- percent mixture of these substances the vapor-pressure of the acetone is 141 mm. and that of the chloroform is 113 mm. Assuming that the apparent deviation from Raoult's law is wholly due to the formation of a non-volatile compound (CH 3 ) 2 CO.CHC1 3 , calculate the mol-percent of this compound and the mol-percents of uncombined (CH 3 ) 2 CO and CHC1 3 , a, from the vapor-pressures of the acetone; and b, from those of the chloroform. Ans. (CH 3 ) 2 CO.CHC1 3 : a, 18.2, b, 22.8. In connection with the deviations from Raoult's law it should be borne in mind that, as the mol-fraction of any component approaches unity, so that the solution becomes dilute with respect to the other component, its partial vapor-pressure always approaches that re- quired by Raoult's law, and the partial vapor-pressure of the other component conforms to Henry's law, however great the deviation may be when both components are present in large proportion. Vapor-pressure-composition diagrams for solutions whose com- ponents do not conform to Raoult's law can be constructed only with the aid of experimentally determined data. By reference to such diagrams the behavior of the solutions when submitted to distillation at constant temperature can be predicted. Prob. 36. Construction of Vapor-Pressure-Composition Diagrams. a. Draw on a large-scale diagram vapor-pressure curves representing the partial vapor-pressure at 35.2 of carbon bisulfide and of acetone in solutions of these components throughout the whole range of composi- tion. Assume that Raoult's law and Henry's law hold with this pair of components up to 5 mol-percent; and make use of the following values in millimeters of the vapor-pressures at 35.2: Mol-percent of CSj. o i 20 40 60 80 99 100 Vapor-pressure of CS 2 o 17.8 274 377 425 460 518 Vapor-pressure of (CH 3 ) 2 CO 353 289 255 228 187 20.1 o b. On the same diagram draw a curve representing the total vapor- pressures of the solutions. Draw on the diagram dotted lines showing what the partial and total vapor-pressures would be if the solutions behaved as perfect solutions, c. Calculate the mol-percents of CS 2 in the vapor in equilibrium with the 5, 20, 40, 60, 80, and 95 mol-percent liquid solutions; and on the same diagram plot these vapor-compositions against the total pressures and draw a dotted line through the points. 43. Boiling-Point of Concentrated Solutions in General in Relation to their Composition. The boiling-point-composition diagrams for solutions whose vapor-pressures do not conform to Raoult's law can be based upon direct experimental determinations of the boiling-point VAPOR-PRESSURE AND BOILING-POINT 73 of solutions of known composition and upon analyses of the correspond- ing distillates. All that can be done, in the way of generalization, is to consider the different types of curves to which different pairs of substances conform. The figure shows the three types of curves exhibited by substances miscible in all proportions. In each case the solid curve shows the compositions (expressed as mol-fractions) and corresponding boiling-points of the liquid solutions at one atmos- phere; and the broken curve shows the composition of the vapor that is in equilibrium with these solutions. Thus any point on a broken curve represents the composition of the vapor of the liquid solution whose point lies in the same horizontal line. 76 0.6 0.5 0.4 Mol-Fraction 0.3 0.2 0.1 Curve I is the experimentally determined curve for solutions of carbon tetrachloride (b. pt., 76.7) and carbon bisulfide (b. pt., 46.3). Curve II is that for solutions of acetone (b. pt., 56.2) and chloroform (b. pt., 61.3). Curve III is that for solutions of acetone (b. pt., 56.2) and carbon bisulfide (b. pt., 46.3). 74 THE MOLAL PROPERTIES OF SOLUTIONS Solutions of type I, when subjected to fractional distillation, behave like perfect solutions (which form a special case of this type), and may like them be finally resolved into the pure components. The behavior of solutions of types II and III on fractional distillation is shown in the following problems. Fractional Distillation of Solutions. Prob. 57. a. Determine from curve III in the figure the boiling- point and composition of the first portion of distillate obtained by dis- tillation of solutions of acetone and carbon bisulfide containing 70 mol- percent of acetone, 15 mol-percent of acetone, and 35 mol-percent of acetone, respectively. Tabulate the boiling-point and composition of each solution and distillate, b. State in what respects each distillate and e,ach residue differs from the solution from which it was obtained. c. If each of the solutions were submitted repeatedly to fractional dis- tillation, what would be the composition of the products finally obtained as distillate and as residue? d. If of the 70 mol-percent solution 1000 g. were so fractionated, what weight of each product would be obtained? Ans. d, 508 g. of distillate, 492 g. of residue.. Prob. 38. By reference to curve II in the figure answer the same ques- tions for solutions of acetone and chloroform as are asked in Prob. 37&j and 37^, for solutions of acetone and carbon bisulfide. Just as the solid lines in such temperature-composition diagrams show the boiling-points of any liquid solution, so the broken lines show the condensation-point at one atmosphere of vapor of any composition ; the composition of the liquid which first condenses out of it being given by the corresponding point on the solid line. Such diagrams therefore serve to predict the behavior of vapors when subjected to fractional condensation, as illustrated by the following problem. The processes used in chemical practice for the separation of volatile liquids, such as alcohol and water or benzene and toluene, involve fractional distillations and condensations taking place in accordance with the principles here considered. Prob. 39. Fractional Condensation of Vapors. a. A vapor com- posed of equimolal quantities of carbon tetrachloride and carbon bisulfide is cooled at i atm. till condensation begins. By referring to curve I in the figure find the temperature at which condensation begins, and the composition of the condensate. b. The vapor is gradually cooled, removing the condensate as it forms, till the temperature falls to 60. Find the composition of the condensate which is now separat- ing, and that of the residual vapor, c. Tabulate the composition (50 mol-percent) of the original vapor, the average composition of VAPOR-PRESSURE AND BOILING-POINT 75 the condensate obtained from it in b, and the composition of the residual vapor. Include in the table also the composition of the liquid which upon distillation would furnish the original vapor. 76 THE MOLAL PROPERTIES OF SOLUTIONS V. DISTRIBUTION BETWEEN PARTIALLY MISCIBLE SOLVENTS 44. Determination of Equilibrium-Conditions by the Perpetual- Motion Principle. Before considering other molal properties of solutions it is desirable to present an energy principle which is of great value in predicting the equilibrium conditions of systems consisting of two or more phases. This principle is illustrated by Prob. 40 and discussed in the succeeding paragraph. Prob. 40. Condition of Equilibrium between Two Phases. A volatile substance S is dissolved in each of two non-miscible solvents, the two solutions A and B being shaken to- gether at some definite temperature till equi- librium is reached. They are placed, as in the figure, in contact with the vapor-phase contain- ing the substance S at a pressure equal to its partial vapor-pressure in solution A. Show that if this pressure were greater or less than the partial vapor-pressure of S in the solution B, the substance S would pass continuously through the three phases of the system under a difference of pressure, whereby work could be produced. Perpetual motion (of the kind here considered) signifies in general an ideal process by which an unlimited amount of work might be pro- duced by a system operating in surroundings of constant temperature and drawing from them no work. Thus, in the case considered in Prob. 40, if the vapor-pressures of the substance in the two solutions were different, a current of its vapor would flow continuously from one surface to the other, and work could be obtained from the moving vapor for an unlimited period of time, for example, by placing a wind- mill in the vapor-space. Even though a quantity of heat equivalent to the work produced were taken up from the surroundings, the process would still be a kind of perpetual motion which is impossible, as will be seen later in the discussion of the second law of thermodynamics. The principle that perpetual motion of this kind is impossible is often employed, as in this instance, for determining the conditions of equilibrium between the different phases of a system. It leads in such cases to the general conclusion that, if two phases are each in equilibrium with a third phase, they must be in equilibrium with each other, with respect to every substance that is present. This principle will hereafter be called simply the perpetual-motion principle. 45. Distribution of a Solute between Two Non-Miscible Solvents. There has been experimentally established another important law DISTRIBUTION BETWEEN SOLVENTS 77 of perfect solutions, as follows. At any definite temperature the ratio between the equilibrium concentrations C A and C B of a chemical substance S in two non-miscible solvents (A and B) is constant, whatever be the initial concentrations^ That is c A /c B =K, where K is a constant, called the distribution-ratio, determined by the nature of the substances A, B, and S, and by the temperature. This law may be generalized so as to be applicable to the distribu- tion of a definite chemical substance between any two kinds of phases. Thus in its general form it includes Henry's law, since the pressure of a gas at any definite temperature is proportional to its concentra- tion; that is, since p = (N/v)RT = cRT. It is called the law of distri- bution between phases, or simply the distribution law. This distribution law, like Raoult's law and Henry's law, is a limit- ing law which becomes more exact as the concentrations approach zero, but which holds true approximately up to moderate concentrations such as i molal. The distribution-ratio of a solute between water and another solvent is decreased by the addition of a salt to the water in accordance with the principles of the salting-out effect stated in Art. 39. The distribution law between liquid phases can be derived from Henry's law and the perpetual-motion principle (as in Prob. 41) for a volatile solute. This derivation is general, since every solute has a vapor-pressure, even though it may be too small to measure. In considering the distribution of solutes between liquid phases, the concentration is usually expressed in mols per 1000 ccm. of solvent, instead of in mols per 1000 grams of solvent; and the value of the distribution-ratio varies correspondingly. In the problems of this article such volume concentrations are employed. Prob 41. Derivation of the Distribution Law. Derive the law of the distribution of a volatile solute between two solvents from Henry's law and the conclusion reached in Prob. 40. Prob. 42. Evaluation of the Distribution-Ratio from Vapor-Pressure Data. At 25 the vapor-pressure of ammonia above a o.i volume- molal (vm.) solution of it in chloroform is 33.25 mm., and above a 0.5 vm. solution of it in water is 6.65 mm. a. What is the distribution- ratio of ammonia between water and chloroform? b. What would be its distribution-ratio between a 0.5 n. NaCl solution and chloroform? Prob. 43. Extraction of Solutes from Aqueous Solutions by Organic Solvents. The distribution-ratio of an organic acid between water and 78 THE MOLAL PROPERTIES OF SOLUTIONS ether at 20 is 0.4. A solution of 5 g. of the acid in 100 ccm. of water is shaken successively with three 2o-ccm. portions of ether, a. How much acid is left in the water? b. How much acid would have been left in the water if the solution had been shaken once with a 6o-ccm. portion of ether? Ans. a, 1.48; b, 2.00. Prob. 44. Determination of Complex Salt Formation. An aqueous solution containing 0.25 KC1 and 0.20 HgCl 2 in one liter of water is shaken with an equal volume of benzene at 25. The benzene phase is found by analysis to contain 0.0057 m l of HgCl 2 per liter, but no KC1. The distribution-ratio of HgCl 2 between water and benzene at 25 is 13.3. a. Calculate the total concentration of mercuric chloride in the aqueous phase and the concentration of the part of it which is combined with the potassium chloride (neglecting the salting-out effect), b. The complex salt has been shown by other measurements to be KHgCk Tabulate its concentration and the concentrations of the (uncom- bined) KC1 and HgCl 2 . Ans. 0.1185 KHgQ 3 per liter. *46. The Lowering by Solutes of the Solubility of One Solvent in Another Solvent. The solubility of one solvent, such as water, in another solvent, such as ether, is found to be lowered by dissolving in the first solvent a solute, such as sodium chloride. The general law expressing this effect corresponds in form to Raoult's law of vapor-pressure lowering. This law, called the law of solubility lowering, may be stated as follows. If in a solvent A, which has a small solubility SQ in another solvent B, there is dissolved a solute S (not soluble in the solvent B) whose mol-fraction in A has any small value x, the solubility s of A in B is now less than its original value S Q to the extent required by the law that the fractional lowering of solubility is equal to the mol-fraction of the solute. That is: S Q -S _ x. So The law of solubility lowering can be derived from Raoult's law, Henry's law, and the perpetual motion principle, as shown in the following problem. Prob. 45. Derivation of the Law of Solubility Lowering. Show how the law of solubility lowering can be derived from Raoult's law of vapor-pressure lowering with the aid of the perpetual-motion princi- ple (applied to the vapor of the substance A above the two liquid phases) and of Henry's law (applied to the equilibrium of the substance A between the vapor phase and the solution of A in B). Prob. 46. Application of the Law of Solubility Lowering. The solubility of isoamyl alcohol (C 5 H n OH) in water at 25 is 0.280 mol per 1000 g. Calculate its solubility in water when 0.5 mol of chloro- form (CHCls) is dissolved in 1000 g. of isoamyl alcohol. DISTRIBUTION BETWEEN SOLVENTS 79 47. The Vapor-Pressure Relations of Partially Miscible Liquids. Raoult's law, Henry's law, and the perpetual-motion principle make it possible to predict in part the vapor-pressure relations of liquids which have limited solubilities in each other, as is illustrated inProb.47. Pr ob. 47. The Vapor-Pressure-Composition Diagram for Partially Miscible Liquids. When isoamyl alcohol (C 6 HnOH) and water are shaken together at 25 till equilibrium is attained, two liquid phases result, one containing 36.84 mol-percent, and the other 99.50 mol- percent of water, a. Calculate the vapor-pressure of the water in each of the phases from the vapor-pressure, 23.8 mm., of pure water at 25, regarding the isoamyl alcohol present in small proportion in the aqueous phase as a perfect solute, b. Construct a diagram showing the vapor- pressure of water in all mixtures of these two substances at 25, with the aid of the above values and the following data. The vapor-pres- sure of water at 25 is 19.1 mm. in a 20.5 mol-percent, 14.5 mm. in 13.1 mol-percent, and 8.4 mm. in a 6.9 mol-percent solution of it in isoamyl alcohol. Note that the water must conform to Henry's law so long as it is present at small concentration. Prob. 48. Steam Distillation of Liquids Partially Miscible with Water. Aniline (C 6 H 5 NH 2 ) is distilled with steam at i atm. The vapor-pressure of aniline is 46 mm. at 100 and 40 mm. at 97. Water and aniline have limited solubilities in each other; one phase containing at 100 1.5 mol-percent of aniline, and the other 68 mol-percent of aniline. Find the boiling-point of the mixture, and the number of grams of aniline distilling over with each gram of water. Assume that the solubilities do not change appreciably within the small temperature interval, and that the vapor-pressure of the aniline in the 68 mol- percent solution is lowered only half as much as Raoult's law requires. Ans. 0.26 g. 8o THE MOLAL PROPERTIES OF SOLUTIONS VI. FREEZING-POINT OF SOLUTIONS 48. Freezing-Point and Its Relation to Vapor-Pressure and Molal Composition. The freezing-point of a liquid is that temperature at which the solid solvent and the liquid coexist in equilibrium with each other. The solid which separates from the solution commonly consists of the pure solvent; and this is assumed to be the case through- out the following considerations. Jn a.ny such case the freezing-point of a solvent is lowered by dissolving another substance in it. At the freezing-point the vapor-pressure of the solvent in the solu- tion and of the pure solid solvent must be equal (as shown in Prob. 49); and accordingly, from the vapor-pressure-temperature curves of the liquid solvent and solid solvent the freezing-points of solutions can be derived from their vapor-pressures (as in Prob. 50). Prob. 4Q. Equality of the Vapor-Pressures at the Freezing-Point. Prove that at the freezing-point of a solution its vapor-pressure and that of the solid which separates from it must be equal, by showing that otherwise perpetual motion would result. Prob. 50. Evaluation of Freezing-Points from Vapor-Pressure Data. The vapor-pressures of ice and (supercooled) water are as follows: o -i -2 -3 -4 Water . . . 4.579 4.255 3.952 3.669 3.404 mm. Ice . . . . 4.579 4.215 3.879 3.566 3.277 mm. a. Calculate by Raoult's law the vapor-pressure at each of these temperatures of a solution consisting of 3 mols of solute and 97 mols of solvent. Plot on a diagram the vapor-pressures of this solution, of water, and of ice as ordinates against the temperatures as abscissas, using a scale large enough to enable o.ooi mm. to be estimated (by including on th plot pressures ranging only from 3.20 to 4.60 mm.). b. Determine from the plot the freezing-point of the solution. The general relations between freezing-point and vapor-pressure are illustrated by the diagram on the following page. A consideration (as in ProJ), 51) of the geometrical relations of the diagram /shows tha^t the vapor-pressure lowering pop caused by a solute is for a dilute solution related to the freezing-point lowering To T in the way expressed by the equation: pop dpi _ dpi ,v r -r dT dT In this expression the two derivatives represent the rate of change with the temperature of the vapor-pressure of the solid solvent (ice) and of the liquid solvent, respectively. FREEZING-POINT 81 Prob. 51. Derivation of the Relation between Freezing- Point and Vapor-Pressure. From the geometrical relations of the following diagram derive equation (i) given in the text. o T T O Freezing-Point From the approximate Clapeyron equation (Art. 33) expressions for the temperature-coefficients of the vapor-pressures of the solid solvent and liquid solvent can be obtained in terms of the molal heats of vaporization A# s and A# L of the solid solvent' and the liquid solvent, respectively. Since the difference between these two heats of vaporization can be shown by the law of initial and final states (Art. 23) to be equal to the fcnolal heat of fusion AZTojbf the solid solvent, the following expressions, identical in form with those for the boiling-point raising (Art. 37), are readily obtained (as in Prob. 52) for the freezing-point lowering of a perfect solution: . RT * p Q -p RTJ In these equations T is the freezing-point of the solvent, T that of a solution containing the solute at the mol-fraction x, and A#o is the heat absorbed by the fusion of one mol of the solvent at TV Prob. 52. Derivation of the Freezing-Point Equations for Perfect Solutions. Derive equation (2) from equation (i), indicating the principle involved in each step of the process, and proving that the heat of fusion is equal to the difference in the two heats of vaporization. The quantity /JTV/A&o occurring in equation (2) is evidently a constant characteristic of the solvent. Representing this quantity, which may be called the freezing-point constant, by a single letter F , and replacing the mol-fraction x by the mol-ratio N/^Vo, which becomes 82 THE MOLAL PROPERTIES OF SOLUTIONS identical with it as its value approaches zero, the law of freezing-point lowering for perfect solutions may be expressed by the equation: Instead of this freezing-point constant, the related mold freezing- point lowering, denned in analogy with the molal boiling-point raising (Art. 37) to be the freezing-point lowering per mol of solute in 1000 grams of solvent, is commonly recorded in chemical literature. Pr ob. 53. Evaluation of the Freezing-Point Constants. When one gram of ice at o melts, the heat absorbed is 79.7 cal. a. What is the freezing-point constant for water? b. What is its molal freezing-point lowering? Ans. 6,1.858 (1.86). *The preceding equations are rigorously exact only when the lower- ings of the freezing-point and of the vapor-pressure of the solvent are infinitesimal; for in the derivation of equation (i) all the vapor- pressure curves were regarded as straight lines, and those for the solvent and solution were regarded as parallel. A consideration of the figure at the beginning of this article shows, however, that the- following differential equation, corresponding to equation (i) above, is exact for any solution whatever containing the solute at any mol- fraction x whatever: *By substituting for the first of these derivatives the expression given by the corrected Raoult equation (Art. 36), and for the last two derivatives the expressions given by the exact Clapeyron equation (Art. 33), neglecting only the volume of the solid or liquid in compari- son with that of the vapor, there is obtained the following equation : In this equation dT denotes the lowering produced in the freezing- point T of a solution by increasing the mol-fraction x of the solute by dx, and A# denotes the heat absorbed by the fusion of one mol of the solid solvent into an infinite volume of the solution at its freezing- point T. *Prob. 54. Freezing-Point of Solutions Conforming to Raoult's Law at Any Concentration. Derive equation (5) from equation (4) by the method indicated in the text. FREEZING-POINT 83 This equation will be seen to correspond in form to equation (5.) of Art 37 for the boiling-point raising. The equation is exact (except for the error caused by neglecting the volume of the solid or liquid in the Clapeyron equation) at any concentration up to which the corrected Raoult equation holds true. It may be integrated, usually with negligible error, under the assumption that, within the limits of temperature and concentration involved, AH does not vary with the temperature or with the mol-fraction of the solute, and that it has the same value as the heat of fusion A# of the pure solvent at its freezing-point TV *The freezing-points of solutions that do not conform to Raoult's law cannot be calculated from the molal composition. The relation between freezing-point and molal composition can, of course, be experimentally determined; and the results may be represented by freezing-point-composition diagrams, analogous to the boiling-point- composition diagrams of Art. 43. Such diagrams are considered in Chapter VIII. 49. Determination of Molecular Weights. With the aid of the laws of freezing-point lowering stated in Art. 48 the molecular weigtfts of substances dissolved in various solvents can be calculated. Prob. 55. Variation of Molecular Weight with the Nature of the Solvent. A solution of 0.60 g. of acetic acid in 50.0 g. of water freezes at 0.376. A solution of 2.32 g. of acetic acid in 100 g. of benzene freezes 0.970 lower than pure benzene. The freezing-point constant for benzene is 65.4. Calculate the molecular weight of acetic acid in each of these solvents, and state what the results show in regard to its molecular formula in each solvent. Determinations of the molecular weights of dissolved substances by the freezing-point method have confirmed the conclusions derived from boiling-point determinations, and stated in Art. 38, as to the dissociating effect of various solvents. 84 THE MOLAL PROPERTIES OF SOLUTIONS VII. OSMOTIC PRESSURE OF SOLUTIONS 50. Osmotic Pressure. When a solution S is separated from a pure solvent W, as illustrated in the figure, by a wall aa which allows this solvent to pass through it, but prevents entirely the passage of the solute, the solvent is drawn through the wall into the solution. This flow of solvent may, however, be pre- vented by exerting a pressure fa on the solution greater by a definite amount than the pressure pi upon the pure solvent; and the solvent may be forced out of the solution by exerting a still greater pressure upon the solution. The difference of pressures on solution and solvent which produces a condition of equilibrium such that there is no tendency of the solvent to flow in either direction is called the osmotic pressure P of the solution. Walls of the kind just described are known as semipermeable walls. Certain animal membranes, such as parchment or bladder, are per- meable for water, but no^|r certain solutes of high molecular weight (the so-called colloids). The walls of some animal and plant cells are very perfect semipermeable walls. The most satisfactory artificial semipermeable walls have been made by precipitating copper ferro- cyanide within the pcres of an unglazed porcelain cell, which gives the precipitate sufficient rigidity to withstand high osmotic pressures. The cell, filled with the solution and immersed in pure water, is con- nected with a manometer whose mercury column is in direct contact with the solution. By means of such cells exact measurements of the osmotic pressure of aqueous solutions of cane-sugar and glucose have been made up to pressures of 230 atmospheres. Osmotic pressure plays a very important part in the physiological processes taking place in the bodies of animals and plants. In the study of the general principles of chemistry, it is of value because it is a property which, like vapor-pressure, enables the various properties of solutions to be correlated and their energy relations to be treated upon the basis of one simple fundamental concept. The correspond- ence shown in Art. 5 2 to exist between the laws of the osmotic pressure of dilute solutions and those of the pressure of perfect gases makes possible, moreover, a closely analogous treatment of these two states. OSMOTIC PRESSURE 85 *51. Relation of Osmotic Pressure to Vapor-Pressure. In the osmotic arrangement represented in the figure of Art. 50, when the pressures pi and p 2 are such that there is equilibrium and therefore no tendency for the solvent to pass through the semipermeable wall, it follows from the perpetual motion principle that the vapor-pressure of the solvent in the pure state must be equal to its vapor-pressure in the solution. For, if a vapor phase were in contact (through walls per- meable only for the vapor) both with the solvent and the solution, there would evidently be a continuous flow of the solvent-substance through the vapor-phase and back through the wall between the two liquid phases, unless its vapor-pressures in these liquid phases were equal. That this condition is realized results from the fact that the vapor- pressure of a liquid is increased by increasing the pressure upon it. Thus the larger pressure p% on the solution increases its vapor-pressure up to the value of the vapor-pressure of the solvent under the smaller pressure pi. The effect of pressure on vapor-pressure may be now considered from a quantitative standpoint. It can be shown (as in Prob. 56) that the increase dp in the vapor-pressure p of a liquid caused by an increase dp^ in the pressure p^ on the liquid is expressed by the follow- ing equation, in which z> L represents the volume of any definite weight of the liquid at the pressure L , and v represents the volume of the same weight of the vapor at the pressure p: (i) In case the vapor may be regarded as a perfect gas, this equation takes the following form, in which z> L represents the volume of one mol of the liquid at the pressure p^: By integration of this equation under the assumption that the volume of the liquid. V L does not change with the pressure there results the following expression, in which p and p denote the vapor- pressures of the solvent when the pressures on it are p L and zero, respectively: _ /> " RT 86 THE MOLAL PROPERTIES OF SOLUTIONS ?ELJ& Effect of Pressure on the Vapor-Pressure of Liquids. Prob. 56. Consider, as illustrated in the figure, a column of a pure liquid contained in a porous tube impermeable to it, but permeable to its vapor; and consider this tube to be surrounded by the vapor of the liquid; the whole system being at a constant tempera- ture T. When there is equilibrium the vapor-pressure of the liquid at any level must evidently be equal to the pressure of the vapor at that level. The pressure of the liquid or of the vapor must, however, be greater at a lower than a higher level by the (hydrostatic) pressure of the intervening column of liquid or vapor. a. Formulate an expression for the increase dpi. of the pressure pi. of the liquid, and one for the increase dp of the pressure p of the vapor, corresponding to an in- crease dl in the distance / beneath the surface, b. By combining these expressions, derive equation (i) given in the text. c. Integrate this equation, assuming that the vapor conforms to the perfect-gas laws and that the volume of the liquid does not vary with the pres- sure upon it, so as to obtain a relation between the vapor-pressures po and p of the liquid at the top and bottom of the column in terms of its hydrostatic pressure pi,. Prob. 57. Find the ratio of the vapor-pressure of water at 4 and 10 atm. to that at 4 and the pressure (6 mm.) of the saturated vapor. It is evident now that, by applying a sufficient negative pressure (or suction) to a liquid solvent, its vapor-pressure can be reduced from its normal value p Q to the value p which it normally has in any definite solution, and that then it would be in equilibrium with that solution if placed in communication with it through a semipermeable wall. (Liquids can in fact with suitable precautions be subjected to negative pressures of many atmospheres, without the column breaking.) The difference in the pressures on the solution and the solvent, defined to be the osmotic pressure P, is evidently numerically equal to this pressure /> L applied to the solvent, but opposite in sign ; that is, P= pi.. Hence it follows from equation (3) that the osmotic pressure at the temperature T of a solution in a solvent whose vapor- pressure is p in the solution and p in the pure state, and whose molal volume is V L , is given by the expression: This expression is exact even at large concentrations provided the vapor conforms to the perfect-gas laws, except for the usually negli- gible effect of pressure on the volume z> L of the solvent. OSMOTIC PRESSURE 87 Relation of Osmotic Pressure to Vapor-Pressure. Prob. 58. a. How much must the pressure on water be reduced in order that it may be in equilibrium through a semipermeable wall with an aqueous solution which has at 25 a vapor-pressure 98% as great as that of water? b. What is the osmotic pressure of this solution at 25? Prob. 59. At 100 the vapor-pressure of a solution consisting of 28 g. of NaCl and 100 g. of water is 624 mm. What is its osmotic pressure? The specific volume of water at 100 is 1.043. Ans. 320 atm. When the vapor does not conform to the perfect-gas laws an exact expression may be readily obtained by substituting for v in equation (i) the expression RT(i + a p)/p given by the equation of state for gases at moderate pressure (Art. 15), and integrating as before. This leads to the result that the osmotic pressure equation may be corrected for deviation of the vapor from the perfect-gas laws by writing it in the form: P=^(log>-+a(^)). (5) 52. Relation between Osmotic Pressure and Molal Composition. The osmotic pressure P of concentrated solutions whose vapor-pres- sure conforms to Raoult's law is expressed by the following equation, which may readily be derived (as shown in Prob. 60) from equation (4) or (5) of Art. 51. (i) In this equation L represents the molal volume of the solvent, and N denotes the number of mols of solute associated with No mols of solvent. For dilute solutions, for which N/ N is small, this equation can be simplified (as shown in Prob. 61) by expanding the logarithm into a series, neglecting all the terms except the first one, and transforming. The following expression then results: Pv =NRT. (2) In this expression P is the osmotic pressure at the temperature T of a solution consisting of N mols of solute and N Q mols of solvent whose volume in the pure state is i> , and R is the gas-constant. This equation is seen not only to be identical in form with the perfect-gas equation, but to contain the same constant R. This shows that a substance in dilute solution produces an osmotic pressure which is equal to the pressure it exerts at the same temperature as a perfect gas when present at the same volume concentration. 88 THE MOLAL PROPERTIES OF SOLUTIONS The osmotic pressure of concentrated solutions to which Raoult's law does not apply cannot be calculated from the composition of the solution. It may, however, be determined not only by direct measure- ment, but also by calculation from vapor-pressure measurements, by equations (4) and (5) of Art. 51. Derivation of the Osmotic-Pressure Equations. *Prob. 60. Derive equation (i) of this article, a, from equation (4), and b, from equation (5) of Art. 51. Prob. 61. Derive equation (2) from equation (i) in the way in- dicated in the above text. Prob. 62. Osmotic Pressure in Relation to Hydrostatic Pressure. The lower end of a vertical tube is closed with a semipermeable wall and is dipped just beneath the surface of pure water. A o.i molal solution of cane-sugar (CiaHaOu) of density 1.014 is poured into the tube until the hydrostatic pressure at the semipermeable wall is sufficient to prevent water from entering the solution. The temperature is 4. Find the height of the column in meters. Ans. 23.6 REVIEW OF THE PRINCIPLES 89 VIII. REVIEW OF THE PRINCIPLES RELATING TO THE MOLAL PROPERTIES OF SOLUTIONS 53. Review Problems. The following problems afford a review and give additional applications of the principles relating to the molal properties of solutions. Prob. 63. a. Summarize the equations expressing for dilute solutions the approximate relations between molal composition and (i) vapor- pressure, (2) boiling-point, (3) freezing-point, (4) osmotic pressure. State explicitly what each symbol signifies. *Prob. 64. Summarize the corresponding equations which hold true for perfect solutions of any concentration. Prob. 65. Calculate by the laws of dilute solutions the fractional vapor-pressure lowering, the boiling-point raising, the freezing-point lowering, and the osmotic pressure at 5.5, of a solution containing o.i mol of a nonvolatile solute in 1000 g. of benzene, with the aid of the following data. The heat of fusion of one gram of benzene at its freezing-point, 5.5, is 30.2 cal. The heat of vaporization of one gram of benzene at its boiling-point, 80. i, is 93.0 cal. The density at the freezing-point is 0.895. *Prob. 66. a. Calculate by the laws of concentrated perfect solu- tions the fractional vapor-pressure lowering, the boiling-point raising, the freezing-point lowering, and the osmotic pressure at 5.5, of a solution containing 2 mols of a non-volatile solute in 1000 g. of benzene, using the data of Prob. 65, and assuming that the vapor conforms to the perfect-gas laws. b. Calculate the ratio of each of these values to the corresponding one for the solution of Prob. 65. (Note that the value of this ratio would be 20.0 if the equations for dilute solutions were applicable to this concentrated solution.) c. Assuming the solute to be volatile and to have a partial vapor-pressure of 20 mm. in the solution at its boiling-point, calculate this boiling-point. Ans. c, 84.2. Prob. 67. a. Carbon bisulfide boils at 46 at i atm. Its molal heat of vaporization at this temperature is 6430 cal. What is its boil- ing-point constant? b. A solution of 15.5 g. of phosphorus (at. wt., 31.0) in 1000 g. of CS 2 boils 0.300 higher than pure CSa. What is the molecular weight and what is the molecular formula of phosphorus in this solvent? -Prob. 68. If 127 g. of iodine (at. wt., 127) were added to the solution of Prob. 676, how much higher than the boiling-point of pure CSa would that of the solution be: a, in case the iodine remained uncom- bined in the form of I 2 ; &, in case it all combined with the phosphorus forming P 4 I 8 ; c, forming P 2 I 4 ; d, forming PI 2 ? *Prob. dp. Human blood freezes at -0.56. Find its osmotic pressure at 37. Ans. 7.6 atm. - Prob. 70. At 25 the distribution-ratio of Br 2 between carbon tetra- chloride and water is 38 expressed in volume concentrations; and the pressure of bromine above a 0.05 molal solution of Br 2 in water is 90 THE MOLAL PROPERTIES OF SOLUTIONS 50 mm. Assuming that one liter of this solution is shaken with 50 ccm. of carbon tetrachloride, calculate the pressure of the bromine over the carbon- tetrachloride phase. (Bromine exists in all three phases only % as Br 2 .) Ans. 17.2 mm. Prob. 71. At i atm. pure nitric acid has a boiling-point of 86. A solution of nitric acid and water of the composition HN0 3 + i.6H 2 O distils at i atm. at a constant temperature of 121. Make a diagram showing the character of the boiling-point-composition curve for this pair of substances. Draw in on the diagram a curve representing in a general way the composition of the vapor in equilibrium with any solu- tion at its boiling-point. State what products would finally result as distillate and residue from the fractionation of the three solutions, HN0 3 +H 2 O, HNO 3 + i.6H 2 O, HNO 3 +3H 2 O. Prob. 72. Upon partial distillation at constant pressure a solution of two components A and B containing 20% of A, having a boiling- point of 80, yields a residue containing 15% of A; and a mixture con- taining 75% of A, having a boiling-point of 60, yields a distillate containing 70% of A. Draw a diagram which will show in a general way the character of the liquid-composition and of the vapor-composi- tion curve. Predict what products would finally result as distillate and residue from the fractionation of the 20% and of the 75% solution. CHAPTER IV THE ATOMIC PROPERTIES OF SOLID SUBSTANCES I. THE HEAT-CAPACITY OF SOLID SUBSTANCES 54. Properties in the Solid State. Many of the properties of substances in the gaseous and dissolved states are primarily deter- mined, as has been shown in the preceding chapters, by the number of molecules present; but in the solid state such molal properties are met with only in the case of substances present at small concentration in solid solutions (considered in Art. 123). In the solid state there are, however, certain properties, here called atomic properties, which depend primarily either on the number of atoms present in the unit- weight of the substance, or on the number of atoms present in each molecule of the substance. The heat-capacity of solid substances is a property of the first of these types; and isomorphism, or the power of two substances of assuming nearly the same crystal structure and of forming with each other a continuous series of solid solutions, is a property of the second of these types. The heat-capacity of solid substances, which is of importance not only in its relations to the atomic theory, but also in thermodynamic considerations, will alone be considered here. 55. The Heat-Capacity of Solid Elementary Substances. Meas- urements have shown that at room temperature all elementary sub- stances with atomic weights above 35, and the metallic elementary substances with smaller atomic weights than 35, have values of the atomic heat-capacity (defined as in Art. 25) not differing greatly from each other. This approximate principle is known as Dulong and Petit 1 s law. The average value of the atomic heat-capacity at constant pressure at 20 for these elements is 6.2 calories per degree. Deviations of + 0.4 unit are not uncommon; and deviations of +0.5 to +0.9 unit are exhibited by some elements (for example, by sodium, potassium and iodine) which at room temperature are not much below their melting-points. In the case of the non-metallic elements with smaller atomic weights than 35 the atomic heat-capacity has a value much smaller than 6.2 at room temperature; thus the value for boron is 2.6, for graphite 1.9, for silicon 4.8, for phosphorus 5.6, and for sulphur 5.5. 91 92 ATOMIC PROPERTIES OF SOLID SUBSTANCES The corresponding average value of the atomic heat-capacity at constant volume at room temperature is 5.9; and from this simpler, more fundamental quantity the values for the separate substances show considerably smaller deviations. It is interesting to note that this average atomic heat-capacity is very nearly twice the translatory kinetic energy (2.98 cal.) taken up per degree by the molecules of a gaseous or liquid substance (Art. 19). The kinetic factors determining this roughly constant value for solid elementary substances, and causing the marked variation of the heat- capacity with the temperature to be now mentioned,. are too complex and hypothetical to be here considered. The effect of temperature on the atomic heat-capacity at constant pressure is shown in the figure. It will be seen that at very low temperatures (below 200 A) the atomic heat-capacity of solid elemen- tary substances varies much with the nature of the substance and is always much smaller than at room temperature; that with rising temperature it increases at different rates in the case of different substances; and that at a temperature, varying with the different substances but usually not far from the room temperature, it attains the average value, and then as a rule increases only slowly with further increase of temperature. 50 100 150 200 250 300 350 400 450 500 Absolute Temperature HEAT-CAPACITY OF SOLID SUBSTANCES 93 56. The Heat-Capacity of Solid Compound Substances. Another roughly approximate principle, called Kopp's law, has been found experimentally to express the heat-capacity of solid inorganic com- pounds at room temperature. It has been found, namely, that the formal heat-capacity is approximately an additive property, that is, one whose value can be approximately calculated by adding together certain values representing the heat-capacity of the elements con- tained in the compound. This principle is expressed by the following equation, which shows at the same time the values of the constants (the so-called atomic heat-capacities) for the common elements: In this equation Cp represents the formal heat-capacity of the com- pound at constant pressure at room temperature; no, nu, nc, n$, WB? np, and nsi are the number of atomic weights of oxygen, hydrogen, carbon, sulfur, boron, phosphorus, and silicon present in one formula- weight of the compound; and HE is the number of atomic weights of any other element so present. The values given for the constants are average values derived from heat-capacity measurements with solid compounds. They do not accurately represent the atomic heat-capacities of the solid elementary substances. The following table illustrates the degree of correspondence which exists between the values of the formal heat-capacity so calculated and those measured experimentally. It will be observed that differences of ten percent between the calculated and measured values are not uncommon. HEAT-CAPACITIES OF SOLID COMPOUND SUBSTANCES Substance H 2 (ice) A1 2 0, Fe 2 8 80,8, KC1 PbCl 2 CaCOs KClOs Prob. i. Specific Heat-Capacities Calculated by the Additivity Principle. Calculate an approximate value at 20 of the specific heat-capacity at constant pressure of: a, platinum; b, silver bromide; c, potassium sulfate. Find the percentage deviations of these values from the measured values, which are, a, 0.032; b, 0.074; c, 0.190. Calc. Meas. Substance Calc. Meas. 8.6 9.1 PbN 2 O 6 42.6 38.8 24.4 20.5 CaSiOs 22.0 21.3 24.4 25.6 K 4 Fe(CN) 6 79-o 79.0 28.6 28.7 CuSO 4 27.6 25.8 12.4 12.4 CuSO 4 .5H 2 O 70.6 72.7 18.6 18.2 AlK(S0 4 ) 2 .i2H 2 O 158. I6 5 . 2O.O 20. 2 NH 4 C1 21.6 19.9 24.4 24.1 H 2 C2O 4 24.2 2S.I 94 ATOMIC PROPERTIES OF SOLID SUBSTANCES 57. Determination of Atomic Weights. The law of Dulong ana Petit, even though it is only an approximate principle, may evidently be employed for determining what multiple of the combining weight of an element is its atomic weight; and the application of this law was in fact one of the most important methods by which the present system of atomic weight values was established. Prob. 2. Determination of Atomic Weights. Calculate the exact atomic weight of an element whose specific heat-capacity is 0.092 and whose oxide contains 88.82% of the element. GENERALIZATIONS RELATING TO ATOMIC WEIGHTS 95 H. GENERALIZATIONS RELATING TO ATOMIC WEIGHTS 58. The Methods of Atomic Weight Determination and the Periodic Law. The important methods already considered for determining what multiple of the combining weight is the atomic weight may now be briefly summarized. It was shown in Art. 14 that the atomic weight of an element, or at any rate a maximum value for its atomic weight, can be obtained by finding the smallest weight of the element contained in one mo- lecular weight of any of its gaseous compounds, as derived from the density and Avogadro's principle. Molecular weights derived from the laws of perfect solutions (Arts. 36, 38, 49) may also be used for the same purpose. It was shown in Art. 27 that the molal heat-capacity of perfect gases is dependent on the number of atoms in the molecule, that this property has a definite value predicted by the kinetic theory for gases with monatomic molecules, and that therefore the atomic weight of elements which form gases having this molal heat-capacity can be immediately derived from their gas densities and the corresponding molecular weights. It was shown in Art. 57 that the atomic weight of elements can be directly determined from the principle that the atomic heat- capacities of solid elementary substances at room temperature have approximately the same value. Finally, atomic weights can also be derived from the phenomenon of isomorphism mentioned in Art. 54. By a combination of all these methods, which have in general given concordant results, the present system of atomic weight values has been derived. When the elements are arranged in the order of the so determined atomic weights, it is found that there is a progressive .change in the various properties of the elementary substances and their compounds, and a periodic recurrence of similar properties. This generalization is called the periodic law; and many systematic arrangements of the elements have been proposed for the purpose of bringing out as fully as possible the relations between their properties. These relations are most appropriately considered in the detailed study of inorganic chemistry, and are therefore only briefly referred to here. CHAPTER V THE ELECTROLYTIC BEHAVIOR OF SOLUTIONS AND THE IONIC THEORY I. THE MOLAL PROPERTIES OF SOLUTIONS OF IONIZED SUBSTANCES 59. Effects of Salts on the Molal Properties of Aqueous Solutions and Their Explanation by the Ionic Theory. In dilute solution the effect of salts of the uniunivalent type, such as sodium chloride or silver nitrate, on the vapor-pressure of water and on the other related molal properties is nearly twice as great, and the effect of salts of the unibivalent type, such as potassium sulfate and barium chloride, is nearly three times as great, as it would be if each formula- weight yielded a single mol in the solution. Strong acids and bases show a similar behavior. The following values illustrate the effect of some typical ionized substances on the freezing-point of water at a concentration o.i formal. The value given is the ratio, called the moil-number (i), of the effect produced by one formula- weight of the substance to that produced by one mol of a perfect solute. NaCl, 1.87; NaNO 3 , 1.83; KC1O 3 , 1.80; HC1, 1.92; MgSO 4 , 1.32; BaCl 2 , 2.58; Pb(N0 3 ) 2 , 2.30; Na 2 S0 4 , 2.47; K 4 Fe(CN) 6 , 3.32. As the concentration decreases the value of the mol-number ap- proaches 2 for the uniunivalent substances, and 3 for the unibivalent substances. These and other facts have led to the conclusion that these sub- stances are largely dissociated in aqueous solution. For example, NaCl dissociates into Na + and Cl~; HNO 3 into H + and NO 3 ~; H 2 S0 4 into H + , H + , and SO 4 =; and Ba(OH) 2 into Ba ++ , OH~, and OH~. Tri-ionic substances may also dissociate partially into intermediate ions; thus H 2 SO 4 into HSO 4 ~ (and H+). The electrical behavior of the solutions indicates that these dis- sociation-products differ from ordinary substances in that their mole- cules are electrically charged. These charged molecules are called' ions; and to their formulas + or signs are attached, as in the above examples, to indicate the nature and magnitude of the charge. The fraction of the salt dissociated is called its ionization y. This fraction always decreases with increasing concentration. 96 * MOLAL PROPERTIES OF IONIZED SUBSTANCES 97 From the effect of ionized substances on the molal properties their ionization has often been calculated. The evidence, however, seems to show that ions produce on the vapor-pressure, boiling-point, and freezing-point of water, even at moderate concentrations such as o.i molal, effects which differ considerably from those produced by perfect solutes at the same molal concentration. These deviations, which will be more fully considered in Art. 79, make unreliable the exact computation of the ionization of salts from the vapor-pressure and freezing-point lowering. In spite of the deviations a rough estimate of the number of mols of solute present in solutions of ionized sub- stances can be derived from the results of measurements of the molal properties, with the aid of the laws of perfect solutions; and such measurements often furnish useful information as to the forms in which substances exist in the solution. Prob. i. Effect of Salts on the Molal Properties. A solution of 0.65 formula- weight of KC1 in 1000 g. of water has at 100 a vapor-pressure of 744.8 mm. a. Calculate the mol-number i, which expresses the ratio of the observed lowering to that which would be produced by 0.65 mol of a perfect solute in 1000 g. of water, b. Find the ionization of the salt to which this value of i would correspond if the ions acted as perfect solutes. Prob. 2. Determination of the Chemical Substances Present in Solutions. The freezing-point of a solution containing 0.05 formula- weights of H 2 S0 4 per 1000 g. of water is -o. 2 1 5. a. Find the correspond- ing value of the mol-number i. b. Derive from this value an estimate of the secondary ionization of the HSO 4 ~ into H + and SO 4 = , assuming that the primary ionization of the H 2 S0 4 into H + and HS0 4 ~ is complete and that the ions act as perfect solutes. 98 ELECTROLYTIC BEHAVIOR OF SOLUTIONS II. ELECTROLYSIS AND FARADAY'S LAW 60. Electrolytic Conduction. Conductors are divided into two classes with reference to the changes that are produced in them by the passage of electric currents. Those which undergo no changes except such as are produced by a rise in temperature are called metallic conductors. Those in which the passage of a current is attended by a chemical change are called electrolytes. Aqueous solutions of salts, bases, and acids, and melted salts at high temperatures, are the most important classes of the well-conducting electrolytes. The most obvious chemical changes attending the passage of a current through an electrolyte are those that take place at the surfaces of the metallic conductors where the current enters and leaves the electrolyte. The production of such chemical changes by a current from an external source is called electrolysis. The occurrence of such changes, when they themselves give rise to an electric current, is called voltaic action. Those portions of the metallic conductors that are in contact with the electrolyte are called the electrodes; the one at which the current leaves the electrolyte, or the one towards which the positive electricity flows through the electrolyte, is designated the cathode; the other, the anode. 61. Chemical Changes at the Electrodes. The chemical change produced at the cathode is always a reduction; that at the anode, an oxidation. The following table exemplifies types of chemical chajiges- commonly occurring in electrolysis. It shows the products resulting when aqueous solutions of certain typical salts, bases, and acids are electrolyzed between electrodes which are unattacked, for example, between platinum or carbon electrodes. THE PRODUCTS OF ELECTROLYSIS Solute Cathode products Anode products Cu(N0 3 ) 2 or AgN0 3 Cu or Ag O 2 and HN0 3 KNO, H 2 and KOH O 2 and HNO 3 Na 2 SO 4 H 2 and NaOH O 2 and H 2 SO 4 KOH or Ba(OH) 2 H 2 O 2 H 2 SO 4 or H 3 PO 4 H 2 O 2 Dilute HC1 or HNO 3 H 2 I Oj Concentrated HC1 H 2 C1 2 When the anode is a metal which can react with the anion of the solute, the change at the anode may consist only in the dissolving of the metal; thus when a nitrate or sulfate is electrolyzed with a ELECTROLYSIS AND FARADAY'S LAW 99 copper anode, copper passes into solution, forming copper nitrate or sulfate. When the cathode is coated with a reducible solid substance, such as silver chloride or lead dioxide, or is surrounded with a solution con- taining a reducible solute, such as a ferric salt or a chromate, this substance is reduced and there may be no hydrogen evolved. In any definite electrolytic cell, however, the nature of the changes at the electrodes often varies with the conditions of the electrolysis, such as the applied electromotive force, current-density, concentration, and temperature. In the case of voltaic actions, the chemical changes are of a similar character, most commonly consisting in the solution of the metal composing the anode, and in the deposition of another metal or of hydrogen on the cathode or in the production of a soluble reduced substance at the cathode. Thus, in the J^anieJL cell, which consists of a copper electrode in a copper sulfate solution and of a zinc electrode in a zinc sulfate solution, the two solutions being in contact and the two electrodes connected by a metallic conductor, the zinc dissolves and the copper precipitates; and, in the Bunsen cell, consisting of a zinc electrode in dilute sulfuric acid and a carbon electrode in strong nitric acid, zinc dissolves at the anode, and the hydrogen primarily produced at the cathode reduces the nitric acid to lower oxides of nitrogen. Prob. 3. Changes at the Electrodes. State what substances are produced or destroyed at each electrode when electricity passes through: o, a concentrated solution of NaCl between a carbon anode and an iron cathode; b, a solution of NaCl between a silver anode and a metal cathode coated with AgCl; c, dilute H 2 SO 4 between a lead anode and a cathode of lead coated with PbO 2 ; d, a solution of ZnSO 4 between a zinc-amalgam anode and a cathode of mercury covered with Hg2SO 4 ; e, through a solution of K 2 SO 4 between platinum electrodes. 62. Faraday's Law. The passage of electricity through an elec- trolyte is attended at each electrode by a chemical change involving a number of chemical equivalents N strictly proportional to the quan- tity of electricity Q passed through and dependent on that alone. That is: Q = FN, where r is a constant with respect to all variations of the conditions, such as temperature, concentration, current-strength, current-density, etc. Such variations often influence the character of the chemical change, but not the total number of equivalents ioo ELECTROLYTIC BEHAVIOR OF SOLUTIONS involved. The law is applicable to concentrated, as well as to dilute solutions, and to fused salts. It is known as Faraday's law. The constant F evidently represents the quantity of electricity pro- ducing a chemical change involving one equivalent. It is called a faraday, and has the value 96,500 coulombs. One coulomb is the quan- tity of electricity flowing per second when the current is one ampere. The term chemical equivalent in the above statement of Faraday's law signifies the oxidation or reduction equivalent of the substance in the sense in which it is used in volumetric analysis. That is, one equivalent of any substance is that weight of it which is capable of oxidizing one atomic weight of hydrogen, or which has the same reducing power as one atomic weight of hydrogen. Applications of Faraday's Law. - \ ) that it was justifiable to neglect it. Ans. b, 2.5 Xio-. *Prob. 24. Calculate the ionization-constant at 18 for the third hydrogen of H 3 PO 4 (that for the ion HPO 4 = ) from the data of Prob. 16 and from the ionization-constants of water and of ammonium hydroxide, which at 18 are 0.5X10-14 and i.7Xio~ 5 , respectively. 107. Displacement of One Acid or Base from its Salt by Another. One of the most important types of equilibrium in aqueous solution is the partial displacement of one acid or base from its salt by another; for example, that of acetic acid from sodium acetate by formic acid, or that of ammonium hydroxide from ammonium chloride by sodium hydroxide. Before the mass-action relations involved were fully understood, the extent of the displacement was taken as a measure of the relative strengths of different acids or bases; those which are largely displaced from their salts being called weak acids or bases, and those which cause such displacement being called strong acids or bases. It is shown by Probs. 25-28 that the mass-action law and ionic theory give a comparatively simple explanation of this phenomenon in the case of not largely ionized univalent acids or bases; also that the relative strengths of different acids or bases, as shown by their dis- placements, are determined by their ionization, weak ones being those which are slightly ionized and strong ones those which are largely ionized. Displacement of Acids or Bases from their Salts. Prob. 25. To a o.i formal solution of KNO 2 is added at 25 an equal volume of a o.i formal solution of acetic acid. a. Calculate the frac- tion of the potassium nitrite that is converted into potassium acetate. b. Calculate the fraction that would be so converted if the acetic acid solution were i.o formal (instead of o.i formal). Prob. 26. a. For the general case expressed by the equation B A + HA' =B + A'~+HA where the solution is originally c formal in B + A~ and c r formal in HA', derive an expression for the fraction x of the salt B + A~ 160 EQUILIBRIUM OF CHEMICAL CHANGES converted into the salt B + A'~, assuming that the ionization-constants of both acids are so small that the ionization of them in the presence of their salts is negligible, b. Derive from this expression the relation be- tween the ionization-constants of the acids and the fractions of the basic constituent that are combined with the two acidic constituents for the case that c =c r ; and state the principle fully in words. Prob. 27. To a liter of o.i formal HC1 solution is added at 25 a liter of o.i formal NH 4 OH solution and a liter of o.i formal CH 3 NH 3 OH solu- tion. Calculate the fraction of the acid which combines with each base. Prob. 28. A o. i formal solution of acetic acid is added to an equal volume of o.i formal NaHCO 3 solution at 25, the carbon dioxide pro- duced being kept above the solution at a pressure of i atm. a. Calcu- late the concentrations of the two salts and two acids in the resulting solution, taking 0.0338 as the solubility-constant (Art. 39) for CO 2 in water at 25. b. Repeat the calculation for the case where no carbon dioxide is allowed to escape from the solution; and tabulate the two results. The displacement of acids and bases from their salts can often be determined by measuring the heat effect or the change in volume (as in Prob. 30) attending the addition of one acid to the salt of the other acid, or of one base to the salt of the other base. In cases where one of the acids has a color different from that of its salt (as in Prob. 29), the displacement can be determined by measuring the attendant color change. Prob. 29. Determination of Displacement by Color Measurements. Violuric acid is a slightly ionized acid whose solution is colorless, while solutions of its salts have a violet color which increases proportionately with the quantity of salt. 25 ccm. of a o.oi formal solution of violuric acid are placed in each of two tubes of the same diameter; to one tube are added 25 ccm. of a o.oi formal solution of potassium acetate; and to the other tube is added o.oi formal KOH solution until on looking down through the tubes the colors are seen to be the same, 13.75 ccm - of the KOH solution being required, a. Calculate the fraction of the violuric acid which exists in the first mixture in the form of its salt. b. Calculate the ionizat ion-constant of the violuric acid. * Prob. 30. Determination of Di placement by Volume Measure- ments. When 1000 g. of a solution containing iKOH is mixed with 1000 g. of a solution containing iHC 2 H 3 O 2 there is an increase of volume of 9.52 ccm. When the former solution is mixed with 1000 g. of a solu- tion containing iHCHO 2 there is an increase of volume of 12.39 ccm. When 2000 g. of a solution containing iKCHO 2 is mixed with 1000 g. of one containing iHC 2 H 3 O 2 there is a decrease of volume of 0.74 ccm. a. Determine what fraction of the formic acid is displaced from its salt by the acetic acid. b. Calculate from the ionization-constants the fraction displaced, and compare the results. Ans. a, 0.26. REACTIONS BETWEEN SOLUTES 161 108. Neutralization-Indicators. An acid neutralization-indicator (such as litmus, paranitrophenol, or phenol phthalein) is a mixture of two isomeric acids (HIn' and HIn '0 in equilibrium with each other, one of which (HIn") is present in much smaller proportion, but is so much more ionized than the other (HIn') that when a base BOH is added the salt produced is almost wholly of the form B + In "~. The substances HIn' and B+In'~ are different in color from the substances HIn" and B+In"~, the color being determined only by the molecular structure of the group In. It follows from these conditions that the indicator acid changes color when converted into its salt. For ex- ample, phenolphthalein exists in two isomeric forms to which the following molecular structures are assigned: and C 6 H 4 -C |\C 6 H 4 OH | \CeH4 =O CO - O COOH The first is the (colorless) form HIn' which is present in predominating proportion in acid solution; the second is the (red) form HIn", which has a much larger ionization-constant, and hence on the addition of alkali is converted into its (red) salt in much larger proportion than the other form. It can be shown by the mass-action law that the indicator behaves as if it were a single acid HIn whose salt has a different color from the acid itself; and its behavior can be represented by a single constant called the indicator-constant Ki. Namely, by combining the two equilibrium equations: (Kin") (H+)(In-) (HIS 7 )-* (Kin") there results the equation (HInO Since according to the above statements (In'-)is negligible in compari- son with (In""), and (HIn") is negligible in comparison with (HIn'), the total indicator acid is substantially equal to (HIn'), and the total indicator salt is substantially equal to (In"-). Hence in the above expression the ratio (In"-)/(HIn') may be replaced by the ratio 1 62 EQUILIBRIUM OF CHEMICAL CHANGES x/(i x), where x represents the fraction of the total quantity of the indicator which is in the colored form that exists in alkaline solution. Making this substitution, there results the following fundamental indicator equation: The relations in the case of basic indicators (existing almost wholly in the two differently colored forms In' OH and In" + A~) are entirely similar to those in the case of acid indicators. By formulating the mass-action equations, it can, in fact, be shown that the relation just derived between the hydrogen-ion concentration and the fraction x of the indicator transformed holds true also in the case of basic indi- cators. A basic indicator may therefore be treated as an acid indi- cator by adopting as the value of its indicator-constant that calculated by the above-given indicator equation; and it will be so treated in the following problems. Prob. 31. Determination of the Indicator-Constant, To 100 ccm. of a solution o.i normal in potassium acetate and o.oi normal in acetic acid are added 10 ccm. of a o.oi normal solution of paranitrophenol. This solution is found to have the same yellow color as a solution made by adding 0.50 ccm. of o.oi normal paranitrophenol solution and 9.5 ccm. of o.oi normal KOH solution to 100 ccm. of water, a. Calculate the indicator-constant, neglecting the quantity of the acetic acid dis- placed from its salt by the small proportion of paranitrophenol present. 6. Show what percentage error is made in (H + ), and therefore in K x by neglecting this displaced quantity. Prob. 32. Determination of Small Hydrogen-Ion Concentrations by Means of Indicators. When a small proportion of phenolphthalein (Ki = io~ ltf ) is added to a o.i formal solution of NaHCO 3 at 25 the indicator is found by color comparisons to be 6.0% transformed into its salt. Calculate the hydrogen-ion and hydroxide-ion concentrations in the solution. It is evident from the indicator equation that in the titration of an acid with a base an end-point will be reached when the hydrogen-ion concentration (H+) becomes so reduced that a considerable fraction x of the indicator is transformed into its salt; also that the observed end-point will be correctly located only if this transformation occurs when equivalent quantities of acid and base are present, and that the sharpness of the end-point will depend on how fast the hydrogen-ion concentration changes in this region. Hence, in order to determine how an acid of any known ionization-constant will behave on titration REACTIONS BETWEEN SOLUTES 163 and in order to select the most suitable indicator, it is necessary to know how the hydrogen-ion concentration varies when a largely ionized base is progressively added. The following table contains data of this kind, calculated as shown in Prob. 33. Namely, it shows the hydrogen-ion concentrations prevailing at 25 in solutions of acids of ionization-constants io~ 3 , io~, 5 io~ 7 , and io~ 9 when the ratio B/A of the quantity of NaOH added to the quantity of the acid present has the values given in the first column (the original concentration of the acid and that of the standard base being 0.2 normal). CHANGE OF THE HYDROGEN-ION CONCENTRATION ON NEUTRALIZATION Ratio Values of the hydrogen-ion concentration for B/A K A = zo- 3 K A = /o- 5 K A = io- 7 K = jQ-9 0.980 2XIO- 6 2XIO- 7 2XIO- 9 2.4X10-" 0.990 iXio- 6 iXio- 7 iXio- 9 1.6 Xio-" 0-995 SXio- 8 SXio- 8 SXio- 10 1.2X10-" 0.998 2 XIO- 6 2X10-8 2.4 Xio- 10 1. 1 Xio-" .000 iX to- 8 iXio~ 9 i Xio- 10 1.0X10-" .002 SXio- 11 SXio- 11 4 Xio- 11 0.9X10-" .005 2 XIO- 11 2 Xio- 11 2X10-" 0.8X10-" .OIO i Xio- 11 i Xio- 11 i Xio- 11 0.6 Xio-" .020 5 Xio- 12 5 Xio- 12 5 Xio- 12 0.4X10-" These numbers are also the values of the hydroxide-ion concentra- tion for the case that a 0.2 normal solution of a base having an ionization-constant equal to lo" 3 , io~ 5 , io~ 7 , or icr 9 is titrated with a 0.2 normal solution of a strong acid (like HC1), provided the numbers in the first column denote the ratio (A/B) of the quantity of acid to the quantity of base. Thus, when the ratio A/B is 0.98, the value of (OH~) is 2 X lo" 5 for a base for which K = lo" 8 . Titration of Acids and Bases. Prob. jj. 100 ccm. of a 0.2 normal solution of an acid whose ioni- zation-constant is io- 5 are titrated at 25 with 0.2 normal KOH solu- tion. Calculate the hydrogen-ion concentration (H + ) in the mixture when 99.0, 99.5, 99.8, 100.0, 100.2, 100.5, an d 101.0 ccm. of the KOH solution have been added, using i X io~ 14 as the ionization-constant of water. Prob. 34. a. Plot the common logarithms of the values of the hydrogen-ion concentrations given in the preceding table as ordinates against the corresponding values of the ratio of base to acid as abscissas, for each of the four acids, b. On the right-hand side of the same diagram write in a scale of values of logio(OH~) corresponding to the values of logio(H+) on the left-hand side; and at the top of the diagram write in a scale of ratios of acid to base (A/B) corresponding to the 1 64 EQUILIBRIUM OF CHEMICAL CHANGES scale of ratios of base to acid at the bottom (thus, B/A=o.g8 corre- sponds to A/B = 1.02). Now make plots on the same diagram, showing how (H + ) or (OH~) varies in titrating bases of ionization-constants io~ 3 , io- 6 , io- 7 , and io~ 9 with HC1 at 25, similar in all respects to the plots previously made for the four acids. Prob. 35. a. From a study of the diagram of Prob. 34 tabulate the values between which the indicator-constant must lie in order that the titration of each of the four acids and four bases may be correct within 0.2%, assuming that the indicator is 9% transformed, b. Show from the plot what percentage error would be made in using phenol phthalein (Ki = io~~ 10 ) hi titrating an acid for which K A io~ 5 when the fraction of the indicator transformed is i%, 9%, and 50%; also in titrating an acid for which J A = io- 7 when the fraction of the in- dicator transformed is 9% and 50%. c. If the acid for which JK" A = io- 9 were titrated with the aid of an indicator for which KI = io- 12 (which is not far from the value for trinitrobenzene), what error would be made when the fraction of the indicator transformed is 5%, 9%, and I 5%? (Note that hi a titration carried out in the usual way the fraction transformed is not determined more closely than this. Note also that an error in the assumed value of the indicator-constant would affect the results in the same way as a variation in the fraction transformed.) Prob. 36. Calculate the value of (H + ) at the end-point in titrating with phenolphthalein (K I = io~ 10 } when the fraction x of it trans- formed is 5% and 20%; with rosolic acid ( J fiT I = io~ 8 ) when x is 5% and 20%; with paranitrophenol (^i = io~ 7 ) when x is i% and 20%; and with methyl orange (K I = 5XicT*') when x is 80% and 95% (these being about the ranges practicable hi a titration). Draw in on the diagram made in Prob. 34 horizontal lines representing these limiting values of (H + ) for the four indicators. Letter the curves and lines on the diagram so as to show what each represents. Prob. 37. With the aid of the diagram show which of these indi- cators would give a result accurate within 0.2-0.3% in titrating a, NH 4 OH with HC1; b, HNO 2 with KOH; c, aniline (C 6 H 5 NH 3 OH) with HC1. Prob. 38. Titration of the Successive Hydrogens of Poly basic Acids. A solution 0.2 formal in H 3 P0 4 is titrated with 0.2 formal NaOH solu- tion. With the aid of the ionization-constants, calculate the H + ion concentration after the addition of enough base to be equivalent, a, to 99% of the first hydrogen of the acid, b, to i% of the second hydrogen, c, to 99% of the second hydrogen, and d, to i% of the third hydrogen. e. By reference to the diagram determine what indicators could be used for titrating the first hydrogen and the second hydrogen of the acid. REACTIONS INVOLVING SOLIDS 165 IV. THE MASS-ACTION LAW FOR REACTIONS INVOLVING SOLID PHASES 109. Form of the Mass-Action Expression. When a substance present as a solid phase is involved in a reaction with gaseous sub- stances at small pressures, that substance has in the gaseous phase of all equilibrium mixtures at any definite temperature the same pressure, namely, one equal to the vapor-pressure of the solid substance. Simi- larly, when a substance present as a solid phase is involved in a re- action with dissolved substances at small concentrations, it has in the liquid phase of all equilibrium mixtures at any definite tempera- ture the same concentration, namely, one equal to its concentration in a solution saturated with the solid substance and containing no other solutes. Hence the pressure in the gaseous phase or the con- centration in the liquid phase of any substances which are also present as solid phases may be left out in formulating the mass-action expres- sion, their constant pressures or concentrations being understood to be included in the equilibrium-constant. Thus, p*2/p&o = K is the mass- action expression for the reaction Fe(s)+H 2 O(g) = FeO(s)+H 2 (g). As here illustrated, solid, liquid, and gaseous substances occurring in chemical equations are indicated by attaching to their formulas (s), (1), and (g), respectively, when it is important to show the state of aggregation. Substances in solution are written either without any such addition to their formulas or with the addition of a parenthesis showing the concentration or composition; for example, NH 3 (o.i f.), H+Cl-(in 100 H 2 O). 110. Reactions involving Solid and Gaseous Substances. The simplest type of the reactions involving solid and gaseous substances is that in which only one substance is present in appreciable quantity in the gaseous phase. Examples of this type are: 2 Ag 2 0(s) = 4 Ag(s)+0 2 (g); CaCO 3 (s) =CaO(s)+C0 2 (g); CaS0 4 .2H 2 O(s) =CaSO 4 (s) + 2H 2 0(g). The mass-action expression for this case is simply p = K, which sig- nifies that at any definite temperature there is only one pressure of the gas at which there can be equilibrium. This pressure is called the dissociation-pressure of the substance undergoing decomposition (thus of the silver oxide, the calcium carbonate, or the gypsum). If the pressure of the gas is kept larger than this pressure, the reaction takes place completely in one direction, with the result that the gas is entirely absorbed; and if the pressure is kept smaller, the reaction 1 66 EQUILIBRIUM OF CHEMICAL CHANGES takes place completely in the other direction, with the result that the dissociating substance completely decomposes. This important char- acteristic of reactions of this type will be fully considered in Art. 119. Other types of reactions, involving equilibrium between solid substances and two or more gaseous substances, are illustrated by the following problems. Dissociation of Solid Substances Producing Two Gaseous Substances. Prob. 39. When solid NH 4 SH is -placed in a vacuous space at 25 a pressure of 500 mm. is developed, owing to the dissociation of the salt, which is itself not appreciably volatile, into NH 3 and H 2 S. If solid NH 4 SH were introduced into a space which already contained H 2 S at a pressure of 300 mm., what increase of pressure would result? Prob. 40. The equilibrium-constant (expressed in atmospheres) of the reaction 2NaHCO 3 (s) =Na 2 CO 3 (s)+CO 2 (g)+H 2 0(g) at 100 is 0.23. A current of moist carbon dioxide is passed at i atm. over solid sodium hydrogen carbonate (in order to free it from adhering water). How great must the mol-fraction of the water in the gas be to prevent decomposition of the salt? Prob. 41. When solid mercuric oxide is heated in an evacuated tube to 357 in the vapor of mercury boiling at i atm., a pressure of 70 mm. is produced in the tube owing to dissociation of the oxide into mercury vapor and oxygen. Assuming that a mixture of solid mercuric oxide and liquid mercury is heated in the same way, calculate the partial pressure of the oxygen in the tube. Ans. 0.088- mm. Prob. 42. Equilibrium of the Producer Gas Reaction. In one stage of the manufacture of producer gas a current of air at i atm. is passed through coke at a high temperature. Calculate the mol-fractions of CO 2 and of CO in the gas issuing at 1000, assuming equilibrium to be reached. The equilibrium-constant for the reaction C(s)+C0 2 (g) = 2CO(g) at 1000 is 140. Dissociation-Pressure in Relation to Equilibrium-Constants. Prob. 43. At 1 1 20 the gaseous mixture in equilibrium with iron and ferrous oxide consists of 54 mol-percent of H 2 and 46 mol-percent of H2O. Calculate the dissociation-pressure of ferrous oxide, referring to Prob. 8 for the additional data needed. Prob. 44. In the blast-furnace process iron is reduced by the reac- tion FeO(s) +C0(g) =Fe(s) + CO 2 (g). Calculate the least quantity of I carbon monoxide that could reduce one formula-weight of FeO at 1120, using the dissociation-pressure of ferrous oxide found in Prob. 43. 111. Solubility of Unionized and Slightly Ionized Substances. The fundamental principle determining the solubility of a unionized or slightly ionized substance in the presence of other solutes is that the concentration of this substance, or of its unionized part if its REACTIONS INVOLVING SOLIDS 167 ionization is not negligible, has the same value at any definite tempera- ture in any solution saturated with respect to this substance, whatever other solutes may be present at small concentrations. The application of this principle is illustrated by the following problems. Solubility-Increase Due to Complex Formation. Prob. 45. At 25 the solubility of iodine in water is 0.0013 molal, and its solubility in o.i formal KI solution is 0.0517 molal, the increase being due to the reaction K + I~+I 2 (s) =K + I 3 ~. Calculate its solubility in o.oi formal KI solution at 25. Prob. 46 Calculate the solubility of HgCl 2 in o.i formal NaCl solu- tion at 25 from the results of the distribution experiments of Prob. 44 of Art. 45 and from the fact that its solubility in water at 25 is 0.263 molal. 112. Solubility of Largely Ionized Substances. The mass-action law evidently requires, in all dilute solutions saturated with the same solid substance, that the concentration of the unionized substance have the same value, and also that the product of the ion-concentra- tions raised to the appropriate power have the same value, whatever other substances may be present (at small concentrations). Thus at any definite temperature in any aqueous solution saturated with silver sulfate, whether it is present alone or with another solute, such as silver nitrate or ammonium hydroxide, the first of these principles requires that (Ag 2 S04) have a definite value; and the second principle requires that the product (Ag + ) 2 X(SO4 = ) also have a definite value. This saturation value of the ion-concentration product is commonly called the solubility-product] and the principle expressing its con- stancy is called the solubility-product principle. The principle relating to the concentration of the unionized part cannot be applied to largely ionized substances; for the degree of ionization of such substances is so imperfectly known that it is not possible to estimate even roughly the concentration of the unionized part. In the case of largely ionized substances the solubility-product principle is therefore employed; but even this gives only approximate results, since, as shown in Art. 103, ions deviate considerably from the laws of perfect solutions even at fairly small concentrations. Never- theless, the solubility-product principle, supplemented by the hy- potheses that largely ionized substances are completely ionized and that their ions act as perfect solutes, yields roughly approximate results in the treatment of the solubility effects of uniunivalent sub- stances up to moderate concentrations, such as 0.1-0.2 formal, as 1 68 EQUILIBRIUM OF CHEMICAL CHANGES will be shown in the following problems by comparison of the observed solubility values with those calculated from the solubility in pure water. The effects of uniunivalent substances on the solubility of unibivalent substances, and the effects of unibivalent substances on the solubility of uniunivalent substances, are also in roughly approxi- mate agreement with the requirements of these principles. Very large deviations are met with, however, in evaluating the solubility of one unibivalent substance in the presence of another. Thus the effect of Ag + NO 3 ~on the solubility of Ag + 2 SO 4 = , or the effect of Ca- H -Cl~ 2 on the solubility of T1+C1~, may be estimated by these principles; but the effect of Na+ 2 SO 4 = on the solubility of Ag+ 2 SO 4 = is very different from that calculated. This very abnormal effect exerted by bivalent ions may arise either from great deviations in their behavior from that of perfect solutes, or from their conversion into intermediate ions (such as AgSO 4 ~ in the case of Ag 2 SO 4 ). In view of this great divergence it is not practicable to predict even approximately the effect of unibivalent salts on the solubility of one another, except at very small concentrations (for example, below o.oi formal). An exact method of treatment of solubility effects in the case of largely ionized substances is described in Art. 113. Solubility-Decrease by Substances with a Common Ion. Prob. 47. The solubility of AgBr0 3 in water at 25 is 0.0081 formal. Calculate its solubility in a solution 0.0085 formal in AgNO 3 , and com- pare the result with the value 0.0051 formal experimentally determined. Prob. 48. The solubility of KC1O 4 in water at 25 is 0.148 formal. Calculate its solubility in a solution o.ioo formal in KC1. Compare the result with the value 0.112 formal experimentally determined. Prob} 49. a. Derive an expression for the solubility s of AgCl in a dilute NaCl solution of concentration c in terms of its solubility s in water, b. Calculate the ratio S/SQ for c = s , for c - 2% and c - IOS Q . Prob. 50. The solubility of Ag 2 SO 4 in water at 25 is 0.0268 formal. Calculate its solubility, a, in a solution 0.050 formal in AgNO 3 , b, in a solution 0.025 formal in K 2 SO 4 , solving in each case the cubic equation by trial, c. Tabulate the results with those actually determined, which are 0.0142 and 0.0247, respectively. Prob. 51. The solubility of Mg(OH) 2 in water at 18 is i.4Xio~ 4 formal, a. Calculate its solubility in 0.002 formal NaOH solution. b. Calculate its solubility in o.ooi formal MgCl 2 solution. Prob. 52. Solubility-Increase through Metathesis. The solubility of Mg(OH) 2 in water at 18 is 1.4X10"* formal, a. Calculate its solubility in 0.002 formal NH 4 C1 solution. Neglect the concentration REACTIONS INVOLVING SOLIDS 169 of OH~ In comparison with that of NH 4 OH. b. Calculate the ratio of (OH~) to (NH 4 OH) in the saturated solution in order to test the simpli- fying assumption above made. Ans. a, 8.6Xio~ 4 . Prob. 55. Solubility-Increase through Complex-Formation. The solubility of AgCl in water at 25 is i.3oXio~ 5 formal. Calculate its solubility in o.i formal NH 3 solution. There is formed a complex ion by the reaction Ag + -f 2NH 3 =Ag(NH 3 ) 2 + , its equilibrium-constant (commonly called the complex-constant) having the value i.4Xio 7 . Ans. 0.0044. Conversion of One Solid Substance into Another. Prob. 54. The solubility of silver thiocyanate is i.2Xio~ 6 formal and that of silver bromide is 0.7 X io~ 6 formal at 25. a. Calculate the equilibrium-constant of the reaction AgSCN(s)+K + Br~=AgBr(s)-f K + SCN~ in dilute solution, b. If 8.3 g. of solid AgSCN are treated with 200 ccm. of o.i formal KBr solution, what proportion of the silver salt is converted into bromide? c. What volume of the o.i formal KBr solution would convert the solid AgSCN completely into AgBr? d. With what mixtures of potassium thiocyanate and potassium bromide could the silver thiocyanate be treated without any change taking place? Prob. 55. Determine the ratio of carbonate to hydroxide in the solution obtained by digesting at 25 a o.i formal Na 2 C0 3 solution with excess of solid Ca(OH) 2 (as in the technical process of causticizing soda). The solubility of calcium hydroxide is 0.020 formal, and the product (Ca ++ )X(C0 3 = ) has the value 3Xio~ 9 in water saturated with cal- cium carbonate. *113. The Mass-Action of Imperfect Solutes. The Concept of Activity. The mass-action law is a limiting law rigorously exact only for perfect gases or perfect solutes, but holding true with reason- able accuracy in the case of most gases at moderate pressures (such as i to 5 atmospheres), and in the case of solutes with electrically neutral molecules up to moderate concentrations (such as i molal), but show- ing large deviations in the case of ions even at small concentrations (such as o.i molal). For the mass-action and thermodynamic treatment of solutes at concentrations larger than those at which they can be regarded as perfect solutes, it has been found convenient to introduce a new concept, which will now be described. The mass-action of a perfect gas or perfect solute, as the mass- action law states, is represented by its concentration; but when a chemical substance is not a perfect gas or solute or when its concen- tration is unknown, its mass-action is expressed by the term activity (a), by which is meant that quantity which, when substituted for the concentration of the substance in mass-action equations, expresses 170 EQUILIBRIUM OF CHEMICAL CHANGES its effect in determining the equilibrium. Hence the activity of a substance represents its effective concentration from a mass-action viewpoint; and the factor by which theactual concentration c must be multiplied to give the activity is called the activity-coefficient a, that is, a = a c, or a = a/c. In accordance with these general definitions, the most obvious method of determining the relative activities of a substance in two solutions of different concentrations, or in general in any two phases, is to find its concentrations hi gaseous phases in equilibrium with the two solutions, or in general with the two phases; for the activity of a perfect gas is placed equal to its concentration, and the activity of a gas at small pressure can ordinarily be considered to be substantially equal to its concentration. This method affords a definition of activity, which is more specific and quantitative than the general statement in regard to it made above. Namely, the ratio of the activities of a substance hi two phases is equal to the ratio of its vapor- pressures in the two phases, provided the vapor can be regarded as perfect gas. That is, ai/O2=pi/pz- For example, the ratio of the activities of unionized hydrochloric acid in its 10 formal and 7 formal aqueous solutions at 25 is 30.0, since the partial vapor-pressures of the hydrochloric acid in those solutions have been found to be 4.2 and 0.14 mm. of mercury. This quantity is also the ratio of the products of the activities of hydrogen-ion and chloride-ion in the two solutions, since these products are by definition the quantities that must be substituted in the mass-action expression for the equilibrium of the reaction HC1=H + +C1~. These statements are expressed by the equations: (0HCl)l (flH* Calculation of Activity Ratios from Vapor-Pressure. Prob. 56. a. Calculate the relative activities of NH 3 and H 2 O in i.o formal and in o.i formal solution at 25 from the facts that the vapor- pressures of NH 3 in these solutions are 13.52 mm. and 1.334 mm., and that the vapor-pressures of H 2 O are those required by Raoult's law. b. Calculate the ratio of the activities of the NH 4 OH in these two solu- tions; also the ratio in each solution of the activity of the NH 4 OH to that of the NH 3 . c. Calculate the ratio of the products of the activities of the NH 4 + and OH~ ions in the two solutions. Prob. 57. a. Calculate the ratio of the activities of H 2 O in pure water and in i formal NaCl solution at o, whose mol-number (Art. 59) REACTIONS INVOLVING SOLIDS 171 is 1.79, as determined from its freezing-point, b. Find also the ratio of the product of the activities of the H + and OH~ ions in pure water to the product in i formal NaCl solution at o. This direct method is of very limited applicability to largely ionized substances, since they seldom have appreciable vapor-pressures. There is, however, a simple thermodynamic relation between activity and electromotive force (considered in Arts. 146 and 147) which furnishes a means of determining the change in the product of the activities of the ions with the concentration of the salt; and, by assuming that at very small concentrations the activity of the ions is equal to the concentration of the ion-constituent, absolute values of the product of the activities of the ions at various concentrations of the substance are obtained. The change in the activity of solutes with the concentration can also be derived from the freezing-point lowering with the aid of thermodynamic relations, and from the effect of salts on the solubility of one another. A number of substances have already been studied by these methods. The following table contains some typical values of the activity- coefficient at 25. (These values are in reality the square-root of the product of the activity-coefficients of the two ions of the substance, or the geometrical mean of the activity-coefficients of the two ions; for it is only this product that can be determined.) These values are so expressed as to be the quantity by which the concentration of the substance must be multiplied to give the activity of the ions. For comparison the values of the conductance- viscosity ratio are also given in the table. Potassium chloride and ammonium chloride have been found to have substantially the same activity-coefficients and the same conductance-viscosity-ratios as sodium chloride, at any rate up to 0.2 formal. ION-ACTIVITY COEFFICIENTS I on- A ctimty-Coefficients Conductance- Viscosity-Ratios Formality NaCl LiCl HCl KOH KCl LiCl HCl KOH O.OOI 0.98 0.98 0.99 0.98 0.98 0.99 0.99 O.OIO 0.92 0.92 0.92 0.92 0.94 0-93 0.97 0.96 0.02O 0.89 0.89 0.89 0.89 0-93 0.92 0.96 0-95 0.050 0.84 0.84 0.86 0.84 0.89 0.88 0.94 0.92 O.IOO 0.8o 0.8o 0.81 0.80 0.86 0.85 0.92 0.91 0.200 0-75 0.77 0.78 0-75 0.83 0.81 0.91 0.89 0.500 0.69 o-7S 0.76 0-73 0.78 0.77 0.89 0.88 I.OOO 0.65 0.78 0.82 0-75 0.74 0.74 0.84 0.88 3.000 0.70 i. 20 i-3S I.I 172 EQUILIBRIUM OF CHEMICAL CHANGES The table shows that with increasing concentration the ion-activity - coefficient of all four substances decreases at first much more rapidly than the conductance- viscosity-ratio. Thus at o.i formal the co- efficient is less than this ratio by 6 to 12 percent. Moreover, the coefficient, unlike the ratio, passes through a minimum at about 0.5 formal (except in the case of sodium chloride where the minimum occurs between i and 3 formal). This comparison shows quantita- tively the error in assuming that the conductance-viscosity-ratio is a measure of the ion-activity. The activity-coefficient, however, does not show, any more than the conductance-ratio, the ionization of the substance. For just as the conductance-ratio affords no means of determining the separate influences of ionization and ion-mobility, so these activity-coefficients do not show to what extent the change of activity with increasing concentration is due to decreasing ionization and to what extent it is due to an increasing deviation in the behavior of the ions from that of perfect solutes. In other words, though the activity-coefficient is an exact expression of the mass-action of the ions, it shows nothing about their concentration or about the concentration of the unionized sub- stance present with them. In most applications of the mass-action law it suffices to know the activities of the ions involved, a knowledge of the actual ionization being superfluous. In other applications where the concentration of the unionized substance must be evaluated, it is necessary to estimate the ionization. As there is no experimental basis for making this estimate, some supplementary hypothesis must be employed in these cases, for example, the hypothesis stated in Art. 79 that the ionization of largely ionized substances is complete. Applications of the Activity Principles and Comparison of the Results with Those Obtained by the Usual Methods. Prob. 58. a. Calculate the activity-coefficient of the OH ion-con- stituent, that is, the ratio of its activity to the activity or concentration of the NH 4 OH, in a solution o.i f. in NH 4 OH and 0.2 f. in NH 4 C1 at 25, assuming that the NH 4 + and Cl~ ions have equal activities in the solution, and taking i.8Xio~ 6 as the ionization-constant of NH 4 OH. b. Calculate the ionization of the NH 4 OH in the solution under the assumptions that the conductance-viscosity-ratio for NH 4 C1 is equal to its ionization and that its ions are perfect solutes, c. Calculate the ionization of the NH 4 OH under the assumptions that the NH 4 C1 is completely ionized and that the ions are perfect solutes, d. Tabulate the results of these three calculations so as to show the different conse- quences of the activity principle and of the two simplifying assumptions. REACTIONS INVOLVING SOLIDS 173 Prob. 59. When a 0.06 formal solution of C1 2 in water at o has come to equilibrium, it is found (by the method illustrated in Prob. 36 of Art. 80) that 31% of the C1 2 has undergone change according to the equation Cl 2 +H 2 O=H + Cr +HOC1. a. Calculate the equilibrium- constant of this reaction assuming that the HC1 is completely ionized and that its ions are perfect solutes, b. Calculate under these assump- tions the percentage of the C1 2 that would undergo change in a solution that is 0.06 formal in C1 2 and 0.2 formal in HC1 at the beginning, first simplifying the equation by omitting negligible quantities, c. Calcu- late the equilibrium-constant using the true activity-coefficient for HC1 obtained from the above table, d. Solve the problem stated in 6, using true activity-coefficients, e. Tabulate the results of these four calculations, so as to show the differences resulting from the use of the approximate and exact methods of treatment. Prob. 60. The solubility of T1C1 in water is 0.0161 formal at 25. Calculate its solubility hi a 0.05 formal KC1 solution under the following assumptions: a, that the two salts are completely ionized and that their ions are perfect solutes; b, that the ionization of each salt is that given by the conductance-viscosity ratio for KC1 at a concentration equal to the total salt concentration in the solution; c, that the activity-co- efficient for each salt is equal to that for KC1 at a concentration equal to the total salt concentration in the solution, d. Tabulate the results of these three calculations together with the experimentally deter- mined solubility, which is 0.0059 formal. (The deviation of the value calculated with the activity-coefficients from the observed value is probably due to inaccuracy in the assumption that the activity- coefficient for T1C1 is equal to that of KC1 at the same concentration. CHAPTER VIII EQUILIBRIUM OF CHEMICAL SYSTEMS IN RELATION TO THE PHASES PRESENT I. GENERAL CONSIDERATIONS 114. General Considerations. In this chapter are considered the principles relating to the number, state of aggregation, and composi- tion of the phases (denned as in Art. 35) which coexist in equilibrium with one another when systems composed of one substance or of two or more substances in various proportions are subjected to different temperatures and pressures. The kinds of phenomena to be considered are illustrated by the following examples. The state in which the substance water exists is determined by the temperature and pressure. Thus these conditions determine whether it exists in the form of a single phase as ice, as liquid water, or as vapor; in the form of two phases as ice and liquid water, as ice and vapor, or as liquid water and vapor; or in the form of the three phases, ice, liquid water, and vapor. So also in the case of two substances, such as carbon bisulfide and acetone, there are, as shown by the vapor-pressure-composition and boiling-point-com- position diagrams of Arts. 42 and 43, definite conditions of pressure and temperature at which any definite mixture of the two substances forms two phases a liquid phase and a vapor phase; and under these conditions the composition of each phase is also definite. A system (Art. 23) is determined by its composition, that is, by the nature and quantities of its components; the components being any substances out of which the systems under consideration can be made up, through the use of the smallest possible number of such substances. The state of a system is definite when it has certain definite phases and when the specific properties of each phase, such as its density, specific conductance, index of refraction, etc., have definite values. In order that the specific properties of any one phase of a system may be fully determined, it is necessary to specify, in addition to the propor- tions of its components, any external factors which affect these proper- ties. The only external factors which commonly have an appreciable influence are the pressure and temperature; and, in the following considerations relating to the equilibrium of phases, these factors 174 GENERAL CONSIDERATIONS 175 alone are taken into account, and their values are assumed to be uni- form throughout all the phases of a system. It is a fundamental law of the equilibrium between phases that the absolute quantity of the different phases does not influence their com- position. Thus the composition of a solution in equilibrium with a solid salt is not dependent on the quantity of the solid in contact with the solution. The composition of the vapor in equilibrium with a definite liquid solution is not dependent on the quantities of liquid and vapor in contact with one another. The proportion of hydrogen and water- vapor in equilibrium with solid iron and solid ferrous oxide is not dependent on the quantities of these solid phases in contact with the gaseous phase. The rale at which equilibrium is established between phases is, however, greatly increased by increasing the extent of the surfaces between them. The purpose of this chapter may now be more definitely stated. Its purpose is to show how the nature, the quantity, and the composi- tion of the phases of a series of systems of the same qualitative com- position vary when the composition of the system as a whole and the pressure and temperature are varied. One-component systems are first considered; then a general principle, known as the phase rule, applicable to systems with any number of components, is presented; and finally two-component and three-component systems are dis- cussed. i 7 6 EQUILIBRIUM BETWEEN PHASES H. ONE-COMPONENT SYSTEMS 115. Representation of the Equilibrium-Conditions by Diagrams. In the case of one-component systems the conditions under which the different phases exist in equilibrium with each other are fully represented by pressure- temperature diagrams; for the state of any phase of such systems is evidently determined when the pressure and temperature are specified. The figure shows a part of the temperature-pressure diagram for the component sulfur, which forms not only liquid and gaseous phases, but also two solid phases, known from their crystalline forms as rhombic and monoclinic sulfur. In the diagram the (vapor-pressure) curve AB represents the pressures at which rhombic sulfur and sulfur- vapor are in equilibrium at various temperatures; the (vapor-pressure) curve BC represents the pressures at which monoclinic sulfur and sulfur- vapor are in equilibrium at various temperatures; and the (transition-temperature) curve BE represents the temperatures at which monoclinic and rhombic sulfur are in equilibrium at various pressures. Temperatures, like these, at which two solid phases are in equilibrium with each other are called transition-temperatures. 0.05- 0.04- 0.03- 0.02- 0.01- Vapor 90 95 115 120 125 100 105 110 Temperature The point of intersection B of these three curves shows the only temperature and pressure at which rhombic sulfur, monoclinic sulfur, ONE-COMPONENT SYSTEMS 177 and sulfur-vapor are in equilibrium with one another. A point, like this, at which three phases coexist is called a triple point. The (vapor-pressure) curve CD represents the pressures at which liquid sulfur and sulfur-vapor are in equilibrium with each other at various temperatures; and the (melting-point) curve CF represents the temperatures at which monoclinic sulfur and liquid sulfur are in equilibrium at various pressures. The point C is evidently a second triple point at which monoclinic sulfur, liquid sulfur, and sulfur-vapor coexist. As indicated on the diagram, the fields between the different lines show the conditions under which sulfur exists as a single phase. Prob. i. Behavior of Sulfur Derived from the Diagram. Describe with the aid of the diagram the changes that take place, a, when sulfur is heated in an evacuated tube in contact with its vapor from 90 to 125; b, when sulfur is allowed to cool from 125 to 90, the pressure being kept constant at 0.04 mm. The curves BE and CF in the figure are very nearly vertical lines; for increase of pressure always produces a relatively small change in the transition or melting temperature. Thus the transition-tem- perature of the two solid forms of sulfur increases about 0.04, and the melting-point of monoclinic sulfur increases about 0.03, per atmosphere of pressure. With certain substances the effect of pres- sure is to decrease the transition or melting temperature; thus the melting-point of ice is lowered by 0.0076 by an increase of pressure of one atmosphere. The direction and magnitude of this effect can be derived from thermodynamic considerations, as shown in Art. 165. Unstable Forms. Prob. 2. a. To what equilibria do the curves BG, GC, and GH and the point G in the figure correspond? b. Considered with reference to these equilibria, to what form of sulfur do the fields EBGH, HGCF, and GBC correspond? c. In what sense are these equilibria unstable? Prob. j. Draw a sulfur diagram extending to pressures above the triple-point for rhombic, monoclinic, and liquid sulfur, which lies at 151 and i28oatm. Prob. 4. It will be noted that the unstable form at any temperature has the greater vapor-pressure. Prove that this must be so by showing what would happen if the two forms were placed beside each other in an evacuated apparatus. Prob. 5. Prove by a similar consideration that the unstable form must also have the greater solubility in any solvent, such as carbon bisulfide. (Note that a substance is commonly present in the same molecular form in the solutions produced by dissolving its different solid forms.) 178 EQUILIBRIUM BETWEEN PHASES How great the tendency is for a substance to remain in the same form after passing through a melting-temperature or transition-tem- perature and thus to exist in an unstable form depends in large measure on the nature of the substance. The following general statements in regard to it can, however, be made. A crystalline solid cannot as a rule be heated appreciably above its melting-point; thus, ice always melts sharply at o (under a pressure of i atm.). On the other hand, a liquid (like water) can ordinarily be cooled to a temperature con- siderably below the freezing-point if agitation and intimate contact with solid particles, especially with the stable solid phase, is avoided. Still more pronounced is the tendency of solid substances to remain in the same form upon being heated or cooled through a transition- temperature; thus rhombic sulfur can be heated to its melting-point (110), although this is about 15 higher than the transition- tempera- ture (95.5) at which it should go over into monoclinic sulfur; and monoclinic sulfur can be cooled to room temperature without going over into the rhombic form, provided this be done quickly and without agitation. The rate at which an unstable phase goes over into the stable one tends to increase with the distance from the transition- temperatures; but when the substance is below its transition-temperature this tendency may be more than compensated by the greatly reduced rate of reaction which a considerable lowering of temperature always pro- duces; thus white phosphorus is an unstable form, but the rate at which it goes over into the stable red form at room temperature is so small that it may be preserved unchanged for years; similarly, diamond is an unstable form of carbon at room temperature, but it does not go over into graphite or amorphous carbon. An effective means of causing an unstable form to go over into a stable one is to mix it intimately with the stable form. The transition may also be accelerated by moistening the mixture of the two forms with a solvent in which they are somewhat soluble. These facts are made use of in the determination of transition-temperatures. Thus the transition-temperature of sulfur has been determined by charging a bulb with a mixture of rhombic and monoclinic sulfur, filling it with carbon bisulfide and oil of turpentine, keeping it for an hour first at 95 and then at 96, and noting whether the liquid rose or fell in the capillary stem attached to the bulb. The volume was found to decrease steadily at 95 (owing to the transition of the mono- ONE-COMPONENT SYSTEMS 179 clinic into the rhombic form), and to increase steadily at 96 (owing to the reverse transition), showing that the transition-temperature lies between 95 and 96. Prob. 6. Transition in Relation to Solubility. a. Suggest an explanation of the catalytic action of the solvent in accelerating the transition of the sulfur, b. Outline a method by which the transition- temperature of sulfur could be determined by quantitative solubility measurements. i8o EQUILIBRIUM BETWEEN PHASES in. THE PHASE RULE 116. The Concept of Variance and Inductive Derivation of the Phase Rule. There will next be considered a principle, called the phase rule, which serves as a basis of classification of different types of equilibrium and enables the number of phases that exist under specified conditions to be predicted. There must, however, first be presented a concept involved in that principle that of variance. Prob. 7. Number of Factors Determining the State of One-Component Systems. Assuming that sulfur is kept at a specified pressure of 0.04 mm., state at what temperatures it is stable, a, in a single phase as rhombic sulfur, as monoclinic sulfur, and as liquid sulfur; b, in two phases, as rhombic and monoclinic sulfur, and as monoclinic and liquid sulfur; c, in the three phases, rhombic sulfur, monoclinic sulfur, and sulfur-vapor. It will be noted that, in order to determine the position on the diagram and therefore the state of the system, the values of two determining factors, namely, the values of both the pressure and the temperature, must be specified when there is only one phase; that the value of only one of these factors, either the temperature or pres- sure, need be specified when any two phases coexist; and that no condition can be arbitrarily specified when any three phases coexist. The number of determining factors whose values can and must be specified in order to determine the state of a system consisting of definite phases and components is called its variance or number of degrees of freedom; and, corresponding to the number of such factors, systems are said to be nonvariant, tmivariant, bivariant, etc. It is evident from the preceding statements that when a one-com- ponent system consists of only one phase the system is bivariant^ when it consists of two phases it is univariant, and when it consists of three phases it is nonvariant. In other words, the sum of the variance and number of phases is always three for a one-component system. Relation of Variance to Number of Phases. Prob. 8. Discuss with reference to the principle just stated and to the sulfur diagram the possibility of the coexistence, a, of the three phases, rhombic, monoclinic, and liquid sulfur; b, of the four phases, rhombic, monoclinic, liquid, and gaseous sulfur. Prob. p. Consider the following systems existing in the phases in- dicated: THE PHASE RULE 181 (1) Solution of C 2 H 6 OH and H 2 O. (2) Two solutions, Br 2 in H 2 O, and H 2 in Br 2 . (3) Two solutions, Br 2 in H 2 O, and H 2 O in Br 2 ; and their vapors. (4) Two solutions, Br 2 in H 2 O, and H 2 O in Br 2 ; their vapors; ice (5) Solution of H 2 O, C 2 H 5 OH, and HC 2 H 3 O 2 . (6) Solution of H 2 O, C 2 H 6 OH, and HC 2 H 3 O; and their vapors. a. Upon the basis of the principles already considered relating to solubility, vapor-pressure, and freezing-point, name definitely certain factors the specification of whose values will fully determine all the properties of each of these systems, b. Make a table showing the number of components, the number of phases, the variance, and the sum of the number of phases and the variance, for each of these systems. Note that the specific properties of any one phase, such as its density, specific heat-capacity, or refractive index, have definite values only when it is at some definite temperature and pressure and when it has a definite composition, which may be determined by specifying the mol- fractions of all but one of its components. These problems show that in every case the sum of the number of phases (P) and of the variance (V) i$ greater by two than the number of the components (C); that is, P-f V=C+2. This principle, which is called the phase rule, is a general one, applicable to systems consist- ing of any number of components and of any number of phases. The phase rule furnishes a basis for the classification of different types of equilibrium. It also enables the number of phases that can exist under specified conditions to be predicted. The usefulness of the phase rule itself is, however, often exaggerated. Of primary importance in the treatment of the equilibrium conditions of systems in relation to the phases present are the methods of representing those conditions by diagrams, as described in later articles of this chapter. Prob. 10. Number of Phases Permitted by the Phase Rule. Sodium carbonate and water form solid phases of the composition Na 2 CO 3 .H 2 O, Na 2 CO 3 .7H 2 O, and Na 2 CO 3 .ioH 2 O. a. State how many of these hydrates could exist in equilibrium with the solution and ice under a pressure of i atm. b. State how many of these hydrates could exist in equilibrium with water-vapor at 30. *117. Discussion of the Concept of Components. The substances to be regarded as the components, and correspondingly the number of components, are obvious in cases, like those already considered, in which there is no transformation of the components into other chemical substances. In cases where a chemical action occurs, the term com- ponent requires to be more specifically defined, which may be done 1 82 EQUILIBRIUM BETWEEN PHASES as follows. It is evident from the preceding derivation of the phase rule (and from that in Art. 118) that in general any substances may be selected as the components, the specification of the relative quantities of which in any phase suffices to determine fully the composition of that phase; the components, moreover, being such that this is true whatever phase of the system is considered. This definition is illus- trated by the following examples. The composition of a gaseous phase containing hydrogen, oxygen, and water-vapor in equilibrium with one another at any definite temperature and pressure is fully determined by specifying the relative quantities of total hydrogen and total oxygen present, either as the elementary substances or as constituents of the water; for then the proportions of the three substances adjust themselves so as to satisfy the conditions of the equilibrium between them, for example, so as to satisfy the mass-action law if the substances are perfect gases. The number of components in this case is therefore two. The composition of a gaseous phase containing hydrogen, oxygen, and water-vapor at low temperatures where there is no reaction be- tween the substances is evidently determined only when the relative quantities of hydrogen, oxygen, and water in the phase are specified. The number of components is therefore three under these conditions. It is obvious from these examples that the number of components is determined not only by the substances present, but also by the equi- libria that are established between them. It is therefore advisable to write chemical equations expressing all the equilibria actually established, so as to make sure that the proportion of every chemical substance present in any phase would be determined by specifying that definite relative quantities of the components adopted are present in that phase, and to make sure, on the other hand, that more com- ponents have not been adopted than are necessary. Another aspect of the matter may be considered. In a gaseous phase where hydrogen, oxygen, and water are in chemical equilibrium, one particular system could be produced by taking water as the only component; but the phase considerations of this chapter have refer- ence always to a series of systems of the same qualitative, but varying quantitative composition; and such a series containing hydrogen, oxygen, and water-vapor in every possible proportion cannot be pro- duced out of water alone, but can be produced out of hydrogen and oxygen. These systems are therefore properly regarded as consisting THE PHASE RULE 183 of two components. As another example consider a gaseous phase containing HC1, O 2 , C1 2 , and H 2 O. By taking HC1 and O 2 as two components, the other two substances can be produced out of them by the reaction O 2 +4HC1 = 2C1 2 +2H 2 O, but only in equivalent proportions. To produce a system containing the four substances in any proportion whatsoever, it is necessary to make use of a third component, either C1 2 or H 2 O. Thus, either by adding C1 2 to a mixture containing O 2 and HC1 in any proportions and C1 2 and H 2 O in equiva- lent proportions, or by removing C1 2 from such a mixture, any composi- tion-whatever can be secured. The systems are therefore said to consist of three components. It may be further noted that, though the number of components is fixed, there is a certain arbitrariness in their selection. Thus, although in this example O 2 , HC1, and C1 2 have been used as the components, any other three of the four sub- stances might be employed. It is, however, usual to select as com- ponents the simplest substances. Determination of Components. Prob. n. Specify the components which will produce phases containing the following chemical substances, assuming chemical equilibrium to be established between them. In answering this ques- tion, write chemical equations expressing all the equilibria that may be established between these substances, making sure that all the substances can be produced in any proportion from the specified compo- nents, and that all those components are necessary, a. Gaseous H 2 , O 2 , CO, CO 2 , H 2 O. b. Solution containing H 2 O, (H 2 O) 2 , NaC1.2H 2 O, NaCl, Na + , Cl~. c. Solution containing H 2 O, NH 4 CN, NH 4 + , CN~, NH 3 , NH 4 OH, HCN. Prob. 12. Specify the components of the systems that exist in the following groups of phases, assuming chemical equilibrium to be estab- lished between the substances named: a. Solid NH 4 C1, gaseous NH 3 and HC1. b. Solid CaCO 3 , solid CaO, gaseous CO 2 . c. Solid MgSO 4 -7H 2 O, solid MgSO 4 .6H 2 O, solution of Mg ++ SO 4 = in water, water-vapor. d. Solid carbon; gaseous H 2 O, H 2 , CO, and CO 2 . e. Solid iron, solid FeO, gaseous CO and CO 2 . /. Solid iron, solid FeO, solid carbon, gaseous CO and CO 2 . *118. Derivation of the Phase Rule from the Perpetual-Motion Principle. In order to determine fully the state of one-phase systems consisting of any number C of components (as defined in Art. 117), evidently the composition of the phase and in addition any external factors that determine its properties must be specified. The compo- sition of the phase is fully determined by specifying the mol-fractions of all but one of the components, that is, by specifying C i quanti- 1 84 EQUILIBRIUM BETWEEN PHASES ties. The only external factors which commonly affect the properties of a phase of specified composition are temperature and pressure; but in special cases certain other factors, some of which are mentioned below, have an appreciable influence. Representing the number of such external determining factors by n, the variance or total number of the independent variables that must be specified is Ci+n. In any system, therefore, in which the number of phases P is one, the sum of the number of phases and the variance V is equal toC + n; that is, P+V = C+w; or, for the common case where pressure and temperature are the only external determining factors, P+V = C + 2. For example, the state of one-phase systems consisting of three com- ponents (i, 2, 3) is fully determined by specifying four quantities, namely, the mol-fractions (x\ and x%) of any two of the components and the pressure (p) and temperature (7 1 ); hence any specific property whatever of the system, such as its density d, can be expressed as some function of these four variables, such as d = f(xi, xz, p, T). In the general case, where the number of components is C and the number of external factors is n, the function becomes d = i(x i} X2,. .x c -i,p, T,. .) The derivation of the phase-rule now consists in showing that the equation P -f V = C + n, which for one-phase systems is a result of the definitions of components and of external factors, still holds true whatever be the number of phases. In order that this may be so, it is evidently necessary only that each new phase introduced into the system shall dimmish the variance by one; for then P+V will still have its former value C+n. That this is the case can be shown as follows. Consider that a system of C components (i, 2, 3,. .C) exists as a gaseous phase, and that a new liquid or solid phase is developed in it (for example, by varying the pressure or temperature). Now the partial pressures (pi, p 2 ,. .p c ) of the separate components in the gaseous phase are, like any other property of the phase, functions only of the mol-fractions (xi, x 2) . .x c -i) and of the external factors (p, T,. .). This conclusion may be expressed mathematically as follows: pi=ii(xi, xz,. .xc-i,p, T,. .). p2 = i*(xi, xz,. .x c -i, p, T,. .). THE PHASE RULE 185 When the new, liquid or solid, phase is present in equilibrium with the gaseous phase, the partial pressure of each component in the gaseous phase is determined by the mol-fractions (#/, 00%, . .x' c -i) of the liquid or solid phase, by the temperature, and by any other external determining factor which has an appreciable influence; for, if a liquid or solid phase of such composition as to be in equilibrium with the gase- ous phase could also be in equilibrium at the same temperature with some other gaseous phase with different partial pressures, perpetual motion of the kind described in Art. 44 could be realized. This con- clusion from the perpetual-motion principle that the partial pressures must be fully determined by the mol-fractions in the liquid phase and by the external factors may be expressed mathematically as follows: p c =f c '(xi',x2 f ,..x' c -i,p, r,..). By equating these two sets of expressions for pi, p%,. .p c , the follow- ing functional relations between the mol-fractions in the gaseous phase and those in the liquid phase are obtained: fife, *2,. .sc-i, p, T r .) = f/fo', xj,. .*' c -i, p, TV .). ft(xi, x*,. .c-i, p, T,. .) = fc'(*i', **',. .*'c-i, p, T,. .). fc(*i, *2,. .sc-i, P, T,. .) = f c '(*i', a*',. .*'c-i, p, T,. .). That is, the new phase gives us C new functional equations in which only C - 1 new variables (namely, xi, x*, . . x' c -i) are introduced. By combining these new equations with one another the new variables may evidently be eliminated, yielding a functional relation of the form: fi(*i,a*,. .x c -i,p, T,. .)=o. This is obviously a relation between the Ci+n variables which determine any property of the gaseous phase, in the way described in the first paragraph of this article. Hence the number of these vari- ables whose values can and must be specified to determine the proper- ties of the gaseous phase is one less than it was when that phase was alone present. And in a similar way this can be shown to be true also of the properties of the liquid or solid phase. In other words, the new phase diminishes the variance of the system by one, and the sum p_j_V retains the value C + n which it had when the system consisted 1 86 EQUILIBRIUM BETWEEN PHASES of a single phase. And evidently, since each additional phase formed within the system will similarly decrease the variance by one, the sum P+V will always have the same value C+n. In the above derivation it was assumed that a gaseous phase was present in the system. It will be noted, however, that the partial pressures in the gaseous phase were employed only as a means of deriv- ing functional relations between the mol-fractions of the components in two different phases, and that the partial pressures disappeared in these relations. This indicates, and it can be rigorously shown, that functional relations between the mol-fractions of the same form as those given above hold true for any pair of phases; for example, for two liquid phases. It follows, therefore, that the phase rule is applicable to any kind of system whatever, in the form P+V = C+w. As has been stated, the value of n in the above derived expression of the phase rule is commonly 2; for the only external factors that ordinarily influence appreciably the state of the system are temper- ature and pressure. In some cases, however, other factors come into play. For example, this is sometimes true of intensity of illumination or of electric or magnetic field. Thus illumination of silver chloride increases its dissociation-pressure; and under certain conditions an electric discharge through an equilibrium mixture of nitric oxide, nitrogen, and oxygen increases the proportion of nitric oxide in the mixture. Another factor which makes the value of n greater than 2 is introduced when different pressures are applied by means of semi- permeable walls to different phases of the system. Thus the pressure of the vapor in equilibrium with a liquid is progressively increased (as shown in Art. 51) when the liquid is subjected to an increasing pressure by means of a piston permeable for the vapor only. A common case of this kind is that where the atmosphere acts as such a piston, exerting a pressure on the liquid and solid phases of the system, but not on the components in the vapor phase. TWO-COMPONENT SYSTEMS 187 IV. TWO-COMPONENT SYSTEMS 119. Systems with Solid and Gaseous Phases. The equilibrium of systems consisting of solid and gaseous phases at constant temper- ature has already been considered from the mass-action viewpoint in Art. no. Their equilibrium in relation to temperature and pressure will now be considered from the phase-rule viewpoint. When a two-component system consists of three phases, the phase rule evidently shows that the specification of one of the determining factors (for example, the temperature) fixes the state of the system and therefore the values of the other factors (for example, of the pres- sure and of the composition of the liquid or the gaseous phase). The pressures at which the different groups of three phases exist in equi- librium at various temperatures can therefore be represented by lines on a diagram in which the pressure and temperature are the coordinates. Dissociation-Pressure and its Relation to Temperature. Prob. 13. Calcium carbonate dissociates at high temperatures into CaO and CO 2 . a. Show from the phase rule that CaCO 8 can be heated in CO 2 gas at a given pressure through a certain range of temperature without any decomposition taking place, b. Show also that there is one temperature, and only one, at which a mixture of CaCOs and CaO can be kept under CO 2 at the given pressure without any change occurring. Prob. 14. The dissociation-pressure of solid calcium carbonate is 10 180 320 580 760 looo mm. at 600 800 840 880 896 910 a. Construct a pressure-temperature diagram for this two-component system, lettering the fields so as to show what phases coexist at different temperatures and pressures, b. State at what temperature the solid begins to dissociate when it is heated in air free from carbon dioxide. c. State at what temperature it would dissociate completely when heated in a covered crucible (so that there is equalization of the pressure, but no circulation of air into the crucible), d. In lime-burning state what temperature would have to be maintained in the kiln if there were no circulation of gases through it. e. State what temperature would have to be maintained if the coal used as fuel were burned to carbon dioxide with the minimum quantity of air and the combustion-products were passed up through the kiln. Prob. 15. The dissociation pressure of silver oxide is o.i atm. at 1 16, 0.2 atm. at 132, i.o atm. at 175, and 2.0 atm. at 197. a. If finely divided silver be heated in the air, what proportion of it will be finally converted into oxide when the temperature is 130? when it is 140? b. How could silver oxide be heated to 1 70 without any decomposition? Prob. 16. Describe a method by which pure oxygen can be prepared from the air with the aid of the reaction 2BaO 2 (s) = 2BaO(s)+O 2 (g). i88 EQUILIBRIUM BETWEEN PHASES Prob. 17. When a precipitate of hydrated manganese dioxide is ig- nited in the air it comes to a constant weight corresponding to the com- position MnO 2 when the temperature is 450, and to another constant weight corresponding to the composition Mn 2 O 3 when the temperature is 500; but when ignited in oxygen it changes to MnO 2 even at 500. State what these facts show as to the dissociation-pressure of MnO 2 into Mn 2 O 3 , and that of Mn 2 O 3 into a lower oxide. Prob. 18. A method has been suggested for the standardization of sulfuric acid solutions which consists in adding an excess of ammonia solution, evaporating, drying the residue at 100, and weighing it. In drying the salt some decomposition according to the reaction (NH 4 ) 2 SO4(s)=NH 4 HSO4(s)-fNH 3 (g) is likely to take place. How might the process be modified so as to hasten the drying and yet en- tirely prevent the decomposition? 120. Systems with Solid, Liquid, and Gaseous Phases. Pressure- Temperature Diagrams. The two-component systems that involve salt-hydrates are of especial importance. The vapor-pressure rela- tions of these substances at different temperatures may be repre- sented by pressure-temperature diagrams, like that shown on the next page. This diagram has reference to systems consisting of the two components disodium hydrogen phosphate (Na 2 HPO4) and water (H 2 O), for the case that the vapor-phase (V), which consists only of water- vapor, is always present. These two components form the following solid phases: ice (I), anhydrous salt (A), dihydrate Na2HPO 4 .2H 2 O (AWj), heptahydrate Na 2 HPO 4 .7H 2 O (AW 7 ), and dodecahydrate Na 2 HPO 4 .i2H 2 O (AWja); also a solution-phase (S) of variable composition, approaching as one limit pure water, whose vapor-pressure is represented by the uppermost curve in the diagram. The phase rule evidently requires that in systems of this kind, in addition to the vapor-phase, two solid phases, or one solid and one liquid phase, must be present in order that the pressure of the vapor may have a definite value at any definite temperature. Prob. IQ. Interpretation of the Vapor-PressureTemperature Dia- gram for Salt-Hydrates. At 30 one formula- weight of Na 2 HPO 4 is placed in contact with a large volume of water-vapor at i mm., and the volume of the vapor is steadily diminished (so slowly that equi- librium is established) until finally there remains hi contact with the vapor only the saturated solution (which contains the components in the ratio iNa 2 HPO 4 : 33.2H 2 O). State, with the aid of the figure, the changes that take place in the pressure of the vapor and the accompany- ing changes that take place in the character of the other phases in contact with it, throughout the whole process. TWO-COMPONENT SYSTEMS 30 Temperature 40 Prob. 20. Vapor-Pressure-Composition Diagram for Salt-Hy- drates. a. Plot the pressure of the vapor (as ordinates) against the number of formula-weights of water absorbed by the salt as abscissas for the process described in Prob. 19 (up to the point where 15 formula- weights have been absorbed). Mark the lines on the plot so as to show what phases are present during each stage of the process, b. Make a similar plot for the case that the process described in Prob. 19 takes place at 38 (instead of at 30). Prob. 21. Experimental Determination of the Hydrates Formed by a Salt. Describe an experimental method of determining what hydrates of copper sulfate exist at 25. Prob. 22. Transition-Temperature of Salt-Hydrates. Describe with the aid of the figure what changes take place when a mixture of heptahydrate and dodecahydrate is heated from 30 to 38 in a sealed tube previously evacuated. I go EQUILIBRIUM BETWEEN PHASES Prob. 23. Conditions under which Salts are Efflorescent or Hygro- scopic. At 30 moist air is in contact with solid Na 2 HPO 4 .7H 2 O. Under what conditions of humidity, a, would the salt remain unchanged? b, would it lose water? c, would it absorb water? (By the humidity of a gas is meant the ratio of the pressure of the water- vapor in the gas to the vapor-pressure of water at the same temperature.) Prob. 24. Drying of Hydrates without Decomposition. a. State the conditions under which moist crystals of Na 2 HPO 4 .i2H 2 O could be completely dried at 30 without danger of decomposition, b. Describe a method by which these conditions could be practically realized. Separation of Hydrates from Solutions. Prob. 25. State what solid phase separates on evaporating a dilute solution of Na 2 HPO 4 , a, at 30, b, at 38. Prob. 26. The equilibrium-pressures for the reaction CaSO 4 .2H 2 O (gypsum) =CaSO 4 (anhydrite) +2H 2 O(g) and the vapor-pressures of pure water at various temperatures are as follows: Temperature . . 50 55 60 65 Gypsum ... 80 109 149 204 mm. Water .... 92 118 149 188 mm. The solubility of calcium sulfate is so small that the vapor-pressure of its saturated solution may be considered to be identical with that of water, a. State what happens on heating gypsum from 50 to 65 in a sealed tube previously evacuated, b. State what solid phase separates when a solution of calcium sulfate is evaporated at 55, and at 65. c. State what solid phase would separate upon evaporating the solution at 55 if, when it became saturated, enough calcium chloride were added to reduce its vapor-pressure by 10%. Give the reasons in each case. Prob. 27. Dehydration of Organic Liquids by Salts. Determine the extent to which the mol-percent of water in isoamyl alcohol can be reduced by shaking at 25 isoamyl alcohol saturated with water with a large excess of each of the following substances: a, Na 2 HPO 4 .i2H 2 O; b t Na 2 HPO 4 .7H 2 O; c, Na 2 HPO 4 .2H 2 O; d, Na 2 HPO 4 . For the data refer to the diagrams of this article and of Prob. 47, Art. 47. ..121. Systems with Solid and Liquid Phases. Temperature- Composition Diagrams. In a two-component system the composi- tion of a solution which coexists with its vapor and a solid phase is evidently fully determined by specification of the temperature, the system being univariant. The relation between this composition and the temperature can therefore be represented by a diagram in which these factors are plotted as coordinates. It is evident that the equilibrium conditions of a two-component system can still be represented by a temperature-composition diagram when the specification that the vapor-phase is present is replaced by TWO-COMPONENT SYSTEMS 191 the specification that the pressure has some definite value (greater than that at which the vapor can exist). Moreover, since pressure has, as illustrated in Art. 115, only a small effect on equilibria in which solid and liquid phases are alone involved, the lines on the temperature-composition diagram have substantially the same posi- tion when the system is under a pressure of one atmosphere as they do when it is under the pressure of the vapor. And in practice com- position-temperature diagrams are ordinarily constructed from data determined under the atmospheric pressure. The form of the temperature-composition diagram varies greatly with the character of the solid phases which the components are capable of producing. The simplest type of such a diagram is that in which the two components A and B do not form any solid compound with each other, but separate from the solution in the pure state. This type is illustrated by the figure, which shows the complete dia- gram for systems composed of acetic acid (A) and benzene (B) at one atmosphere. 10 90 100 Composition The freezing-point curve CD represents the composition of the solutions (S) which are in equilibrium with solid A, and the freezing- point curve DF represents the composition of the solutions which are in equilibrium with solid B at different temperatures. The point D at 1 92 EQUILIBRIUM BETWEEN PHASES which the solution is in equilibrium with the two solid phases A and B is called the eutectic point. When a solution in the condition corre- sponding to this point is cooled, it solidifies completely without change of composition or temperature to a mixture of the two solid phases A and B. This mixture is usually so fine-grained and intimate that it differs markedly in texture from ordinary mixtures of the same solid phases. It is called the eutectic mixture, or simply the eutectic. Prob. 28. Eutectics in Relation to the Phase Rule. Show that the phase rule requires that a solution at the eutectic point solidify without change of temperature or composition when heat is withdrawn from it. Prob. 2Q. Effect of Cooling a Solution Predicted from the Diagram. A tube containing a solution of 75% of benzene and 25% of acetic acid at 10 is placed within an air jacket surrounded by a freezing mix- ture at 20, so that the system slowly loses heat, till its temperature falls to 10. Predict with the aid of the figure the values of the tem- perature and composition of the solution at which any phase appears or disappears. State also the character of the solid mixture finally obtained. Prob. jo. Cooling Curves. a. On a diagram having as ordinates the temperatures in degrees and as abscissas the time of cooling in arbitrary units draw curves representing in a general way the rate at which the temperature decreases when a solution of 75% of benzene and 25% acetic acid is cooled as described in Prob. 29, assuming, first, that the liquid overcools without the separation of any solid phase; and assuming, secondly, that the solid phases separate so that there is always equilibrium. Take into account the fact that on cooling a system there is always an evolution of heat whenever a new phase separates. b. Draw on the same diagram, at the right of these curves, a new cool- ing curve showing how the temperature changes when pure benzene is cooled from 10 to 10. c. Draw a cooling curve also for the case that a solut'on of 64% of benzene and 36% of acetic acid is cooled from 10 to 10. The fields in the diagram also become of much significance, when the composition represented by the abscissas is understood to be that of the whole system not merely that of the liquid phase. Thus, when the system consists of the substances in such a proportion (thus, 25% of B to 75% of A) and is at such a temperature (thus, o) that its condition is represented by a point n within the field CDG, the diagram shows that it consists of the two phases, solid A and solution S of the compo- sition (corresponding to the point p) at which these two phases are in equilibrium at the given temperature. Similarly, when the system has a composition and temperature corresponding to any point within the field GDJH, it consists of solid A and the eutectic mixture E, TWO-COMPONENT SYSTEMS 193 which always has the composition corresponding to the point D. It can readily be shown, moreover, that, when the state of the system is represented by the point n, the weight of the solid A is to the weight of the solution present as the length of the line joining n and p is to the length of the line joining m and n; and similarly, that at any point below r on the same ordinate with it the weight of pure A is to the weight of the eutectic mixture present as the length of the line rD is to that of the line Gr. Prob. 31. Nature of the Solid Phases. How do the two rectangular fields at the bottom of the figure differ with respect to, a, the phases present; b, the texture of the mixture? Prob. 32. Relative Weights of the Separate Phases. Show that the relation stated in the last sentence of the preceding text is true. Prob. jj. Eutectic Composition and Temperature in Relation to the Laws of Freezing-Point Lowering. Lead (which melts at 327) and silver (which melts at 960) form a eutectic which melts at 304. The heat absorbed by the fusion of one atomic weight of lead is 1340 cal. Calculate the composition of the eutectic, taking into account the facts that the first part of a freezing-point curve can be located with the aid of the laws of perfect solutions, and that the molecules of metallic elements in dilute metallic solutions are as a rule identical with their atoms. Compare the calculated composition of the eutectic with that (4.7 Ag to 95.3 Pb) derived from cooling curves. Prob. 34. Metallurgical Process of Enriching Lead-Silver Alloys Resulting from the Reduction of Ores. A technical process of enriching lead-silver alloys has been based on the facts stated in Prob. 33. De- scribe how this process might be applied to a melt containing i at. wt. of silver and 99 at. wts. of lead, and state the extent to which the per- centage of silver could be thereby increased. The types of temperature-composition diagrams to be next con- sidered are those for systems of which the components form one or more compounds which separate as solid phases. The simplest case, illustrated by Prob. 35, is that in which the solid compounds melt on heating. Another important case, illustrated by Prob. 36, is that in which a solid compound separates which, when heated, instead of melting, undergoes transition into another solid compound and a liquid phase. The diagrams for such systems may be constructed either from cooling curves or from the results of solubility and freezing-point measurements. The former method is used in the case of alloys and other high-melting systems (as in Prob. 35); the latter, in the case of systems involving solutions at moderate temperatures (as in Prob. 36). 1 94 EQUILIBRIUM BETWEEN PHASES Prob. 35. Construction of Temperature-Composition Diagrams from Cooling Curves. The figure shows the cooling curves for a series of mixtures of magnesium and lead containing the atomic percentages of lead shown by the numbers at the tops of the curves. On a diagram whose coordinates are temperature and atomic percentages plot points representing the temperature at which the solidification of each mixture begins and ends. Draw in solid lines representing the freezing-point curves. Draw also dotted lines limiting the different fields (as was done in the benzene-acetic-acid diagram), and letter the fields so as to show of what the system consists in each field. 700- 650- 600- 550- 500' 450- 400- 350 300- 250- 25 33.3 40 50 100 -700 -650 -600 -550 500 450 -400 -350 -300 250 Time Construction of a Temperature-Composition Diagram for Salt Hydrates. Prob. 36. Construct a temperature-composition diagram for the sys- tem composed of Na 2 HP0 4 and H 2 O by plotting the following values of the percentage (100 x) of Na 2 HPO 4 in the saturated solution as abscissas against the temperature (/) as ordinates. The solid phase which is in equilibrium with the solution (S) at a pressure of i atm. is indicated by the letters above the data. IOOX Temp. 100* Temp. Ice AWu AWn AWn AWn I.I 2.4 5-5 IQ.2 30.0 0-5 15 30 35 AWi AWi AWt AW 2 AW i 38.0 43-5 44-7 46.6 48.0 43 48 50 55 60 AW-, 33-7 38 TWO-COMPONENT SYSTEMS 195 Draw in on the diagram lines representing the equilibrium conditions, and mark each line with letters indicating the phases which coexist under the conditions represented by the line; and at each triple point designate the phases that coexist there, the temperature, and the composition of the solution. Prob. 37. As a means of determining what the fields hi the dia- gram signify with respect to the phases present, proceed as follows. a. State the solid phases that separate, and the way in which the com- position of the residual solution changes, on cooling from 55 to 5 at i atm. solutions containing i% and 20% of Na 2 HPO 4 . Draw in on the diagram lines bounding the fields within which the phases ice I and solution S coexist, within which AWi 2 and solution coexist, and within which ice and AWi 2 coexist; and letter the fields correspondingly. (Note that at any definite temperature the composition limits of any such field are the compositions of the two phases that are in equilibrium at that temperature, since these are the limiting compositions of mix- tures that can be made up of these two phases.) b. From a similar consideration deduce and state the results of cooling from 55 to 5 solutions containing 36% and 42% of Na 2 HPO 4 . Draw in on the diagram lines bounding the fields within which the phases AW 7 and solution S coexist, and within which the phases AW i2 and AW 7 coexist. (Note that, when the triple point is reached, the phase AW ]2 separates, thereby tending to decrease further the percentage of Na 2 HPO 4 in the solution; but, since the composition must remain constant so long as three phases are present, the phase AW 7 re-dissolves; and this process continues as heat is withdrawn either till the phase AW 7 is consumed or till the solution dries up.) c. Finally state the results of cooling from 55 to 5 a solution containing 46% of Na 2 HPO 4 . Draw in on the diagram lines bounding the fields within which the phases AW 2 and solution S coexist, and within which the phases AW 7 and AW 2 coexist. Prob. 38. Preparation of Different Hydrates by Evaporation at Different Temperatures. Determine from the diagram within what limits of temperature a solution of Na 2 HP0 4 must be evaporated in order to obtain in the pure state, a, Na 2 HPO 4 .i2H 2 O; 6, Na 2 HPO 4 .7H 2 O; c, Na?HPO 4 .2H 2 O. (The conditions under which such salt hydrates can be dried without change in composition were considered in Prob. 24). Conditions of Unstable Equilibrium. Prob. 39. a. By extending by dotted lines the appropriate curves on the diagram, determine the solubilities, in terms of the percentage of anhydrous salt in the solution, of AW 7 at 45 and 50, and of AW 2 at 45 and 50. b. Prove that in contact with the solution AW 7 is unstable at 50 with respect to AW 2 by showing with the aid of these solubility values what happens when AW 7 is placed in contact with a solution saturated at 50 with respect to AW 2 . c. Show that at 45 the more soluble hydrate is again the unstable one in contact with the solution. 196 EQUILIBRIUM BETWEEN PHASES Prob. 40. By extending the appropriate curves on the diagram predict, a, at what temperature salt would begin to separate on cooling a 30% Na 2 HPO 4 solution in case the stable hydrate AW J2 separates, and in case the solution remains supersaturated with respect to this hydrate; b, at what temperature the hydrate AWi 2 would melt if on heating its transition into AW 7 and solution did not take place. Prob. 41. Correlation of Vapor-Pressure-Temperature and Temper- ature-Composition Diagrams. With the aid of the diagram under consideration and that of Art. 120 determine and tabulate for the temperatures 20, 25, 30, 35.5 and 40 the vapor-pressures and per- centage compositions of the saturated solutions and the nature of the solid phases with respect to which the solutions are saturated. (Note that the effect of pressure on the solubility is here neglected.) Brief consideration may also be given to systems involving two liquid phases as well as solid phases. Such systems are occasionally met with among alloys, and frequently among non-metallic substances, such as bromine and water, or isoamyl alcohol and water, which are partially miscible with each other. Their relations are illustrated by the accompanying diagram for the two components lead and zinc. In this diagram the curves CD and DE represent the compositions, expressed in atomic percentages, of the two liquid phases in equi- librium with each other, and the curve BC represents the composition of solutions in equilibrium with zinc. Pb 100 90 80 70 60 50 40 30 20 10 Zn 10 20 30 40 50 60 70 80 90 100 200 U TWO-COMPONENT SYSTEMS 197 Prob. 42. Interpretation of Diagrams for Systems with Solid Phases and Two Liquid Phases. a. With the aid of the diagram state all that would happen on cooling from 1000 to 300 alloys containing 40 and 90 atomic percents of zinc. b. State all that would happen on gradually adding zinc to molten lead at 350, at 700, and at 1000. c. Specify the phases in which the system exists when its composition and temperature are represented by any point within each of the fields, ABF, FBJH, CDE, BCEG, and BGKJ. 122. Systems with Liquid and Gaseous Phases. Systems of two components having a gaseous phase and a liquid phase in which the components are miscible in all proportions have already been con- sidered in Arts. 42 and 43. The boiling-point-composition diagrams of Art. 43, which are temperature-composition diagrams at one atmos- phere, are further considered in Prob. 43 with reference to the signifi- cance of the fields. The vapor-pressure relations of systems whose components have only limited reusability in the liquid state were briefly considered in Art. 47. Prob. 43. Temperature-Composition Diagrams. Reproduce in the form of separate sketches the three diagrams in the figure of Art. 43, and letter each of the fields so as to show the phases in which the system exists when its temperature and composition are represented by any point within the field. 123. Systems Involving Solid Solutions. The components some- times separate from a liquid solution in the form of solid solutions, instead of in the form of the pure solid substances or of solid com- pounds of them. By solid solutions are meant physically homogeneous solid mixtures of two or more substances, that is, solid mixtures which contain no larger aggregates than the molecules of the substances. In their equilibrium relations solid solutions closely resemble liquid solutions; differing from them mainly in the respect that the equi- librium conditions are less readily established, owing to the restricted molecular motion characteristic of the solid state. It is often true in solid systems that equilibrium is not established and that the solid phases fail to attain the uniform composition corresponding to equilib- rium, unless the systems are kept for a time at a high temperature or at a temperature not far below the melting-point, where the molecules still retain sufficient mobility. The purpose of the process of temper- ing or annealing solid substances is to allow time for the establishment of equilibrium at temperatures so high that it may be rapidly attained. In solid solutions in which equilibrium is really attained, each 198 EQUILIBRIUM BETWEEN PHASES component lowers, as in liquid solutions, the vapor-pressure, solution- tendency, and mass-action or activity of the other component. As a result of these effects the freezing-point of a substance may be either raised or lowered by the addition of another substance, according as the solid solution which separates is more or less concentrated than the liquid solution, as is shown by the following problem. Prob. 44. Raising of Freezing-Point by Solutes. The addition of benzene to thiophene raises the freezing-point of the thiophene, owing to the formation of solid solutions, a. Show how this can be accounted for by sketching a diagram, similar to that of Art. 48, representing the vapor-pressures of pure thiophene in the liquid and solid states and of a liquid solution of benzene in thiophene, and draw in on the sketch a line showing the vapor-pressure of thiophene at various temperatures in a solid solution of benzene and thiophene of the composition of that which separates from the liquid solution, b. Show that the mol-fraction of benzene must be greater in the solid solution than in the liquid solution in order that the freezing-point may be raised, provided Raoult's law holds for the solid solution as usually seems to be approximately true when the proportion of solute is small. The two components are sometimes soluble in each other in the solid state in all proportions, forming a complete series of solid solu- tions. In most cases, however, each component has only a limited solubility in the other solid; so that two series of solid solutions result, each covering only a limited range of composition. The temperature-composition diagrams representing the equi- librium conditions between solid solutions and liquid solutions corre- spond completely with those for liquid solutions and their vapors, since in both cases there are involved two phases of variable composi- tion. Thus systems in which there are solid solutions containing the components in all proportions may have diagrams of any of the three types represented in the figure of Art. 43 ; two of which are illustrated by Probs. 45 and 46. Systems in which the components have only partial miscibility in the solid state are illustrated by Prob. 47, the diagram of which represents also certain other effects which have been already considered. Alloys with Complete Series of Solid Solutions. Prob. 45. The cooling curves show that on cooling molten mixtures of nickel and copper solidification begins and becomes complete at the following temperatures, a solid solution separating in each case: Percentage of nickel o 10 40 70 100 Solidification begins 1083 1140 1270 1375 1452 Solidification ends 1083 1100 1185 1310 1452 TWO-COMPONENT SYSTEMS 199 a. Draw on a composition-temperature diagram two continuous curves corresponding to these data. Letter each field so as to show of what the system consists at any point, b. State what happens on slowly cool- ing a 50% mixture from 1400 to 1200, giving the compositions of the liquid and solid solutions in equilibrium with each other at the temperatures at which solidification begins and ends, and at 1275. (Note that the compositions of the liquid and solid solutions at any temperature are given by the points corresponding to that temperature on the upper and lower curves, since at any temperature the composi- tion of the liquid phase is identical with that of a system from which at that temperature solid just begins to separate, and the composition of the solid phase is identical with that of a system which at that tem- perature just completely solidifies.) Prob. 46. Gold and copper form a complete series of solid solutions. The mixture containing 60 atomic percent gold has a constant melting- point of 880. Gold melts at 1063, and copper at 1083. Sketch the temperature-composition diagram. Prob. 47. Alloys Forming Compounds and Limited Series of Solid Solutions. In the figure, constituting the diagram for silver-magne- sium alloys, in which the composition is expressed in atomic percentages, lOOOn 900- 400-p 1 -! '"i ''i r Aq 10 20 30 40 50 60 70 80 90 100 Mg 100 90 80 70 60 50 40 30 20 10 400^ the curves KD, DL, and MF represent the composition of solid solu- tions in equilibrium with the liquid solutions whose composition is represented by the curves CD, DE, and EF, respectively, a. State what equilibria are represented by the curves AB and BC. b. State what compounds are indicated by the diagram, c-h. State what 200 EQUILIBRIUM BETWEEN PHASES happens on cooling slowly till complete solidification results a liquid mixture containing the following atomic percentages of silver: c, go; d, 70; e, 55; /, 50; g, 25; h, 20. i. Specify the phases in which the system exists when its composition and temperature are represented by any point in each of the fields CKD, DLE, EMF, QKDLR, RLMT, andBCJH. THREE-COMPONENT SYSTEMS 201 V. THREE-COMPONENT SYSTEMS 124. Systems with Liquid and Solid Phases. Temperature-Com- position Diagrams. In systems of three components existing in one solid phase and one liquid phase the composition is not determined by specification of the pressure and temperature, but becomes definite when the percentage or mol-fraction of one of the components is also specified. With such systems there must therefore be employed diagrams which show how the percentages or mol-fractions of the three components are related to one another. The most common form of diagram, one which has the advantage of treating the three components symmetrically, is an equilateral triangle, along the three 90 80 70 30 20 10 60 50 40 Percent Sn sides of which are plotted the atomic percentages or mol-fractions of the three components, as in the above figure. In such a diagram the vertices then represent the pure components, points on the sides represent mixtures of each pair of components, and points within the triangle represent mixtures of all three components. Thus the upper vertex would represent pure cadmium ; the point H, a mixture consist- ing of 42% of Cd and 58% of Sn; and the point G, a mixture consisting of 22% of Cd, 57% of Sn, and 21% of Pb. 202 EQUILIBRIUM BETWEEN PHASES At any one temperature the various compositions of the liquid phase with which any definite solid phase is in equilibrium are repre- sented on such a diagram by a line. Thus the lines HJ, JK, and LM represent at 190 the compositions of the liquid with which solid cadmium, solid lead and solid tin, respectively, are in equilibrium. The point J then evidently represents the composition of the liquid with which the two solid phases Cd and Pb are in equilibrium at 190. Similar lines can be drawn corresponding to various other temper- atures, thus giving a series of isotherms representing the effect of temperature on the equilibrium of the phases. The dotted line FG evidently shows the variation with the temperature of the composition of the liquid phase in equilibrium with the two solid phases Cd and Pb; the line DG shows the same thing for the two solid phases Cd and Sn; and the line EG for the two solid phases Sn and Pb. And the point G shows the only temperature and liquid composition at which the three solid phases Cd, Pb, and Sn coexist in equilibrium with the liquid phase. The mixture of three solid phases separating at this point is called the ternary eutectic, and the point itself the ternary eutectic point. A more complete representation of the effect of temperature is secured by plotting temperatures along an axis perpendicular to the plane of the composition triangle. A model in the form of a triangular prism thus results, whose horizontal sections are the isotherms repre- sented in a plane triangular diagram, like that in the figure. Prob. 48. Behavior of a Liquid Ternary Alloy on Cooling Predicted from its Temperature-Composition Diagram. A completely liquid mixture of 50 Cd, 30 Sn, and 20 Pb is cooled till it wholly solidifies. a. State the temperature at which solidification begins and the nature of the solid phase which then separates, b. State the direction on the diagram which on further cooling the changing composition of the liquid phase follows. (Note that the separation of the solid phase does not change the ratio of the atomic quantities of the other two com- ponents in the liquid phase.) c. State the temperature at which a second solid phase begins to separate, the composition of the liquid phase, and the nature of the solid phase, d. Describe what happens on further cooling, e. Describe the texture of the solid alloy. 125. Systems with Gaseous, Liquid, and Solid Phases in Relation to the Phase Rule and Mass-Action Law. Three-component systems existing in gaseous, liquid, and solid phases will be here considered only with reference to the phase rule, and incidentally to the mass- action law in order to show the relations between these two principles. THREE-COMPONENT SYSTEMS 203 Applications of the Phase Ride and Mass- Action Law. Prob. 49. a. Show by the equilibrium laws applicable to dilute solutions how the solubility of CaCO 3 in water saturated with CO 2 gas at a given temperature is quantitatively related to the pressure of the CO 2 gas, considering that the salt in the solution is substantially all Ca ++ (HC0 3 ~)2. b. Show that the phase rule requires that the solubility of CaCO 3 in water saturated with CO 2 gas is definitely deter- mined when the pressure and temperature are specified. Prob. 50. Silver chloride forms with ammonia two solid compounds AgCl.i|NH 3 and AgC1.3NH 3 , which exist in contact with aqueous solutions. Determine by the phase rule the variance of systems in which the following phases are present, and illustrate what this variance signifies by specifying in each case factors that would determine the state of the system: a, solid AgCl, aqueous NH 3 solution, vapor; b, solid AgCl, solid AgCl.i|NH 3 , solution, vapor; c, solid AgCl, solid AgCl.i^NHs, solid AgC1.3NH 3 , solution, vapor, d. At 25 ammonia vapor is in equilibrium with the solids AgCl and AgCl.i|NH 3 at 10 mm. and with the solids AgCl.i|NH 3 and AgC1.3NH 3 at 105 mm. State under what conditions each of these two pairs of solid phases would exist in equilibrium with an aqueous solution, e. With the aid of the data of Prob. 23, Art. 39, determine the concentration of free NH 3 in the aqueous solution at which AgCl is converted into AgCl.i|NH 3 at 25. /. Assuming that the AgCl in the solution is substantially all in the form of completely ionized Ag(NH 3 ) 2 + Cl~, derive from the equi- librium laws of dilute solutions a relation between the solubility s of AgCl and the partial pressure p of NH 3 in the vapor, first, when the solid phase is AgCl, and secondly, when it is AgCl.i^NH 3 . The preceding problems illustrate the characteristic differences between the conclusions to be drawn from the phase rule and from the mass-action law. The phase rule is qualitative, its application pre- supposes no special knowledge beyond that of the number of compo- nents and phases, and it is applicable without any limitation of con- centration. The mass-action law is quantitative, its application presupposes knowledge of the molecular species present, and, if nu- merical values are to be computed, also of their dissociation-con- stants; and it is applicable in exact form only to solutions or gases at small concentrations. PART III THE ENERGY EFFECTS ATTENDING CHEMICAL CHANGES, AND THE EQUILIBRIUM OF CHEMICAL CHANGES IN RELATION TO THESE EFFECTS CHAPTER IX THE PRODUCTION OF HEAT BY CHEMICAL CHANGES I. THE FUNDAMENTAL PRINCIPLES OF THERMOCHEMISTRY 126. The First Law of Thermodynamics. The branch of chemistry dealing with the heat-effects and changes in energy-content attending chemical changes is called thermochemistry. The first section of this chapter is devoted to a consideration of the fundamental principles underlying the subject and of the methods of determining and express- ing thermochemical quantities. In the second section of the chapter are presented some of the more important generalizations derived from the results of thermochemical measurements. The general principles relating to energy which form the basis of thermochemistry have already ^>een considered in Arts. 23 to 25. It will suffice to recall here the following fundamental laws derived from the law of the conservation of energy: (1) The energy-content U of a system js fully determined by the state of the system; and correspondingly, the change in the energy- content of a system attending any change in its state is determined solely by the initial and final states of the system irrespective of the process by which the change in state takes place. (2) The decrease U\ U z in the energy-content of a system attend- ing any change in its state is equal to the sum of the quantities of heat Q and work W produced in the surroundings. These principles constitute the first law of thermodynamics. They are expressed by the following equation known as the first-law equation: It will be noted that, though the change in energy-content is definite for a specified change in state, the relative quantities of heat and work produced may vary with the process by which the change in state is brought about. The change in energy-content is therefore the more fundamental quantity. The law of initial and final states is constantly employed in thermo- chemistry to calculate the heat-effects attending chemical changes that do not take place under conditions which make possible exact calorimetric measurements. This is illustrated by Prob. i below and more fully by the problems of Art. 130. 207 208 PRODUCTION OF HEAT BY CHEMICAL CHANGES Prob. i. Calculation of Heat of Reactions with Aid of the Law of Initial and Final States. When at 20 12 g. of graphite are burned with oxygen to form COa at 20 within a closed vessel so that no work is produced, 94,200 cal. are evolved; and when 28 g. of CO are so burned with oxygen, 68,000 cal. are evolved. Show by the law of initial and final states how the heat evolved by the burning of 12 g. of graphite to CO at 20 within a closed vessel can be calculated. The relations between energy-content, heat, and work were illus- trated in Prob. 38, Art. 23, by applying them to the phenomenon of vaporization. They are applied to a chemical change in Prob. 5, Art. 128. 127. Heat-Effects Attending Isothermal Changes in State. The heat-effect attending a change in the state of a system at a constant temperature is ordinarily experimentally determined by a combination of the following processes. The change in state is first caused to take place within a calorimeter (as nearly as possible) adiabatically; and the change in temperature of the calorimeter is exactly measured. The quantity of heat which must be imparted to or withdrawn from the calorimeter and its contents to restore them to the initial temper- ature is then accurately determined, either by direct measurement or by calculation from the known heat-capacities of the substances present. This quantity of heat is evidently equal to the heat-effect that would attend the change in state if it took place isothermally at the initial temperature. This method is illustrated in principle by the following problem. Prob. 2. Experimental Determination of the Heat-Effects Attending Isothermal Changes in State. Into a calorimeter containing 5oH 2 O at 20.00 iKCl at 20.00 is introduced, and the temperature falls to 15.11. a. What change in state takes place in this process, considering the calorimeter to be a part of the system; and what is the heat-effect, assuming that any loss of heat by radiation has been corrected for? b. What change in state takes place when iKCl is dissolved in 5oH 2 O at 20 and the resulting solution is brought to 20? c. In order to cal- culate the heat-effect attending this change in state, what other change in state must be combined with that occurring in the cal- orimeter? d. Calculate the val.ue of this heat-effect with the aid of such of the following data as may be needed: the heat-capacity of the calorimeter is 19 cal. per degree; the specific heat-capacity of solid potassium chloride is 0.166, of water is i.oo, and that of the solution of iKCl in 50 H 2 O -is 0.904 cal. per degree. Changes in state at constant temperature may take place without change of volume or without change of pressure. The different heat- THE PRINCIPLES OF THERMOCHEMISTRY 209 effects attending these two different changes in state are known as the heat-effect at constant volume Q v and the heat-effect at constant pressure Q p . The heat-effect at constant pressure is the one which is usually called the heat of reaction, and the one which is commonly recorded in tables of constants. There is difference of usage regarding the algebraic sign of the heat of reaction: when heat is actually evolved by the reaction, the heat of reaction is usually taken positive in thermo- chemical considerations, but negative in thermodynamic considera- tions. In this book a uniform convention, corresponding to the thermochemical one, is employed throughout; heat-effects being always considered positive when heat is evolved, as in the combustion of hydrogen and oxygen, and negative when it is absorbed from the surroundings, as in the vaporization of water. The law that the energy-content of a perfect gas at a definite tem- perature is independent of its pressure (Art. 26) leads to the conclusion that in the case of reactions involving perfect gases the heat-effect Q v at constant volume is greater than the heat-effect Qp at constant pressure by an amount equal to the work produced when the reaction takes place at constant pressure. This work can readily be shown to be equal to (N z Ni)RT, where N 2 denotes the number of mols of gaseous substances present in the final state, and Ni the number in the initial state. That is, Q v = Q p +(N2-Ni)RT. The derivation and application of this principle are illustrated by the following problems. Prob. 3. Relation between the Heats of Reaction of Gaseous Substances at Constant Pressure and Constant Volume. Show that RT is the difference between the quantities of heat evolved when at T two mols of CO and one mol of O 2 at i atm. unite to form CO 2 in one case at constant pressure (for example, in an open calorimeter) ; and in another case at constant volume (for example, in a bomb calorimeter). Note that the change of state which results when the reaction takes place at a constant pressure of i atm. could also be brought about by a process consisting of two steps, namely, by causing the mixture of CO and O ? at i atm. to change at constant volume to CO 2 (whereby the pressure would become f atm.), and then compressing it till its pressure becomes i atm. Prob. 4. Heats of Combustion at Constant Pressure and Constant Volume. When i formula-weight of solid naphthalene (CioH 8 ) is burnt with oxygen at 20 in a bomb calorimeter 123,460 cal. are evolved. Calculate its heat of combustion at constant pressure. 128. Change in Energy-Content and in Heat-Content. The decrease in the energy-content attending any change in the state of a system is found simply by adding together the heat evolved and the 210 PRODUCTION OF HEAT BY CHEMICAt CHANGES work produced in any process by which that change is brought about. When the change in state takes place at constant volume, no work is involved, and therefore the heat Q v imparted to the surroundings is equal to the decrease U\ U 2 in the energy-content of the system. When, on the other hand, the change takes place at constant pressure, there is not only a quantity of heat Qp imparted to the surroundings, but also a quantity of work produced in them equal to p(v z vi), as shown in Art. 24. The decrease Ui U 2 in the energy-content of the system is then equal to Qp + />fe~ ^0- Instead of determining and recording the values of U\ C/ 2 for various changes of state, it is generally more convenient to employ the values of the quantity (Ui-\-piVi) (Uz+pzVz). The quantity U-\-p v, like the quantity U, is a property of the system which always has a definite value when the system is in a definite state, and which always changes in value by a definite amount when the system changes from one state to another, whatever be the process by which the change in state is brought about; for it is evident that the pressure p and the volume v, as well as the energy-content U, have values which are determined by the state of the system. In other words, a law of initial and final states applies to the change in the quantity U+pv, just as it does to the change in the quantity U. For brevity, this quantity U-\-p v will be represented by a single letter H, and will be called the heat-content of the system; it being understood that this term is a purely conventional one which does not imply that the energy quantity denoted by it is a heat quantity, any more than the term energy-content implies it. The relation between the decrease in heat-content and the decrease in energy-content attending any change in state is evidently expressed by the equation: Hi -H 2 = Ui- Ut Or writing, as will be done throughout the remainder of this book, A#, A7, A(z>), AN, etc., for the decrease in the value of any quantity attending any change in state, that is, for the difference between its value in the initial state (denoted by subscript i) and its value in the final state (denoted by subscript 2), this equation may be written in the form THE PRINCIPLES OF THERMOCHEMISTRY 211 Substituting for A/ the expression for it given by the first-law equation, there results For a change in state for which the initial pressure pi and the final pressure p z have the same value p the decrease in heat-content is equal to the heat evolved when the change takes place at the constant pressure p; for the work W produced in this process is p(v 2 vi) and hence equal to A (/>*>) That is, -A# = Q 99 where Q 9 is the heat of reaction at constant pressure. The fact that the decrease of heat-content is equal to the commonly considered heat of reaction at constant pressure is one of the impor- tant reasons for employing the heat-content, instead of the energy- content, in thermochemical and thermodynamic considerations. Another advantage in using heat-content is that it makes it unneces- sary to evaluate the change in volume attending the change in state. The employment of heat-content is preferable also because it is related more closely than the energy-content to the concept of free energy considered in the next chapter, and because many of the thermo- dynamic relations assume a simpler form when heat-content and free energy are employed. The relations between energy-content or heat-content and the quantities of heat and work produced are illustrated by the following problem. Prob. 5. Calculation of Changes in Heat-Content and Energy-Content from the Heat and Work Produced. a. When the reaction 2CO+O 2 = 2C0 2 takes placfc without change of temperature or pressure in a system consisting of 2 mols of CO and i mol of O 2 at 20 and i atm., 136,000 cal. are evolved. What is the decrease in the heat-content, and what is the decrease in the energy-content, of the system? b. When the reaction takes place in the same mixture without change of temperature or volume 135,420 cal. are evolved. What is the decrease in the heat-content, and what is the decrease in the energy-content in this case? c. In what respect does the final state of the system in a differ from that in &, and what principle previously considered accounts for the fact that the changes in energy-content or in heat-content are equal in the two cases. 129. Expression of Changes in Heat-Content by Thermochemical Equations. In order to express the changes in heat-content that attend changes in state, especially those involving chemical reactions, at any constant temperature and pressure, equations are conveniently 212 PRODUCTION OF HEAT BY CHEMICAL CHANGES employed in which the heat-contents of the various substances involved are represented by their chemical formulas, and in which the change in heat-content is shown by placing a numerical term on the right-hand side of the equation. For example, the expression Fe20 3 (s)-f3CO(g) = 2Fe(s)+3C0 2 (g)+ 9 ooocal. (at 20) signifies that at 20 and i atm. (this pressure being understood unless some other pressure is stated) the heat-content of one formula-weight of solid ferric oxide plus that of three formula- weights of gaseous carbon monoxide is equal to the heat-content of two formula-weights of solid iron plus that of three formula-weights of gaseous carbon dioxide plus 9000 cal. ; 9000 cal. being the decrease ( A H) in the heat-content of the system, which is equal to the heat evolved by the system when the reaction takes place at a constant temperature and pressure. Such expressions are called thermochemical equations, or specifically, heat- content equations. As indicated in the preceding equation, the states of aggregation of substances are shown by affixing to the formulas letters within paren- theses, as stated in Art. 109. The fact that a substance is dissolved in x formula-weights of water is shown by attaching the symbol #Aq to the formula of the substance. Thus the equation KCl(s)+iooAq = KCl.iooAq-44oo cal. (at 20) signifies that when at 20 one formula-weight of solid potassium chloride is dissolved in 100 formula-weights of water there is an in- crease of 4400 cal. in the heat-content of the system, corresponding to an absorption of 4400 cal. from the surroundings. When the sub- stance is dissolved in so large a quantity of water that the addition of^iore water produces no appreciable heat-effect, the symbol Aq may be attached to the formula of the substance. These thermochemical equations can be treated strictly as algebraic equations, and can be combined with one another by addition or subtraction; for every quantity in them has a definite value (namely, that of the heat-content of the substance represented by the formula), irrespective of the other quantities that occur with it in the equations. Expression of Heat Data and Calculation of Heats of Reactions by Thermochemical Equations. \JProb. 6. The union at 20 and i atm. of i g. of aluminum with oxygen is attended by a heat-evolution of 7010 cal. ; and the union of i g. of graphite with oxygen to form carbon monoxide is attended by a heat- evolution of 2160 cal. Express these data in the form of thermo- THE PRINCIPLES OF THERMOCHEMISTRY 213 chemical equations; and calculate from them the heat of the reaction Al 2 3 (s)+ 3 C(s)=2Al(s)4-3CO(g) at 20. Prob. 7. a. Express the following data in the form of thermochemi- cal equations, employing the conventions described in the preceding text. The heat of formation of i mol of gaseous HC1 from the elementary substances is 22,000 cal. Its heat of solution in 100 formula-weights of water is 17,200 cal. The heat of the reaction between i mol of gaseous chlorine and a solution of 2 formula-weights of HI in 200 formula- weights of water, forming solid iodine and a dilute HC1 solution is 52,400 cal. The heat of solution of i mol of gaseous HI in 100 formula- weights of water is 19,200 cal. b. By combining these equations calculate the heat of formation of i mol of gaseous HI from gaseous hydrogen and solid iodine. Although it is true that only changes in heat-content can be deter- mined, yet it is convenient to employ an arbitrary scale of heat-content which has as its zero-points the heat-contents of the various elementary substances at the temperature under consideration, at a pressure of one atmosphere, and in the form which is most stable at this temperature and pressure. Under this convention the heat-content (H) of any compound substance is evidently equal to the increase in heat-content (A#) which attends its formation out of the elementary substances; and in any thermochemical equation the formula of a substance may evidently be replaced by the numerical value of its heat of formation with opposite sign. For example, the heat-content at 20 of one formula-weight of gaseous hydrogen bromide, or the numerical value of the formula iHBr(g) is 8500 cal.; for 8500 cal. is the heat evolved when it is formed out of gaseous hydrogen and liquid bromine at 20 and i atmosphere. Similarly the heat-content of a potassium chloride solution represented by the formula KCl.iooAq is 101,200 cal.; f ,r 101,200 cal. is the sum of the heat (105,600 cal.) evolved when one formula-weight of solid KC1 is formed out of solid potassium and gaseous chlorine at 20 and of the heat (4400 cal.) evolved when it is dissolved in 100 formula-weights of water at 20. The heat of for- mation of this solution is expressed by the equation K(s)-hja 2 (g)+iooAq = KCLiooAq+ 101,200 cal. This example shows that in thermochemical equations the symbol ffAq (not attached to another formula) has, like the formulas of elementary substances, the value zero; for the water represented by it has not been formed out of its elements. Water which has been so formed is represented by the formula xH 2 O. 214 PRODUCTION OF HEAT BY CHEMICAL CHANGES It is evident that the employment of heats of formation greatly simplifies the task of determining and systematizing thermochemical data; for, instead of measuring and recording the change in heat- content attending every chemical reaction, it suffices to do this for the formation of every compound out of the corresponding elementary substances. The numerical values of the heats of formation so determined, which will be found recorded in tables of physical and chemical constants, like those of Landolt-Bornstein, may then be substituted in any thermochemical equation, and the change in heat- content attending the corresponding reaction may thus be calculated. gnt attenam JProb. 8. Calculation of Heat of Reaction from Heats of Formation. Calculate the heat-effect that attends at 20 the chemical reaction PbS(s)+2PbO(s)=3Pb(s)+SO 2 (g) from the following heats of forma- tion at 20: PbO(s), 50,300 cal.; PbS(s) ? 19,300 cal.; SO 2 (g), 70,200 cal. 130. Indirect Determination of the Heat-Effects of Chemical Changes. On account of radiation errors the heat-effect can be directly determined by calorimetric measurements only for those chemical changes which take place completely within a few minutes, and for such changes only when the temperature is not greatly different from the room temperature. It is, however, possible to calculate the heat-effects of many other changes at or near the room temperature from those which have been directly measured, by applying the law of initial and final states. This has already been exemplified in the preceding articles, and it is more fully illustrated by the following problems. The indirect method commonly employed for determining heat-effects at temperatures much higher or lower than the room temperature is described in the next article. Derivation of Heats of Formation. *Prob. g. At 20 the heat of combustion of one mol of acetylene (C 2 H 2 ) is 313,000 cal. Calculate its heat of formation. The heat of formation of iH 2 O(l) is 68,400 cal., and that of iCO 2 (g) from graphite and oxygen is 94,200 cal. Prob. 10. a. State just what heats of formation are denoted by the second and third terms in the following equation: Zn(s)+2HClooAq=ZnCl 2 ooAq-f-H 2 (g)+34,2oo cal. (at 20). b. Write the complete thermochemical equations which express these heats of formation, taking into account the facts that the heat of forma- tion of i mol of gaseous HC1 is 22,000 cal. and that its heat of solution in a large quantity of water is 17,300 cal. Prob. ii. Calculate the heat of formation of iZn(OH) 2 (s) from the THE PRINCIPLES OF THERMOCHEMISTRY 215 following equation and from the other necessary data, which have been given in preceding problems: Zn(OH) 2 (s) + 2HC1 ooAq = ZnCl 2 Aq + 2 H 2 O + 19,900 cal. Prob. 12. Calculate the heat of formation at 20 of iH 2 SO 4 (l) from the following data at 20 and those given in preceding problems. The heat of solution in a large quantity of water of iSO 2 (g) is 8000 cal., and that of iH 2 SO 4 is 18,000 cal. One mol of gaseous C1 2 acting on a dilute solution of i mol of SO 2 with formation of a dilute solution of HC1 and H 2 SO 4 produces a heat-evolution of 73,900 cal. (Note that when x formula-weights of H 2 O are involved in the chemical reaction the symbol #H 2 O must appear in the thermochemical equation, even though ooAq may also occur in it.) Ans. 192,300 cal. Indirect Determination of Heats of Reaction. Prob. 13. A direct determination of the heat-effect of the reaction CH 3 CH 2 OHcx>Aq+O 2 (g)=CH 3 C0 2 H3Aq-f-H 2 O is not practicable. State what measurements could be made which would enable this heat- effect to be calculated; and show how it would be calculated from the results of such measurements. Prob. 14. Suggest a seiies of chemical reactions from whose heat- effects, which must be readily determinable in a calorimeter, the heat of formation of iNa 2 CO 3 .ioH 2 O(s) at 20 could be calculated. Write the thermochemical equations, and indicate how they would be com- bined to yield the desired result. 131. Influence of Temperature on the Heat-Effects Attending Chemical Changes. The heat-effect at constant pressure of a chemical change taking place at any temperature can be derived from the heat-effect at constant pressure at any other temperature by the following consideration of two different processes resulting in the same change in state. In one process cause the chemical change (for example, the union of i mol of CO with ^ mol of 62) to take place at pressure p and temperature TI, and heat the products (the carbon dioxide) under the pressure p to the other temperature T 2 ; and, in the second process, heat the reacting substances (the carbon monoxide and oxygen) under the pressure p from T\ to 7*2, and cause them to combine (forming carbon dioxide) at the pressure p and the temper- ature T z . Since in each of these processes the system changes from the same initial state (i mol of CO and \ mol of O 2 at p and 7\) to the same final state (i mol of CO 2 at p and r 2 ), the total change in heat- content must be the same in the two processes; and therefore the heat evolved at the temperature T z must differ from the heat evolved at the temperature 7\ by the same amount as the heat absorbed in heating the reacting substances differs from that absorbed in heating the reaction-products from 7\ to T 2 at the constant pressure p. 216 PRODUCTION OF HEAT BY CHEMICAL CHANGES Effect of Temperature on Heat of Reaction. Prob. 15. a. Represent on a diagram, in which the ordinates denote the temperature, and the abscissas the heat-content of the system, how the heat-content changes during each step of the two processes described in the text for bringing about the change in state, iCO+|O 2 at TI and p to iCO 2 at T z and p. Make the arbitrary assumption that the heat-content of the system in the initial state is zero. b. With the aid of the diagram formulate an algebraic relation between the heats of reaction Q Tz and Q Tl at the two temperatures T 2 and 7\ and the heat-capacities Ci and C 2 of the system in its initial and final states respectively, b. Noting that this relation is exact only when the heat- capacities do not vary with the temperature, formulate an exact differential expression, and also an integral which is a general expression 'for the difference in the heats of reaction at two temperatures, when no change takes place hi the state of aggregation between those tem- peratures. Prob. 16. Calculate the heat of formation of iPbO(s) at 200 from its heat of formation (50,300 cal.) at 20 and from the specific heat- capacities at constant pressure of lead (0.032), of oxygen (0.212), and of lead oxide (0.052). Ans. 50,000 cal. Prob. 17. Calculate the heat of formation of iH 2 0(g) at 1000 from the following data. The heat of formation of iH 2 O(l) at 20 is 68,400 cal. Its heat of vaporization at 100 is 9,670 cal. The molal heat- capacity at constant pressure at T is 6.69+0.00077 for oxygen, 6.54 + 0.00077 for hydrogen, and 8.81 0.00197+0.000,002, 227~for water- vapor. Ans. 59,800 cal. THE RESULTS OF THERMOCHEMISTRY 217 II. GENERAL RESULTS OF THERMOCHEMICAL INVESTIGATIONS 132. Heat-Effects Attending Changes in the State of Aggregation of Substances. The heat of vaporization of liquid substances the heat of fusion of solid substances, and the heat of transition of one solid substance into another (as of rhombic into monoclinic sulfur) are quantities which are important in themselves and which are fre- quently involved in calculations of the heat of chemical reactions. The general statement can be made in regard to them that the conver- sion of the form that is stable at lower temperatures into that stable at higher temperatures (for example, of ice into water, or of rhombic into monoclinic sulfur) is always attended by an absorption of heat. The following simple principle, known as Trouton's rule, has been discovered in regard to the values of the heat of vaporization. The ratio of the molal heat of vaporization of a liquid at its boiling-point to its boiling-point on the absolute scale has approximately the same value (namely, about 20.5) for all liquids, except those whose molecules are associated; that is, A&/T = approx. 20.5. The actual values of these quantities in the case of five very different liquids are shown in the following table. HEATS OF VAPORIZATION AND TROUTON'S RULE Substance AH T AH/T Bromine 6760 332 20.4 Benzene 7350 353 20.8 Carbon bisulfide 6380 319 20.0 Ethyl ether 6260 308 20.3 Ethyl formate 7180 327 22.0 Substances containing the hydroxyl group, such as water, alcohols, and acids, whose molecules in the liquid state are for other reasons believed to be associated (see Art. 38), form marked exceptions to Trouton's rule. Thus the value of AH/T is 25.9 for water, 27.0 for ethyl alcohol, and 14.9 for acetic acid. The molal heat of vaporization of a liquid or solid substance can be accurately calculated by the Clapeyron equation (Art. 33) from the change of its vapor-pressure with the temperature. The heat of solution of substances is another important quantity. In determining and expressing it the quantity of solvent in which a definite weight of the substance is dissolved must be taken into con- sideration. The two limiting cases are the heats of solution in a very large quantity of solvent and in that quantity of solvent which forms 2i8 PRODUCTION OF HEAT BY CHEMICAL CHANGES with the substance a saturated solution. These two heat-effects sometimes have different signs. They evidently differ by the heat of dilution of the saturated solution with a large quantity of water. The dissolving of gaseous substances in solvents is always attended by an evolution of heat; and the dissolving of solid substances in solvents is usually attended by an absorption of heat. The mixing of liquids may be attended either by evolution or absorption of heat, as explained in Art. 42. The heat of dilution of substances in solution is also important; for it enables the heat of formation of a solution of one concentration to be calculated from that of a solution of another concentration, and thus enables heats of reaction to be calculated at different concentra- tions. The heat-effect attending the addition of an equal volume of water to a concentrated aqueous solution is often large; but it becomes less as the concentration diminishes; and, after a moderately small concentration (such as 0.2 jJDrmal) has been attained, there is usually only a very small heat-effect on adding even a very large quantity of water. For example, on adding at 18 to i formula- weight of gaseous HC1 or of solid ZnCk successively A7V formula- weights of water, there are evolved the following quantities of heat (Q) in calories: 5 5 10 30 50 100 200 Q for HC1 ..... 14960 1200 .600 360 120 50 Q for ZnCl 2 . . . . 7740 1850 1300 2170 1490 820 390 Prob. 18. Heat of Transition Derived from Heats of Solution. The heat of solution at 20 in a large quantity of chloroform of i at. wt. of rhombic sulfur is 640 cal. and of i at. wt. of monoclinic sulfur is 560 cal. Show, by applying the law of initial and final states, what other heat-effect can be derived from these data, and what its value is. Prob. ig. Change of Heat oj Reaction with the Concentration. The heat of solution at 18 of iZn(s) in HC1.2ooAq is 34,200 cal. Find its heat of solution in HCl.sAq, a, by applying the law of initial and final states, and b, by formulating the thermochemical equations involved. 133. Heats of Reaction in Aqueous Solution. The investigations made of the heat-effects attending chemical reactions in aqueous solution between substances present at fairly small concentrations (0.1-0.3 normal) have established the following principles: (i) On mixing solutions of two neutral salts which do not form a precipitate by metathesis (for example, solutions of potassium chloride and sodium sulfate) there is scarcely any heat-effect. Exceptions to THE RESULTS OF THERMOCHEMISTRY 219 this principle are met with in the few cases in which a unionized salt is produced by the metathesis. Thus the metathetical reaction 2K+Cl- + Hg^(NO 3 -)2 = 2K+N0 3 - + HgCl 2 is attended by a heat- evolution of 12,400 cal. (2) The heat of neutralization of a solution of any largely ionized monobasic acid with a solution of any largely ionized monacidic base (for example, of hydrochloric acid, nitric acid, etc., with sodium hydroxide, potassium hydroxide, etc.) has approximately the same value, whatever be the acid or base. At 18 this nearly constant value averages 13,800 cal. per equivalent when the acid and base solutions are 0.12 to. 0.25 normal. (3) When the base or acid is only partly ionized (as in the case of ammonium hydroxide or hydrofluoric acid), the heat-effect attending its neutralization with a largely ionized acid or base is often much larger or smaller than that observed when both acid and base are largely ionized; thus the heat of neutralization of one equivalent of ammonium hydroxide with one of hydrochloric acid is 12,300 cal., and that of one equivalent of hydrofluoric acid with one of sodium hydroxide is 16,300 cal. (4) When one formula-weight of a dibasic acid is neutralized in steps by adding first one equivalent of a largely ionized base and then a second equivalent, the heat-effects for the two equivalents of base are usually different; for example, at 18 the two heat-effects are 14,600 and 16,600 cal. in neutralizing 0.28 normal sulfuric acid solution with 0.28 normal sodium hydroxide solution, and they are 11,100 and 9,100 cal. in neutralizing carbonic acid with that base. (5) When certain polybasic acids are neutralized, there is sometimes scarcely any heat-effect when the second or third equivalent of base is added. Thus the successive heat-effects when phosphorous acid (H 3 PO 3 ) is treated with sodium hydroxide at 18 are: 14,800 cal. with the first equivalent, 13,600 cal. with the second equivalent, and only 500 cal. with the third equivalent; and with hypophosphorous acid (H 3 PO 2 ) there is a heat-effect of 15,200 cal. with the first equivalent of sodium hydroxide, and only no cal. with the second equivalent. (6) With certain polybasic acids there is a considerable heat-effect when to the solution of the neutral salt another equivalent of base is added; thus, there is a heat-effect of 1200 cal. on mixing a solution containing one equivalent of sodium hydroxide with one containing one formula- weight of sodium phosphate (Na 3 PO 4 ). 220 PRODUCTION OF HEAT BY CHEMICAL CHANGES Interpretation of the Heats of Reaction in Aqueous Solutions. Prob. 20. a. Explain principle (i) stated in the preceding text with the aid of the ionic theory, assuming that the solutions are very dilute. b. What conclusion as to the ionization or as to the heat of ionization of neutral salts can be drawn from the fact that this principle holds true even at fairly high concentrations (such as 0.3 normal)? c. Write a thermochemical equation corresponding to the ionic reaction to which the heat-effect is mainly due in the reaction cited as an exception to the principle. Prob. 21. Show that on mixing dilute solutions of two salts (such as lead nitrate and potassium iodide) which form a precipitate by meta- thesis there must be a heat-effect which is substantially equal, but opposite in sign, to the heat of solution of the precipitated substance (the lead iodide). Prob. 22. a. Write the ionic reaction to which the nearly constant heat of neutralization of largely ionized acids and bases corresponds. b. State what other heat-effect is involved in the neutralization of ammonium hydroxide (a slightly ionized base) with a largely ionized acid, and find its value, c. What is the heat-effect that attends the reaction between iNH 4 Cl and iNaOH in 0.2 normal solution? Prob, 23. Calculate the heat of ionization of iHF from the facts that its heat of neutralization in 0.28 normal solution with 0.28 normal NaOH solution has been found to be 16,300 cal, and its ionization in 0.28 normal solution is estimated to be 5.2%. Prob. 24. Calculate the heats of ionization at 18, a, of H 2 CO 3 , and b } of HCO 3 ~~ from the heats of neutralization of carbonic acid given in paragraph (4). Carbonic acid is a very slightly ionized acid; sodium hydrogen carbonate solution is practically neutral; and the sodium carbonate in the 0.07 formal solution produced by the neutralization is 6.3% hydrolyzed. Prob. 25. Conductance, transference, and reaction-rate measure- ments have shown that sodium hydrogen sulfate in o.i formal solution at 1 8 consists approximately of 52% of HSOr, 48% of SO 4 = , and of the corresponding amounts of Na + and H + . A calorimetric measure- ment at 1 8 has given the result expressed by the equation: NaHSO 4 .6oiAq+NaOH.2ooAq =Na 2 SO 4 .8oiAq+H 2 O+ 16,620 cal. Calculate the heat of the reaction HS0 4 ~=H + +SO 4 = , assuming that the ionization of NaOH, of Na 2 SO 4 , and of NaHSO 4 (into Na+ and HSO 4 ~) is complete. Prob. 26. State what the observed heat-effects, given in paragraph (5), attending the neutralization of phosphorous acid and of hypo- phosphorous acid show as to the existence of salts of these acids in solution. Prob. 27. Explain the fact, stated in paragraph (6), that there is a large heat-effect on adding a solution of NaOH to one of Na 3 P0 4 . THE RESULTS OF THERMOCHEMISTRY 221 Prob. 28. Determination of Chemical Equilibria by Thermochemical Methods. Describe a thermochemical method of determining the extent to which acetic acid displaces hydrofluoric acid from sodium fluoride in dilute solution. The heat of neutralization of acetic acid with sodium hydroxide at 18 is 13,230 cal, and that of hydrofluoric acid is 16,300 caL *134. Applications of Thermochemical Principles. The following problems illustrate some important applications of thermochemical principles to industrial chemical operations. Thus the problems in- clude the estimation of the maximum temperature produced by flames or by the explosion of gaseous mixtures, the determination of the quantity of heat produced in continuous chemical processes (whose removal often constitutes one of the main difficulties of large- scale operation), and a consideration of the effectiveness of hot gases as an evaporating agent. Maximum Temperature Producible by Flames and Explosions. Prob. 29. Calculate the maximum temperature that could theoreti- cally be attained in the flame produced by burning at 20 a "water-gas " consisting of equimolal quantities of hydrogen and carbon monoxide with twice the quantity of air required for complete combustion. As- sume that the reaction-products are not appreciably dissociated, and that air contains iO 2 to 4N 2 . The molal heat-capacity of C0 2 at con- stant pressure at T is 7.0+0.00717 0.000,001, 86 r 2 . The equation H 2 (g) + JO 2 (g) = H 2 O(1) + 68,400 cal. holds true at 20. For the other data needed see Art. 27 and previous problems of this chapter. Ans. i723A. Prob. 30. a. Calculate the maximum temperature and pressure that could be produced by the explosion within a bomb of a mixture consist- ing of i mol of H 2 , % mol of O 2 , and i mol of N 2 at 20 and 100 mm., assuming that the water produced is not appreciably dissociated. b. The maximum pressure produced in an actual experiment was found to be 840 mm. Show how from this result the degree of dissociation of the water-vapor and the temperature of the mixture at the moment of the explosion can be calculated. (The equations should be formulated, but they need not be solved numerically.) Heat Evolved by Continuous Processes at High Temperatures. Prob. 31. In the Deacon process of making chlorine, a mixture of oxygen and hydrogen chloride in the proportion O 2 : iHCl at 20 is passed continuously into a vessel at 386 containing a suitable catalyzer. The gas is passed so slowly that equilibrium is established, 80% of the hydrogen chloride being converted into chlorine and water. Calculate the heat which will be given off from the equilibrium vessel per mol of HC1 passed through, and which must, to maintain a constant 222 PRODUCTION OF HEAT BY CHEMICAL CHANGES temperature, be continuously withdrawn by a suitable system of cooling. For the data needed see Art. 27 and previous problems of this chapter. Ans. 1900 cal. Prob. 32. In the Grillot contact process of making fuming sulfuric acid a mixture of sulfur dioxide and air passes through a chamber con- taining a series of trays charged with the platinum catalyst. This mixture enters the chamber at 380 (the lowest temperature at which a rapid reaction will occur), and rises in temperature as a result of the heat of reaction. This rise in temperature must not exceed 100, if considerable dissociation of the SO 3 is to be avoided. Calculate the smallest volume of air that may be present with one volume of sulfur dioxide in the gas entering the chamber at 380 in order that the rise of temperature may not exceed 100. The heat-effect of the reaction SO 2 (g)+O 2 (g)=SO 3 (g) is 22,000 cal. at 380. Consider the molal heat-capacity of each gas in the mixture to be the same as that of nitrogen or oxygen. Assume that 97% of the SO 2 is .converted into SO 3 , and that there is no loss of heat by heat interchange. Ans. 29 volumes. Prob. 33. Heat-Effects in the Process of Concentrating Sulfuric Acid by Hot Gases. In the Gaillard process of concentrating sulfuric acid a 73% acid (corresponding to iH 2 SO4.2H 2 O) is sprayed at 20 into a brick-lined tower, and in falling through the tower is caused to meet an ascending current of hot producer-gas consisting (mainly) of nitrogen and carbon monoxide, which enters the bottom of the tower at 1100 and leaves the top at 200. Calculate how many cubic meters of producer gas must be supplied per kilogram of the 73% acid to furnish the heat necessary to concentrate it to 91^% H 2 SO 4 (corresponding to 2H 2 SO 4 .iH 2 O), assuming the latter leaves the tower at 200, and that there is no loss of heat by radiation. The heat evolved on adding ]VH 2 O(1) to iH 2 SO 4 (l) at 20 has been calorimetrically determined and found to be expressed by the equation Q = 17860^7 (^V + i. 80). The heat of vaporization of i g. of water at 100 is 537 calories. The mean specific heat-capacity of 91^% H 2 S0 4 between 20 and 200 is 0.4 cal. per degree. In solving this problem write the thermochemical equa- tions by which iH 2 SO 4 .2H 2 O at 20 can be converted into and i|H 2 0(g) at 200. Ans. 3.9. CHAPTER X THE PRODUCTION OF WORK BY ISOTHERMAL CHEMICAL CHANGES IN RELATION TO THEIR EQUILIBRIUM CONDITIONS I. THE SECOND LAW OF THERMODYNAMICS AND THE CONCEPT OF FREE-ENERGY 135. The Second Law of Thermodynamics and its Application to Isothermal Changes in State. This chapter is devoted to a considera- tion of the production of work by chemical changes. Before this can be adequately considered, familiarity with certain aspects of another general principle relating to energy, the so-called second law of thermo- dynamics, is essential. The first law of thermodynamics states that when one form of energy is converted into another the quantity of the form of energy that is produced is equivalent to the quantity of the form that dis- appears; but it does not indicate that there is any other restriction as to the transformability of the different forms of energy. Expe- rience has shown, however, that while the various forms of work can be completely transformed into one another and into heat, thp. transfor- mation of heat into work is subject to certain limitations. Namely, it is found that heat is never transformed into work by any cycle of changes taking place in any system kept in surroundings at any definite temperature; by a cycle of changes being meant any series of changes in state of such a character that the system finally returns to its initial state. If a cycle of changes in state by which heat is transformed into work at constant temperature could be realized, the cycle could be indefinitely repeated and produce work in unlimited quantity, thus constituting a form of perpetual motion, which may be called perpetual motion of the second kind. The impossibility of such perpetual motion is the principle that was stated and illustrated in Art. 44. Perpetual motion of the second kind, by which work is conceived to be produced out of heat at constant temperature, is to be distinguished from perpetual motion of the first kind (described in Art. 23), by which work is conceived to be produced without consuming energy of any form. Experience with processes taking place at different temperatures 223 224 PRODUCTION OF WORK BY CHEMICAL CHANGES has led to the conclusion that this principle is a consequence of a still more general law, known as the second law of thermodynamics, which may be expressed as follows. A process whose final result is only a transformation of a quantity of heat into work is an impossibility. The application of this law to changes of state taking place at constant temperature will be considered in this chapter; its application to processes involving different temperatures, in chapter XII. Prob. i. Production of Work from Heat at Constant Temperature. When a gas that is placed in a reservoir of large heat-capacity and definite temperature expands against an external pressure, it produces work and withdraws from the reservoir a quantity of heat. State why this transformation of heat into work is not a contradiction of the second law of thermodynamics. Since work is never produced out of heat by any isothermal cycle of changes in a system, no change in state at constant temperature can produce more work than must be expended to restore the system to its initial state. In other words, any isothermal change in state taking place in one direction (such as the expansion of a gas from pi, vi, to p2) V2) at T) is capable of producing a definite maximum quantity of work; and to make it take place in the opposite direction (thus, to compress the gas from pz, %, to pi, Vi, at T), there must be expended a quantity of work at least equal to this maximum. Whether this maximum work is actually produced, or whether no more than this minimum work is actually expended, depends on the way or process by which the change in state takes place. Thus, the work produced by the change in state of the gas from pi, vi, to p z , %, at T depends on the process by which this takes place; for example, upon the oppos- ing external pressure, which obviously may have any value from zero up to a value only infinitesimally less than the pressure of the gas itself. If the process is such that the maximum work is actually produced or only the minimum work is expended, it is called a reversible process. If the process is such that less than the maximum work is produced or more than the minimum work is expended, it is called an irreversible process. It will be noted that the criterion of the reversibility of a change in state is whether the system can be restored to its initial state without a net expenditure of work. The following problems illustrate reversible and irreversible pro- cesses and the external conditions which must be fulfilled in order that processes may be reversible. THE CONCEPT OF FREE ENERGY 225 Reversible and Irreversible Processes Contrasted. Prob. 2.N mols of a perfect gas having a pressure of 2 atm. are enclosed within a cylinder placed in a thermostat at a temperature T and provided with a weighted piston. The weight on the piston is reduced so that the piston exerts on the gas a pressure of i atm., and the gas expands till its own pressure becomes i atm. Explain why this process is irreversible; and state what would have to be true of the pressure exerted by the piston on the gas during its expansion in order that the process might be reversible. Prob. 3. a. Derive an expression in terms of N, R, and T for the work produced by the irreversible process described in Prob. 2. b. Derive a corresponding expression for the work produced when the same change in state is brought about by a reversible process, c. Find the numerical ratio of these two quantities of work. Prob. 4. Two Daniell cells, each having an electromotive force of 1. 10 volts, are connected in series and are used for charging a lead storage-cell having a (counter) electromotive force of 2.10 volts. Ex- plain why the process is not reversible; and state how a number of Daniell cells and a number of storage-cells could be so arranged that the latter might be charged reversibly. (In this case the Daniell cells may be regarded as the system, and the storage cells as a part of the surroundings in which the electrical work is produced and stored.) As illustrated by the preceding problems, in order that the process by which a change in state is brought about may be reversible, the pressure externally applied must be substantially equal to the pressure exerted by the system itself, or the applied electromotive force must be substantially equal to the electromotive force of the cell. For the change in state will take place in one direction when the applied pres- sure or electromotive force is only infmitesimally less than that of the system, and in the other direction when it is only innnitesimally greater, and correspondingly, the work produced by the change in state is only infinitesimally less than that expended in restoring the system to its initial state. But, if the applied pressure or electro- motive force were less by a finite amount than that of the system, the quantity of mechanical or electrical work produced in the surround- ings would evidently not suffice to restore the system to its initial state; and if in causing the change to take place hi the opposite direction, the applied pressure or electromotive force were greater than that of the system, more work would be withdrawn from the surroundings than the system would be capable of reproducing on re- verting to its original state. It is to be noted that the term reversible is always employed, in 226 PRODUCTION OF WORK BY CHEMICAL CHANGES the sense above defined, to designate a process of such a character that it is possible to restore the original condition of things both in the system and its surroundings. After an irreversible process has taken place, it is in general possible to restore the system to its original state, but only by withdrawing from the surroundings a larger quantity of work than was produced in them (and imparting to them a corre- sponding quantity of heat), so that the original condition in the sur- roundings is not reproduced. When an irreversible change has once taken place, it is not possible by any means whatever to reproduce in their entirety the conditions that previously existed. Since all actual processes are irreversible, there is constantly taking place a decrease in the work-producing power of the universe. This irreversibility arises from various causes. Thus there may be no device for producing the available work (as when a compressed gas escapes into the atmosphere or when coal burns in the open air) ; the mechanism that is employed may be imperfect owing to frictional or electrical resistance; or the pressure or electromotive force of the system may not be balanced by applying an equal and opposite external pressure or electromotive force. 136. The Concepts of Work-Content and Free Energy. The principles that a definite change in state is capable of producing a definite quantity of work and that this quantity of work is realized when, and only when, the process by which the change is effected is reversible, are further illustrated by the following problem. Prob. 5. Production of Definite Quantity of Work by Reversible Changes in State. In a voltaic cell consisting of one platinum electrode in contact with a o.i f. HC1 solution arfd with hydrogen gas at i atm. and of a second platinum electrode in contact with the same HC1 solu- tion and with hydrogen gas at o.i atm., hydrogen is found to go into solution (as hydrogen-ion) at the first electrode and to be evolved at the second electrode, as a result of the electromotive force which is pro- duced, a. Name the change in state that takes place in such a cell when one faraday of electricity passes through it at 18, specifying all the factors determining the change in state, but disregarding the transference in the solution, which in a cell of this kind will be shown later to be attended by no energy-effect, b. Describe how the same change in state could be brought about reversibly by a process not involving voltaic action; and calculate the work attending this process. c. Show that perpetual motion could be realized if this quantity of work were not equal to the work that would be produced per faraday of electricity when the cell operates reversibly. THE CONCEPT OF FREE ENERGY 227 Since any definite change of state in a system is capable of producing a definite maximum quantity of work (a quantity which is never ex- ceeded, whatever be the process by which the change in state takes place), it is to be inferred that a system in any definite state possesses a certain power of producing work, which may be caUed its jsork^ content^; and that this changes by some definite amount (from A^to~At) when the system undergoes any definite change in state, whatever be the process by which this change takes place, and what- ever be the quantity of work which is actually produced. In other words, the work-content of a system, like its energy-content and its heat-content, is a quantity characteristic of the state of the system; and the change in the work-content of a system is determined solely by its initial and final states. The decrease (Ai- A 2 or -AA) in the work-content of the system attending a definite change in its state is evidently equal to themaxi- mum work which the change is capable of producing, and this is equal to the quantity of work W K which the change actually produces when it takes place reversibly. That is: Ai-At = -AA = W*. It will be noted that, while the First Law requires that there be a quantity of energy produced in the surroundings equal to the de- crease of the energy-content of the system, the Second Law does not require that there be a quantity of work (W) produced equal to the decrease in work-content ( as was illustrated by Prob. 3). The Second Law requires only that the quantity of work produced be not greater than the decrease in work-content. That is, W > AA for no process whatever; W= AA for a reversible process; and W < AA for an irreversible one. Just as it is more convenient in chemical considerations to con- sider the heat-content rather than the energy-content of systems, so there are many advantages in considering in place of the work-content a quantity which differs from it, just as the heat-content differs from the energy-content^5y'the value of_ i^ This quantity, which may be called the free-energy content (F), 01 simply the free-energy of the system, is defined by the equation/? = A_+y. Its value is determined by the state of the system, since tin values of A and of pv are so determined; and the decrease in its value when any change in the state of the system takes place is evidently equal to the work produced when the change takes place reversibly, 228 PRODUCTION OF WORK BY CHEMICAL CHANGES diminished by the increase of the pressure- volume product; that is: F 1 -F 2 = W R -(p 2 v 2 -piV 1 ), or -AF = W R -A(pv). The difference (FiF 2 ) between the free-energy-content of a system in its initial state and that in its final state will be called the free-energy decrease ( AF) attending the change in state, irrespective of the sign of its numerical value, which may be either positive or negative. The change in state is always considered to take place at some constant temperature. Prob. 6. Work-Content and Free-Energy-Content in Relation to the Work Produced. a. What is the decrease in joules in the energy- content, heat-content, work-content, and free-energy-content of one formula-weight of water when it changes from liquid water to gaseous water at 100 against a constant pressure of i atm., referring to Prob. 38, Art. 23, for the data needed? b. What would be the decrease in these four quantities if the liquid water at 100 and i atm. changed to gaseous water at 100 and i atm. by a process which produces no work, for example, by introducing the liquid water into an evacuated vessel having a volume equal to that of the saturated vapor? In cases where different parts of the system are under different pressures, as in the cell of Prob. 5, the decrease in free energy is de- fined to be the quantity obtained by subtracting from the work pro- duced when the change takes place reversibly the difference between the sum of the p v values for all the parts of the system in its final state and the sum of the p v values for all its parts in its initial state. Therefore, in general: Fi-F 2 = TF H -(S^2-S/>i^i); or -AF = W R -A(^pv). The principle that the free-energy decrease attending any process is determined solely by the change in state of the system is the funda- mental principle on which is based the treatment of chemical equi- librium and of electrochemical processes presented in this and the following chapters. The methods of determining the free-energy change attending various kinds of changes in state will be systemati- cally developed, and the applications of the results to chemical and electrochemical problems will be considered. A knowledge of the free-energy changes attending chemical changes is of great importance for the following reasons. In the first place, free energy values give directly the maximum quantity of work obtainable from chemical changes; for example, they show how much work can be produced by the combustion of coal or gasoline or from the chemical THE CONCEPT OF FREE ENERGY 229 reactions taking place in a storage cell; for the decrease in free-energy attending the chemical change, and not its heat-effect, determines the theoretical efficiency of a heat engine or electric battery. Secondly, free^eriergy values enable the equilibrium-constants of "chemical reactions to be calculated; and a knowledge of the equilibrium condi- tions of chemical changes is fundamental both in its scientific aspects and in its technical applications; thus the proper control of a gas producer or of a blast furnace must be based on a quantitative con- sideration of the equilibria of the chemical reactions involved in the V J process. Thirdly, free-energy values form the basis of the science of electrochemistry on the electromotive force side; thus they enable not only the total electromotive force, but also the separate potentials, of voltaic cells of all kinds to be calculated. 230 PRODUCTION OF WORK BY CHEMICAL CHANGES II. FREE-ENERGY CHANGES ATTENDING PHYSICAL CHANGES 137. Free-Energy Changes Attending Changes in Volume and Pressure. In finding the expression for the free-energy decrease attending any kind of change in state, the method of procedure is always as follows. First, the change in state of the system is formu- lated by defining accurately its initial and final states; second, a process is conceived by which the change in state takes place reversibly ; and third, the work attending this reversible process is evaluated. The simplest type of change in state is that in which a system whose pressure is p and volume v changes its state at a constant temperature T so that its pressure becomes p+dp and its volume v-t-dv. This change in state will take place reversibly if a substantially equal external pressure be applied to the system. The work dW B in this process is equal to pdv (Art. 24). The free-energy decrease dF attending such a change in state at a constant temperature is, however, by definition equal to dW^ d(pv). In virtue of the mathematical relation d(pv)=pdv-\-vdp, the free-energy decrease attending an in- finitesimal change in volume or pressure at a constant temperature is therefore given by the expression: -dF=-vdp. For a finite change in volume or pressure the corresponding expression is: /*i -AF= I vdp. Jpt This equation is a general one, applicable to gaseous, liquid, and solid systems. In order to carry out the indicated integration, the functional relation between the pressure and volume of the system at constant temperature must be known. Prob. 7. Free-Energy Decrease Attending the Expansion of Gases. a. Derive an expression for the free-energy decrease attending the change in state of N mols of a perfect gas when at the temperature T its volume and pressure change from v\ and pi to v 2 and p z . b. Derive by integration an expression for the decrease of the free energy of N mols of a gas when at T its pressure changes from pi to p 2 , provided the pressures are moderate so that its pressure-volume relations are ex- pressed with sufficient accuracy by the equation pv=NRT(i+ap) given in Art. 15. c. Calculate by this expression the value in joules of the free-energy decrease attending the expansion of i mol of CO 2 from 4 atm. to i atm. at o-, for which substance at o the deviation-co- efficient a has the value 0.0068 per atmosphere. Ans. c, 3100. FREE ENERGY OF PHYSICAL CHANGES 231 *Prob. 8. Free-Energy Decrease Attending the Compression of Liquids. The change in volume v of liquid or solid substances with the pressure p can commonly be approximately expressed by the equa- tion V=V Q (I kp), where v is the volume at zero pressure and k is a quantity, called the compression-coefficient, which varies with the sub- stance and with the temperature, but may ordinarily be regarded as constant for variations of pressure up to 30 to 50 atm. a. Derive an expression for the decrease in free energy of such a substance when the pressure on it changes from pi to p z . b. Find the free-energy decrease in joules attending the compression from i atm. to 50 atm. at 20 of 10 g. of water, whose compression-coefficient at 20 is 0.000048 when the pressure is in atmospheres, and whose density at 20 and i atm. is 0.998. Ans. b, 49.7. 138. Free-Energy Changes Attending the Transfer of Substances between Solutions of Different Concentrations. As shown in the problems below, there may readily be derived the following expressions for the free-energy decrease which attends the transfer at the tem- perature T of that quantity of a substance which is N mols in the state of a perfect gas from an infinite quantity of a solution in which its vapor-pressure is pi, its mol-fraction xi, and its molality or molal concentration ci, into an infinite quantity of another solution in which its vapor-pressure is p*, its mol-fraction x*, and its molality or molal concentration^: (i) (3) The conditions under which each of these equations is exact will be evident from its derivation. Equation (i) holds true whatever be the mol-fractions or concentrations; but it involves the assumption that the vapor conforms to the perfect-gas law. Equation (2) holds true when, in conformity either with Raoult's law or with Henry's law, the vapor-pressures are proportional to the mol-fractions. Equa- tion (3) holds true when, in conformity with Henry's law, the vapor- pressures are proportional to the mol-fractions and when the latter are small enough to be substantially proportional to the concentra- tions; it is the expression commonly employed when a solute is trans- ferred from one dilute solution to another. 232 PRODUCTION OF WORK BY CHEMICAL CHANGES Derivation of the Free-Energy Equations. Prob. g. a. Formulate an exact expression (not involving the perfect-gas law) for the free-energy decrease attending the introduction at the temperature T of m grams of a pure substance at a pressure equal to its vapor-pressure p Q into an infinite quantity of a solution in which the substance has a vapor-pressure p. Note that this change in state can be brought about by the following reversible process: vaporize the m grams of the substance at the temperature T under a pressure p Q \ change the pressure of this vapor to p; and at this pressure condense the vapor into the solution, b. Derive from this expression an equation which holds true when the vapor conforms to the perfect-gas law. c. Formulate an expression for the free-energy decrease attending the transfer of N mols of a substance from an infinite quantity of a solution in which its vapor-pressure is pi into an infinite quantity of a solution in which its vapor-pressure is p z . Prob. 10. Show how the equation formulated in Prob. gc may be modified so as to contain the mol-fractions x\ and x z of the substance in the two solutions, a, when the vapor-pressures p\ and p z conform to Raoult's law; 6, when these vapor-pressures conform to Henry's law. c. State in the case of a solution consisting of two substances A and B the conditions of composition under which Raoult's law and under which Henry's law (and therefore under which the corresponding free-energy expressions) hold true approximately, d. Show that when the mol- fractions xi and x 2 are small, their ratio may be replaced by the ratio of the molalities or molal concentrations, which signify (Art. 35) the number of mols of the solute per 1000 grams or 1000 ccm., respectively, of the solvent. Since Henry's law applies to the concentrations of a definite chemical substance, it is evident that equation (3) must be applied separately to the transfer of each chemical substance, not to the transfer of the substance as a whole, if it exists in the solution as two or more different chemical substances as is the case with H + C1~ or any other partially ionized substance in aqueous solution, or with acetic acid in benzene solution in which it exists as C^H-A and (CzH-tOz)*. Although equation (3) was derived by the consideration of a process involving the vaporization of the solute and with the aid of the assump- tion that the pressure of the vapor was small enough to conform to the perfect-gas law, yet that equation relates only to the transfer of the substance from one solution to another; and it would be re- markable if its validity depended on whether the substance were volatile or on how large its vapor-pressure might be. In fact, it can be shown, by deriving equation (3) through a consideration of an osmotic process of transferring the substance from one solution to the FREE ENERGY OF PHYSICAL CHANGES 233 other, that the equation is exact, provided only that the substance behaves as a perfect solute, as shown by its conformity to the osmotic- pressure equation Pv Q = NRT considered in Art. 52. Applications of the Free Energy Equations. Prob. ii. a. Calculate the free-energy decrease attending the trans- fer at 20 of iNH 3 from an infinite quantity of a solution of the composi- tion i NH 3 .8|H 2 O in which its vapor-pressure is 80 mm. into an infinite quantity of a solution of the composition iNH 3 .2iH 2 O in which its vapor-pressure is 27 mm. b. Calculate the free-energy decrease attending the dissolving at 20 of iNH 3 at i atm. in an infinite quantity of a solution of the composition iNH 3 .2iH 2 O. Note. Throughout this chapter free-energy values are to be ex- pressed in calories. Prob. 12. Calculate the free-energy decrease attending the transfer at 20 of iNH 3 from an infinite quantity of o.i formal NH 4 OH solution into an infinite quantity of o.ooi formal NH 4 OH solution. The am- monium hydroxide is 1.3% ionized in the o.i formal and 12.5% ionized in the o.ooi formal solution, and the remaining ammonium hydroxide is y per cent dissociated into NH 3 and H 2 O in each solution. Prob. 13. The transfer at the temperature T of one formula-weight of HCO 2 H from an infinite quantity of a dilute solution in which its formality is c\ and its ionization is 71 into an infinite quantity of another dilute solution in which its formality is c 2 and its ionization is y t can be brought about by so transferring either i mol of HCO 2 H or one mol of H + and one mol of HCO 2 ~. a. Formulate an expression for the free- energy decrease attending each of these processes, b. Show that one of these expressions can be derived from the other with the aid of the mass-action equation for the ionization of the salt. c. Calculate the free-energy decrease by both expressions for the case that the tempera- ture is 25, the concentrations are o.i and o.ooi formal, and the ioni- zations are 0.045 an d 0.360. Prob. 14. Formulate an expression for the free-energy decrease attending the transfer of iK 2 SO 4 at 25 from a o.i formal solution to a o.ooi formal solution, assuming that the salt is completely ionized and that the ions act as perfect solutes. Note. The assumptions that largely ionized substances are com- pletely ionized and that their ions are perfect solutes are to be made, unless otherwise stated, also in all later problems. *Owing to the fact that ions at even small concentrations are im- perfect solutes, the free-energy decrease attending the transfer of largely ionized substances is expressed only as a rough approximation by equation (3). An expression exact at any concentration may, however, be formulated with the aid of the concept of activity pre- sented in Art. 113. Namely, it follows from the proportionality 234 PRODUCTION OF WORK BY CHEMICAL CHANGES between the activity of any definite substance in a solution and its vapor-pressure (assuming the vapor to be a perfect-gas) that the ratio of the activities of a substance in two solutions, or in general in two phases, may be substituted for the ratio of the pressures in equa- tion (i). Hence, for the transfer of N mols of a chemical substance at T from a solution or phase in which its activity is a\ to a solution or phase in which its activity is 02 the free- energy decrease is: -AF = NRT log - (4) #2 *This equation has practical significance, not so much because it furnishes a means of evaluating free energies, as because it supple- ments the primary definition of activity (given by the equation a\ja^ = pi/pz) by a secondary definition which affords another method of determining the relative activities of a substance in different solutions. Namely, it enables activities and activity-coefficients to be derived from the free-energy decrease attending the transfer of the substance in cases where this decrease can be obtained from other measure- ments than those of vapor-pressure. It will be shown in Art. 146 that this decrease can be computed from the electromotive force of certain types of voltaic cells; and applications of equation (4) will there be considered. 139. Free-Energy Changes Attending the Transfer of Substances between Different Phases. When the change in state consists in the transfer of a substance from one phase to another phase at the temperature and pressure at which the two phases are in equilibrium, the free-energy decrease is zero. Changes in state of this type are the transition, fusion, or vaporization of a solid substance, and the freezing or vaporization of a liquid substance, provided the two phases are at the temperature and pressure at which they are in equilibrium. This principle follows from the general expression for free-energy decrease, AF = W* A(/rc>), and from the fact that this type of change can be made to take place reversibly at the given pressure, so that the maxi- mum work W R is simply equal to that pressure times the volume- increase, that is, to p(v2vi), which is identical with A(pv). Prob. 15. Vaporization, Fusion, and Transition in Relation to Free Energy. Name a condition of temperature and pressure at which the free energies are equal and a condition under which they are not equal, a, for water as a liquid and as a gaseous phase; b, for water as a solid and as a liquid phase; c, for sulfur in the rhombic and the monoclinic forms (see Art. 115). FREE ENERGY OF PHYSICAL CHANGES 235 Another case of this type is the transfer of a substance from a solid phase to a solution saturated with respect to that substance at a definite temperature and pressure. In order that this change in state may take place reversibly, the solid substance is made to dissolve in an infinite volume of its saturated solution by increasing or decreasing the pressure on the two phases by an infinitesimal amount. The work produced in this process is again simply equal to the pressure times the change in volume (p A), and the free-energy decrease is therefore zero. Prob. 16. Solution in Relation to Free Energy. Find the maximum work and the free-energy decrease attending the melting of iH 2 O as ice at 1.69 into a 0.5 formal NaCl solution at its freezing-point 1.69 at i atm. The molal volume of ice is 19.6 can., and the increase in volume of an infinite quantity of the solution when iH 2 O is added to it is 1 8.0 ccm. Still another case of this type is the transfer at a definite temperature of a substance from one liquid phase to another liquid phase, thus from a solution of it in one solvent to a solution of it in another solvent not miscible with the first solvent. Provided the substance is present in the two phases at the equilibrium concentrations, the free-energy decrease attending such a transfer is zero, as shown in Prob. 17. Prob. 17. Distribution in Relation to Free Energy. A c\ formal solution of NH 3 in chloroform is in equilibrium with a c 2 formal solution of NH 3 in water at T. Show that the free-energy decrease is zero for the transfer of iNH 3 from an infinite quantity of the chloroform solution into an infinite quantity of the water solution, by considering a reversible process involving the vapor phase. When the substance in one phase at any definite temperature is not in equilibrium with the same substance in the other phase, the transfer of it from one phase to the other is attended by an increase or decrease in its free-energy. In determining this free-energy change, a process must as usual be conceived by which the transfer takes place reversibly, and the work attending this reversible process must then be evaluated. This is illustrated by the following problems. Free-Energy Decrease Attending the Conversion of Unstable into Stable Phases. Prob. 18. Find the free-energy decrease that attends the transition at 105 of iS from the rhombic into the monodinic form, each being at a pressure equal to its vapor-pressure, referring to the figure of Art. 115 for the data needed and assuming that the molecular formula of the sulfur vapor is SB. 236 PRODUCTION OF WORK BY CHEMICAL CHANGES Prob. 19. Find the free-energy decrease attending the transition of 32.06 g. of monoclinic to rhombic sulfur at 25 from the facts that at 25 the solubility in benzene of the monoclinic form is 23.2 g. and that of the rhombic form is 18.2 g. per liter of benzene, and that the molecular formula of sulfur in benzene has been shown by molecular-weight determinations to be S 8 . Consider a reversible process that involves dissolving the monoclinic form in its saturated solution, transferring the sulfur from this solution to one saturated with respect to the rhom- bic form, and causing the rhombic form to precipitate. Ans. 18 cal. *Prob. 20. a. Find the free-energy decrease that attends the change of iH 2 O from the state of liquid at 1 and 4.255 mm. (its vapor- pressure at i) to the state of ice at 1 and 4.215 mm. (its vapor- pressure at i) by considering a reversible process similar to that of Prob. 9. b. Find the free-energy decrease attending the same change in state, by considering a reversible process in which the water and ice are brought into equilibrium at 1 by compression. Ice melts at i when the pressure is 130 atm. The mean compression-coefficient (see Prob. 8) at o between zero pressure and this pressure is 0.000051 for water and 0.000034 for ice. The foregoing considerations may be summed up in the following statements: (1) The free energies of a substance in two different phases at any definite temperature are equal when the pressure or concentration is that at which there is equilibrium. (2) When the pressure or concentration is not that corresponding to equilibrium, the substance in the unstable state has the greater free energy; and namely, a free energy greater by the amount given by the expressions NRT log(/>i/^ 2 ) and NRT log (si/s z ), in which pi and p z represent the vapor-pressures of the substance in the unstable and stable states, respectively, and Si and s z represent its solubilities in any solvent in the unstable and stable states, respectively. This second principle evidently corresponds to the principles stated in Art. 115 that the unstable form of a substance has the greater vapor-pressure and the greater solubility in any solvent. These principles show that equality of the free energies of a sub- stance in two phases is a criterion of equilibrium; and that, when the free energies are not equal, a change tends to take place in that direc- tion in which there is an actual decrease in the free-energy. *Since the activity of the substance in any two phases is by defini- tion proportional to its vapor-pressure (Art. 113), it follows that the activity of a substance is the same in different phases which are in FREE ENERGY OF PHYSICAL CHANGES 237 equilibrium with one another. For example, when solid iodine is shaken with water and chloroform at 25 in a closed vessel till equi- librium is reached, the activity of the iodine is the same in. the solid, the gaseous, and the two liquid phases. This is a fundamental principle which is often made use of in applications of the mass-action law. Thus in the mass-action expression (HI) 2 /(H 2 S)(I 2 ) = K, for the equilibrium of the reaction H 2 S + I 2 = 2HI+S(s) in aqueous solution, the concentrations of the H 2 S and HI can be replaced by their vapor-pressures, and that of the I 2 can be replaced by its con- centration in a chloroform phase in equilibrium with the aqueous solution. Such substitutions are useful, either when the substance is an imperfect solute in the aqueous solution, or when its pressure in the gas phase or its concentration in another solvent can be more easily or more accurately determined than its concentration in the aqueous solution. *In case there is not equilibrium it follows from the relation between free-energy decrease and activities derived in Art. 138 that the ratio of the activities of the substance in the two phases is given by the equation: fli -AF 10g 02 ~ NRT Qualitatively this signifies that the activity of a substance is greater in the unstable state; for this state always passes over into the stable state with a positive free-energy decrease. 238 PRODUCTION OF WORK BY CHEMICAL CHANGES m. FREE-ENERGY CHANGES ATTENDING CHEMICAL CHANGES 140. The Free-Energy Equation for Chemical Changes between Perfect Gases. An expression for the free-energy decrease attend- ing a chemical change between perfect gaseous substances in terms of the equilibrium-constant for that change can be derived by the follow- ing considerations. Consider any chemical reaction 0A + 6B. . =eE+fF. .between the gaseous substances A, B,. . E, F,. . ; and consider a change in state which consists in the conversion at the temperature T of a mols of A, b mols of B,. .at pressures p A ', p* r . into e mols of E and /mols of F,. . at pressures E ', py. . This change in state is briefly expressed by the equation: a A (at p/) + 6B (at /) . . = E (at p* 9 ) +/F (at /). . . This reaction would take place continuously in one direction or the other (unless the pressures happened to be those corresponding to equilibrium) if infinite quantities of the gases, each at its specified pressure, were present together in a mixture; and the specified change in state would be realized if the period during which the specified number of mols reacted were considered. However, the change in state would not be taking place reversibly, since no mechanism is provided by which the work which the change is capable of producing would be actually produced. It is therefore necessary to conceive of some process by which the change in state can be made to take place reversibly. Such a process will now be described. In the process an apparatus like that shown in the figure will be employed. This apparatus consists of a reservoir and of a number of cylinders which communicate with it through walls, each of which is permeable for only one of the gases. These walls can be replaced at will by impermeable ones. The cylinders, which serve to hold the Gas A Gas B Gas E Gas F Equilibrium Mixture of A , B, E, and F FREE ENERGY OF CHEMICAL CHANGES 239 separate gases, are provided with weighted frictionless pistons. The reservoir contains a mixture of the gases at any partial pressures PA, ps,- -pE, PY,- -at which the chemical change is in equilibrium. Starting now with a mols of A, b mols of B,. . at pressures p A ', p* . . in their respective cylinders, consider the following process to be carried out at the temperature T under reversible conditions, always keeping the external pressure on any piston equal within an infinitesi- mal amount to that of the gas beneath. 1. The cylinders being closed temporarily by impermeable walls, raise or lower the pistons above the gases A, B,. .till the pressures change from PA, p* r -to those, p A , p s . ., prevailing in the equilib- rium-mixture. 2. Replace the impermeable walls by semipermeable ones, and force the a mols of A, b mols of B,. .into the reservoir, at the same time drawing out into the other cylinders at the pressures p E , p f ,. .pre- vailing in the equilibrium-mixture, the e mols of E, / mols of F,. ; which must form spontaneously out of A, B,. .in order that the equi- librium in the mixture may be maintained. 3. Replace the semipermeable walls under the gases E, F,. .by impermeable ones, and raise or lower the pistons above the gases E, F,. .till the pressures change from the equilibrium-pressures pz, pt,- -to the final pressures p*', />/. . By formulating, as is done in Prob. 21 expressions for the free- energy decrease attending each of the steps in this process and sum- ming up the three values, there is obtained in the case that all the pressures involved are so small that they conform substantially to the perfect-gas law the following expression for the free-energy decrease attending an isothermal chemical change between gaseous substances: Prob. 21. Derivation and Significance of the Free-Energy Equation. a. Formulate expressions for the free-energy decrease attending each of the three steps in the process described in the preceding text. b. By combining these expressions derive the free-energy equation there given, c. State explicitly the change in state to which this equation applies and the difference in the significance of the pressures in the two logarithmic terms; and name a familiar quantity that may be substituted in one of these terms, d. Give the form which the equation assumes when the initial and final partial pressures are all unity. 240 PRODUCTION OF WORK BY CHEMICAL CHANGES Prob. 22. Evaluation of the Free-Energy Decrease. a. Calculate the free-energy decrease in calories attending at 2000 the change 2H 2 (o.i atm.)+O 2 (o.5 atm.) = 2H 2 0(i atm.) from the fact that water- vapor at 2000 and i atm. is 1.85% dissociated into hydrogen and oxygen, b. Calculate also the free-energy decrease at 2000 when the initial pressures of the hydrogen and the oxygen (as well as the final pressure of the water- vapor) are each i atm. Ans. b, 57,120. Prob. 23. Derivation of the Mass- Action Law. Derive the mass" action law expressed in terms of pressures by considering in the deriva- tion of the free-energy equation that the same change in state is brought about by two reversible processes of the type described in the text differing in the respect that different equilibrium mixtures are contained in the reservoir. Prob. 24. Relation of the Free-Energy Change to the Tendency of the Chemical Change to Take Place. a. Show by the mass-action law that; when the pressures of the substances in a gaseous mixture are such that the free-energy decrease attending the reaction as computed by the free-energy equation has a positive value, the reaction as written takes place spontaneously from left to right; and that, when the pres- sures are such that the computed free-energy decrease has a negative value, the reaction as written takes place spontaneously from right to left; and therefore that the reaction always takes place in that direction in which there is a positive free-energy decrease, b. Show that equi- librium prevails when the pressures of the substances in a gaseous mixture are such that the computed free-energy decrease is zero. The conclusions reached in the preceding problem afford another example of the following general principles, already shown in Art. 139 to apply to the transfer of substances between phases: (1) When at any definite temperature the pressures or concentra- tions of the substances present in a system are such that a change in the state would be attended by zero change in free-energy, the substances involved in that change are in equilibrium with each other. That is, the condition AF = o is a criterion of the equilibrium of the sub- stances involved in any change in state. (2) When at any definite temperature the pressures or concentra- tions are such that a change in the state of the system would be attended by a change in its free-energy, the change tends to take place in that direction in which the free-energy actually decreases. That is, a system can spontaneously change its state only in that direction for which AF is positive. 141. The Free-Energy Equation for Chemical Changes between Perfect Solutes. It can be shown, as is done in Prob. 25, that the FREE ENERGY OF CHEMICAL CHANGES 241 free-energy decrease attending at the temperature T the chemical change: a A(at c>!) + &B(at c'). - = *E(at C E ') +/F(at c v '). . , where A, B. .E, F,. .are dissolved substances at the concentrations C A ', c*,. .c E ', c v '.., which are so small that the substances behave as perfect solutes, is given by the equation: In this equation, which is obviously closely analogous to that for gaseous substances, the quantities C A , c*,. .C E , c r ,. .are any small concentrations at which the substances are in equilibrium. Prob. 25. Derivation of the Free-Energy Equation. Derive the free-energy equation given in the preceding text by considering that all the solutes are volatile and that the change in state is brought about by vaporizing A, B,. . out of their solutions, converting them into E, F, . . in the gaseous state by the process described in the text of Art. 140, and condensing E, F, . .into their solutions; each of these changes being carried out under equilibrium-conditions so that it may be reversible. Represent by p A ', PA,. .pE,pE. .the partial vapor-pressures correspond- ing to the concentrations CA ', C A , . . CK, C E , . . , and assume that in ac- cordance with Henry's law the pressures and concentrations are pro- portional. Prob. 26. Evaluation of the Free-Energy Decrease. Calculate the free-energy decrease that attends at 25 the chemical change NH 4 + (at o.i f.) + CN-(at o.i f.) = NH 3 (at o.i f.) + HCN(at o.i f.), assuming that NH 4 CN in o.i formal solution at 25 is 47% dissociated into NH 3 and HCN, and that the remainder of the salt is completely ionized. Although this free-energy equation was derived above under the assumptions that the solutes are volatile and that their concentrations and vapor-pressures are so small that Henry's law is applicable, it can be shown that this equation (like equation (3) of Art. 138) is valid provided only that the concentrations are so small that the solutes conform to the laws of perfect solutions. The mass-action law for perfect solutes can be derived from this free-energy equation just as the mass-action law for perfect gases was derived in Prob. 23 from the free-energy equation of the preceding article. The validity of the mass-action law therefore involves only that of the laws of perfect solutions or of perfect gases; and it will hold true when applied to any definite chemical change with a degree of 242 PRODUCTION OF WORK BY CHEMICAL CHANGES exactness corresponding to that with which the substances involved conform under the given conditions of concentration or pressure to the laws of perfect solutions or of perfect gases. 142. The Free-Energy Equations for Chemical Changes between Solid Substances and Perfect Gases or Perfect Solutes. When pure solid substances are involved in a chemical change with gaseous or dissolved substances it is not necessary to include in the free-energy equation their pressures or concentrations; for these would obviously have the same value (that of the vapor-pressure or solubility of the solid) in the two logarithmic terms corresponding to the equilibrium conditions and to the initial and final conditions. The same is true of the pressure of a pure liquid when it is involved in a chemical change with gaseous substances that are not much soluble in it, as in the formation of liquid water from hydrogen and oxygen. Evaluation of the Free-Energy Decrease. Prob. 27. Calculate from the data of Prob. 39, Art. no, the free- energy decrease attending at 25 the chemical change: NH 4 SH(s) =NH 3 (i atm.)+H 2 S(i a tm.). Prob. 28. Calculate from the data of Prob. 41, Art. no, the free- energy decrease, attending at 357 the change in state: Hg(l) +!0,(i atm.) =HgO(s). , Prob. 29. Calculate from the data of Prob. 54, Art. 112, the free- energy decrease attending at 25 the chemical change: AgSCN(s)+Br~ (o.i f.)=AgBr(s)+SCN- (o.i f.). CHAPTER XI THE PRODUCTION OF WORK FROM ISOTHERMAL CHANGES BY ELECTROCHEMICAL PROCESSES I. CHANGES IN STATE AND IN FREE ENERGY IN VOLTAIC CELLS 143. Introduction. It has been shown in the preceding chapter that the free-energy decrease attending changes in state can be derived from the consideration of certain mechanical processes by which the changes in state take place reversibly; and thereby important rela- tions between free energy and the equilibrium conditions of chemical reactions have been deduced. In this chapter it is shown that electro- chemical processes afford another means of carrying out changes in state reversibly and hence of evaluating the free-energy decrease attending them. The changes in state occurring in voltaic cells are first considered (in Art. 144). The electromotive force of voltaic cells, the production of work by them, and the corresponding free- energy decrease are then treated (hi Art. 145) in their general aspects. In the following articles these general principles are applied to the determination of the electromotive force and of the separate electrode- potentials of special types of cells. 144. Changes in State in Voltaic Cells. A change in state can be made to yield electrical energy when it can be brought about by a process of reduction occurring in one place and a process of oxidation occurring in another place; for it is an inherent characteristic of reduc- tion processes that they are attended by a liberation of positive electricity or by an absorption of negative electricity, and of oxidation processes that they are attended by the opposite electrical effects. Thus the reduction of copper-ion to metallic copper may be repre- sented by the equation Cu++ = Cu + 20; and the oxidation of metallic zinc to zinc-ion, by the equation Zn + 2 =Zn+ + , where the symbol denotes, as in Art. 63, one positive electron or one faraday of positive electricity. If these two processes occur at the same place, as is the case when metallic zinc is placed in a copper-ion solution, no electrical effect is observed. If, however, the zinc is placed in a zinc-ion solution and the copper in a copper-ion solution, and if the two solu- tions are placed in contact, the reduction-process tends to occur at one place and to liberate positive electricity there, and the oxidation- 243 244 PRODUCTION OF WORK IN VOLTAIC CELLS process tends to occur at another place and to absorb positive electric- ity there, thereby producing a difference of potential or an electro- motive force between the two places; and if they are now connected by a metallic conductor, a current of electricity will flow through it, and this current can be made to produce work, for example, by passing it through an electric motor. Such an arrangement as that here described, in which an electromotive force is produced as a result of an oxidation-process and a reduction-process occurring at two different places, is known as a voltaic cell. It has been shown in Art. 136 that the quantity of work that can be produced by any isothermal process is determined solely by the change in state of the system. Hence the maximum work producible by the action of a voltaic cell is fully determined by the initial and final states of the substances of which it consists; and in any case under consideration these states, or the corresponding change in state, must be exactly specified. Thus, there must be stated, in addition to the temperature, not merely the chemical reaction that takes place in the cell, but also the conditions of 'pressure and concentration under which the substances involved in it are produced and destroyed, and any transfers of substances from solutions of one composition to those of another composition. In order that there may not be a finite change in the concentrations of the solutions, and therefore in the electromotive force of the cell, during the occurrence of the change in state, it will always be consid- ered that the solutions are present in infinite quantity, so that when a finite quantity of any substance is introduced into or withdrawn from one of the solutions of the cell there is only an infinitesimal change in its concentration. This method of consideration is much simpler for the treatment of the heat-content, free energy, and electro- motive force of solutes than the closely related one, commonly employed in the formulations of the differential calculus, where the introduction of an infinitesimal quantity of the solute into a finite quantity of the solution would be considered; for it enables equiva- lent or molal quantities of the solute to be directly dealt with. The character of the cell under consideration will be shown by writing the symbols of the pure substances and solutions in the order in which they are actually in contact with one another, commas being inserted to indicate the junctions at which, as will be explained later, an electromotive force is produced. The conventions described in 4 VOLTAIC CELLS IN GENERAL 245 Arts. 109 and 129 will be employed to indicate the state of aggregation of substances and the composition of solutions. Thus a cell consisting in series of metallic zinc, of a zinc chloride solution of the composition ZnCl 2 .iooH 2 0, and of metallic mercury covered with solid mercurous chloride, at a pressure of one atmosphere (as is always understood unless some other pressure is specified), will be represented by the expression: Zn(s), ZnCl 2 .iooH 2 O, Hg 2 Cl 2 (s) ^Hg(l). Similarly, a cell whose electrodes are an inert metal M in contact with hydrogen gas at i atm. and the same metal in contact with chlorine gas at 0.05 atm., and whose electrolyte is a o.i formal HC1 solution, will be represented by the expression: M+H 2 (g) (i atm.), HCl(o.i f.), Cl 2 (g) (0.05 atm.)+M. Or simply: H 2 (i atm.), HCl(o.i f.), Cl 2 (o.o5 atm.). So also a lead storage-cell whose electrodes are lead and an inert metal coated with lead dioxide, and whose electrolyte is a sulfuric acid solution, say of the composition H 2 SO 4 ioH 2 O, saturated with lead sulfate, will be represented by the expression: Pb(s) +PbSO 4 (s), H 2 SO 4 .ioH 2 O, PbSO 4 (s)+PbO 2 (s). /*> The change in state taking place in a cell may be expressed by an equation whose left-hand member represents the initial state, and whose right-hand member represents the final state, of the substances involved in the change. Thus the changes of state occurring when at 25 two faradays pass from left to right (as is always understood unless the opposite is specified) through the first two cells just formu- lated are shown by the equations: Zn(s) + Hg 2 Cl 2 (s) = 2 Hg(l) +Zn++Cl- 2 (in ZnCl 2 .iooH 2 O) at 25. H 2 (i atm.)+Cl 2 (o.o5 atm.) = 2H+Cl-(at o.i f.) at 25. \CThe changes in state resulting from the separate electrode processes /may be expressed by electrochemical equations similar to those de- scribed in Art. 63, but showing also the composition of the solutions in which the ions are produced or destroyed. Thus in the case of the first two of the above cells the changes in state at the anode and cathode may be represented by the equations: Zn(s) + 20 =Zn++ (in ZnCl 2 .iooH 2 O). Hg 2 Cl 2 (s) = 2Hg(l) + 2C1- (in ZnCl 2 .iooH 2 O) + 20. 246 PRODUCTION OF WORK IN VOLTAIC CELLS When, as in these cells, a gaseous or a solid non-metallic substance is in contact with a metal electrode, the adjoining solution is under- stood to be saturated with that substance. In such cells there are therefore in reality two different solutions, even though for the sake of brevity only one may be written in formulating the cell. Thus these cells ought strictly to be represented by the following expressions, in which 5 f . denotes the formal solubility of the substance in the solution in contact with it, thus of Hg 2 Cl2 in a solution of ZnCl2.iooH 2 O: Zn(s), ZnCl 2 .iooH 2 O, ZnCl 2 .iooH 2 O+Hg 2 Cl 2 (s f.), Hg 2 Cl 2 (s)-f-Hg(l). H 2 (i atm.), HCl(o.i f.)+H 2 (> f. or 0.00076 f.), HCl(o.i f.)-f- Cl 2 (s f. or 0.0044 f.), Cl 2 (o.o5 atm.). Pb(s)+PbSO 4 (s), H 2 SO 4 .ioH 2 O+PbSO 4 (s f.), H 2 SO 4 .ioH 2 O+ PbSO 4 f.)+Pb(SO 4 ) 2 (io- 8 f.), PbSO 4 (s)+PbO 2 (s). The recognition of the fact that there are in such cells two solutions of slightly different composition is of great importance in considering the mechanism of voltaic action and in evaluating the electrode-poten- tials discussed in Arts. 150-152. When, as in the three cells just considered, the cell contains two solutions of substantially the same composition (in the respects that in both solutions the ions present at considerable concentrations are the same, that the concentrations of these ions are substantially the same, and that the solvent-medium as a whole is substantially the same), the ion-transferences need not be considered; for, though there is transference from the solution around one electrode to that around the other electrode, there is not an appreciable free-energy change, since the two solutions have substantially the same composition. Changes of State in Cells with a Single Solution. Prob. i. Formulate an electrochemical equation which expresses fully the change in state that takes place at each electrode when two faradays pass through the hydrogen-chlorine cell above mentioned; and derive from these equations the expression for the change in state taking place in the cell as a whole. Prob. 2. Formulate electrochemical equations expressing the changes in state at the electrodes when two faradays pass at 20 through the lead storage-cell formulated above ; and derive from them a corresponding expression for the change of state in the cell as a whole. When the cell contains around the two electrodes solutions of substantially different composition, as in the cells formulated in Prob. 3, it is necessary to consider not only the processes that take VOLTAIC CELLS IN GENERAL 247 place at the two electrodes, but also thejtransference that occurs ')/ inj;he_solutipns^ for in such cells the ion-constitutents are transferred Trom a solution of one composition to a solution of another composition. Changes in State in Cells with Two Solutions. Prob. 3. Write equations expressing (i) the changes in state which take place as a result of the electrolysis, (2) the changes in state brought about by the transference of the ions, and (3) the resultant change in state in the cell as a whole, when one faraday passes at 18 through each of the following cells, taking 0.166 as the transference- number of the chloride-ion in the two HC1 solutions: a. H 2 (i atm.), HCl(o.oi f.), HCl(o.i f.), H 2 (i atm.). b. Cl 2 (i atm.), HCl(o.oi f.), HCl(o.i f.), Cl 2 (i atm.). Note that the transference effects in these cells are similar to those described in Art. 64, the left-hand solution being regarded as the anode- portion and the right-hand solution as the cathode-portion. 145. The Production of Work and the Corresponding Decrease of Free Energy in Voltaic Cells. In general, when a quantity of positive electricity__Q_jlows between two places, such as the electrodes of a voltaic cell, between which there is a fall of potential or electromotive force jjuia the direction of the flow, a quantity of work equal to the product E-ean be produced. Now, according to Faraday's law (Art. 62), the quantity of electricity flowing through a voltaic cell is strictly proportional to the number of equivalents N that are involved in the chemical change at each electrode;. that is,^Q=NF, where F repre- sents the quantity of electricity (96,500 coulombs) that passes when a reaction involving one equivalent of each of the reacting substances takes place at each electrode. The maximum electrical work that can be obtained when N faradays of electricity pass through a voltaic cell which produces an electromotive force E is therefore equal to E N F. When a voltaic cell acts reversibly there is ordinarily produced, in addition to the electrical work, a quantity of mechanical work corresponding to the changes in volume of the different parts of the cell taking place under their respective pressures. The total work W R is therefore ENF-J- S(^Az>). Since by definition (Art. 136) the 'free-energy decrease is equal to W R A(S^), it is equal simply to the electrical work that can be produced. That is, for any change in state taking place in a voltaic cell under constant pressures: AF=ENF. The numerical value of the electromotive force E of the cell will in this book be given a positive sign when the cell tends to produce 248 PRODUCTION OF WORK IN VOLTAIC CELLS a current of positive electricity through the cell in the direction in which it is written, and a negative sign when the cell tends to pro- duce such a current in the opposite direction. The symbol N will denote the number of faradays of positive electricity that are considered to pass through the cell from left to right, its numerical value being given a negative sign when positive electricity is considered to pass in the opposite direction. The maximum electrical work and the free-energy decrease calculated by the equation will then have, as usual, a positive sign when the cell produces electrical work, and a negative sign when external work is expended upon the cell. Its value will be in joules when the electromotive force is in volts and the quantity of electricity in coulombs. The Maximum Work and Free-Energy Decrease Attending Changes in Voltaic Cells. Prob. 4. -The electromotive force at 15 of the Daniell cell, Zn(s) +ZnSO 4 .7H 2 O(s), ZnSO 4 (sf.), CuS0 4 (s f.), CuSO 4 .5H 2 O(s) +Cu(s), is 1.093 volts. Give the value of NF in coulombs and of the electrical work in joules required to deposit iZn on the electrode; and state what the sign of each of these quantities signifies. Prob. 5. a. Calculate exact values of the electrical work and the mechanical work that are produced when i f araday passes under revers- ible conditions through the cell H 2 (i atm.), HCl(o.i f.), Cl 2 (o.o5 atm.) at 25. The electromotive force of this cell is 1.451 volts. The increase in the volume of an infinite quantity of o.i f. HC1 solu- tion caused by introducing iHCl into it is 18.7 ccm. b. Find the corresponding free-energy decrease. CELLS UNDERGOING PHYSICAL CHANGES 249 II. THE ELECTROMOTIVE FORCE OF CELLS UNDERGOING ONLY CHANGES IN CONCENTRATION OR PRESSURE 146. Change of the Electromotive Force of Voltaic Cells with the Concentration of the Solutions. The considerations of the preceding articles make it possible to calculate the change that is produced in the electromotive force of a cell by varying the concentration of the solutions contained in it. Thus the difference between the electro- motive forces of the cells H 2 (iatm.), HCl(o.ooi f.), AgCl(s)+Ag(s) and H 2 (i atm.), HCl(o.i f.), AgCl(s)+Ag(s) can be derived by con- sidering that one faraday passes through these two cells arranged in series in opposition to each other, by noting what the resultant change in state is, and by equating the two expressions for the attendant free-energy decrease derived in Arts. 145 and 138. The exact treatment of such cells is complicated by the uncertainty as to the degree of ionization of largely ionized substances and by the fact that their ions are not perfect solutes. A rough approximation to the truth is, however, secured in electrochemical calculations, as hi those relating to mass-action, by employing the hypotheses that these substances are completely ionized up to moderate concentrations, such as i formal, and that their ions are perfect solutes. These hypotheses are to be applied to these substances in all the problems of this chapter, unless otherwise stated. The errors to which these hypotheses lead will be indicated in some cases by a comparison of the calculated with the observed values. Effect of Concentration on Electromotive Force. Pro}), 6. Calculate the difference between the electromotive forces at 25 of the two cells formulated in the preceding text. Compare the result with the difference between the observed electromotive forces of these cells, which are 0.5789 and 0.3522 volt. p ro b. 7. Calculate the difference between the electromotive force at 25 of the cell H 2 (i atm.), HCl(io f.), Cl 2 (i atm.) and that of the cell H 2 (i atm.), HC1(6 f.), Cl 2 (i atm.). The vapor-pressures of HC1 in the two solutions at 25 are 4.2 and 0.14 mm. p r O b. s. State what data are needed in order to calculate the electro- M motive force EJ at 18 of the cell H 2 (i atm.), H 2 SO 4 (io f.), O 2 (i atm.) from that E 2 of the cell H 2 (i atm.), H 2 SO 4 (o.oi f.), O 2 (i atm.); and formulate an expression by which the calculation could be made. Prob. p. -The cell Ag(s) +AgCl(s), HCl(o.i f.), Cl 2 (i atm.) has at 25 an electromotive force of 1)135 volts. How much would its electro- motive force be changed by substituting HCl(o.oi f.) for the HCl(o.i f.)? 250 PRODUCTION OF WORK IN VOLTAIC CELLS *Measurements with such combined or double cells form the basis of one of the important methods by which values of the activity- coefficients, such as are given hi the table of Art. 113, can be derived. The value of the electromotive force gives directly the free-energy decrease, and from this by equation (4) of Art. 138 the ratio of the products of the activities of the ions in the two solu- tions .and of their activity-coefficients can be computed, as is illus- trated by the following problems. * Derivation of I on- Activity-Coefficients from Electromotive Forces. *Prob. 10. .a. From the observed values, 0.5789 and 0.3522, for the electromotive forces of the cells formulated in the text and considered in Prob. 6, calculate the ratio of the products of the activities of hydro- gen-ion and chloride-ion in o.i formal and in o.ooi formal HC1 solution. b. Find the ratio of the ion-activity-coefficients at o.i formal and o.ooi formal, assuming that hydrogen-ion and chloride-ion have equal activities in the same solution. *Prob. ii. At 25 0.224 volt is the electromotive force of the cell: Hg(l) +HgO(s) , KOH(o. 4 5 f .) , K^l) , KOH(o.oo 4 5 f ) , HgO(s) +Hg(l) , where K w Hg represents a solution of potassium in mercury of definite but unknown concentration. Find the ratio of the ion-activity-coeffi- cients of KOH in the two solutions, assuming that the two ions have equal activities at the same concentration. 147. The Electromotive Force of Concentration-Cells. A cell which consists of two identical electrodes and of two solutions con- taining the same substance at two different concentrations is called a concentration-cell. The cell, Zn(s), ZnCl 2 (o.i f.), ZnCl 2 (o.oo5 f.), Zn(s), is an example of a concentration-cell, as are also the cells for- mulated in Prob. 3, Art. 144. The electromotive force of a concentration-cell containing the solute at small concentrations can be calculated by considering the changes in concentration that occur at each electrode as a result both of the electrolysis and the transference, when N faradays pass through it, by formulating in accordance with Art. 138 an expression for the free-energy decrease attending these concentration changes, and by equating this expression with E N F. Evaluation of the Electromotive Force of Concentration Cells. Prob. 12. Calculate the electromotive force at 18, a, of the cell formulated in Prob. 30 above; b, of the cell formulated in Prob. 36. c. Formulate general algebraic expressions involving the concentrations c' and c" and the transference-number T A or T C , for the electromotive force of cells like these, in which the cation, or the anion, is involved in the electrode process. Ans. a, 0.0192. CELLS UNDERGOING PHYSICAL CHANGES 251 Prob. 13. Calculate the electromotive force at 18 of the following cell, using as the transference-number that derived from the ion-con- ductances given in Art. 76: Zn(s), ZnCl 2 (o.i f.), ZnCl 2 (o.oo5 f.), Zn(s). Ans. 0.0660. *Prob. 14. Calculate the electromotive force at 25 of the cell Ag+AgCl, KCl(o.s f.), KCl(o.osf.), AgCl+Ag, taking 0.504 as the transference-number of the chloride-ion, a, assuming that the salt is completely ionized and that its ions are perfect solutes; b, assuming that the salt has ionization values equal to the conductance-viscosity ratios as given in the table in Art. 113 and that its ions are perfect solutes; c, using the values of the ion-activity-coefficients given in the table in Art. 113. d. Tabulate these three results together with the experi- mentally determined value (0.0536 volt) of the electromotive force. Another type of concentration-cell is that in which there is a single aqueous solution in contact with electrodes consisting of some metal dissolved at two different concentrations in mercury. For example, the cell Zn.iooHg, ZnCl 2 .iooH 2 O, Zn.2ooHg is of this type. Since most of the metals dissolved in mercury at small concentrations have been shown by vapor-pressure and freezing-point measurements to be nearly perfect solutes having monatomic molecules, the electromotive force of cells having such solutions as electrodes can be calculated with the aid of the preceding considerations. A similar type of concentration-cell is that in which the two elec- trodes consist of some inert metal surrounded by the same gas at two different pressures, as in the oxygen cell M+O 2 (i atin.), KOH(i f.), 2 (o.iatm.)-fM. Electromotive Force of Cells with the Electrode Substance at Different Concentrations. Prob. 15. Calculate the electromotive force at 18 of the zinc- mercury cell formulated in the preceding text. Prob. 16. Calculate the electromotive force at 25 of the oxygen cell formulated in the preceding text. 252 PRODUCTION OF WORK IN VOLTAIC CELLS HI. THE ELECTROMOTIVE FORCE OF CELLS UNDERGOING CHEMICAL CHANGES AND ITS RELATION TO THEIR EQUILIBRIUM CONDITIONS 148. Chemical Changes in Voltaic Cells. In the cells thus far considered the change of state consists only in a transfer of one or more of the substances from one pressure or concentration to another. In most voltaic cells, however, a chemical change takes place. Thus in the Daniellcell the reaction Zn(s)+CuSO 4 = Cu(s)+ZnSO 4 occurs. Other examples of cells in which similar chemical changes take place are given in Art. 144. A chemical change somewhat different in character from those just considered takes place in cells whose half-cells consist, not of solid or gaseous elementary substances in contact with solutions of their ions, but of an inert metal electrode in contact with two solutes in different stages of oxidation. Thus the electrode-reactions in the cell M, Fe^Cl- 2 (<:i)+Fe+++a-3fe), Cl-faJ+Ckfa), M, attending the passage of one faraday are Fe++-f- =Fe' HH - and f C1 2 = C1-+ , and the whole reaction is Fe ++ Cl-2+iCl 2 = Fe +++ Cl- 3 . Such cells do not, however, require special treatment; for the principles applicable to other cells can be readily extended to them, as will be seen in the following articles. 149. The Electromotive Force of Cells in Relation to the Equi- librium Conditions of the Chemical Reactions. The free-energy decrease attending chemical changes between substances at any pressures or concentrations is given (as shown in Art. 145) by the general expression AF = ENF, in which E represents the electromotive force of a voltaic cell in which the chemical change takes place re- versibly. Another form of expression for the free-energy decrease attending chemical changes between substances at small pressures or concentrations was derived in Arts. 140-142 from a consideration of their equilibrium conditions. Combination of these two expressions leads to the following relation between the electromotive force E of the cell, the equilibrium-constant K of the chemical change taking place in it, and the actual concentrations, C E ', c r ',. .C A ', C B ', in the cell of the chemical substances involved in the chemical change: -AF-zrt.p-.Rr (logtf-log When a gaseous substance is involved in the chemical change, its actual pressure may be substituted for its concentration in the last CELLS UNDERGOING CHEMICAL CHANGES 253 term of this expression; but in that case its equilibrium pressure must be correspondingly substituted in the expression for the equilibrium- constant. This expression makes possible the calculation of equilibrium- constants from electromotive forces, or conversely that of electro- motive forces from equilibrium-constants, as shown in the following problems. Calculation of Equilibrium-Constants from Electromotive Forces. Prob. 17. Calculate the dissociation-pressure in atmospheres of solid AgCl at 25 from the electromotive force, 1.135 volts, of the voltaic cellAg(s)+AgCl(s), HCl(o.i f.), Cl 2 (i atm.). Ans. 4 .2Xio- 39 . Prob. 18. a. Calculate the equilibrium-constant at 25 of the reac- tion H 2 (g)+2AgCl(s)=2H + +2Cl-+2Ag(s) from the electromotive force, 0.3522 volt, of the cell H 2 (i atm.), HCl(o.i f.), AgCl(s)+Ag(s). b. Calculate the equilibrium pressure in atmospheres of the hydrogen gas that would be produced by the action of metallic silver on i f. HC1 solution. Ans. 6, i.2Xio~ 12 . Prob. ig. a. From the data given in Probs. 17 and i8a find the electromotive force of the cell H 2 (i atm.), H + Cl~(o.i f.), Cl 2 (i atm.). b. Calculate from this electromotive force the equilibrium-constant of the reaction H 2 (g) +Cl 2 (g) = 2H++2C1-./ c. Calculate this equilibrium- constant from the equilibrium-constants found in Probs. 17 and i8a. Ans. b. 1.9 Xio 50 . Calculations of Electromotive Forces from Equilibrium-Constants. Prob. 20. a. Calculate the electromotive force at 25 of the cell H 2 (i atm.), H 2 SO 4 (o.oi f.), O 2 (i atm.) from the equilibrium-constant at 25 of the reaction 2H 2 0(g) =2H 2 (g)+O 2 (g), which has been in- directly derived from equilibrium measurements at higher temperature and found to have the value 9.7 Xio~ 81 . The vapor-pressure of water at 25 is 24 mm. b. Calculate the minimum electrical energy re- quired to decompose iH 2 O by the electrolysis of a o.i f. KOH solution at 25 under a barometric pressure of 784 mm. Ans. &, 1.228. Prob. 21. a. Calculate the electromotive force at 25 of the cell Cl 2 (i atm.), HCl(io f.), O 2 (i atm.) from the equilibrium-constant of the reaction 4HCl(g) + O 2 (g) = 2H 2 O(g) + 2Cl 2 (g), which has been derived from measurements at higher temperatures of the equilibrium of this reaction (involved in the Deacon process of producing chlorine), and found to have at 25 the value i.oXio 13 . The vapor-pressures at 25 of H 2 O and HC1 in 10 f. HC1 solution are 9.4 mm. and 4.2 mm., respec- tively, b. Assuming the process could be made reversible, calculate how much electrical energy might be obtained in producing i mol of chlorine gas by an electric current passing through a 10 f. HC1 solution at 25 between electrodes of an inert metal under a barometric pressure of 750 mm., the solution around the cathode being kept saturated with air. Ans. 0,0.1151. 254 PRODUCTION OF WORK IN VOLTAIC CELLS IV. ELECTRODE-POTENTIALS AND LIQUID-POTENTIALS 150. The Nature of Electrode-Potentials. The electromotive force produced by a voltaic cell is the sum of the electromotive forces produced at the junctions between the electrodes and the solutions and of the electromotive forces produced at the junctions between the different solutions that may be present in the cell. Thus the electro- motive force of the Daniell cell is the algebraic sum of the electro- motive force from the zinc to the zinc sulfate solution, that from the zinc sulfate solution to the copper sulfate solution, and that from the copper sulfate solution to the copper. The electromotive forces at the electrodes are commonly called electrode-potentials; and those at the junctions of the solutions, liquid-potentials (or diffusion-potentials). The consideration of these partial electromotive forces is important in the respects that it shows more clearly the separate factors which determine electromotive force and the mechanism by which it is produced; that it enables the electromotive force of a large number of cells to be calculated from a small number of experimentally deter- mined constants; and that it affords a simple means of predicting the equilibrium conditions of oxidation and reduction reactions. The factors on which electrode-potentials depend will be first considered. Like ordinary chemical changes, the electrochemical reactions that occur at electrodes have definite equilibrium conditions, and take place in one direction or the other till these conditions are satisfied. For example, when a neutral plate of copper is placed in a o.i formal CuSO4 solution at 25, the equilibrium conditions of the reaction Cu(s) + 2 = CU++ require that copper ions deposit on the plate. This gives the plate a positive charge, and raises it to a. higher potential than the solution. This creates a counter electromotive force which tends to drive positive electricity, and therefore the positively charged copper ions, back into the solution. Equilibrium is established when this counter electromotive force just compensates the electromotive force arising from the mass-action tendency of the copper ions to deposit. In other cases the tendency of the reaction to take place may produce an electromotive force directed from the metal to the solution. For example, in order to satisfy the equilibrium conditions of the reaction Zn(s) + 2 =Zn++, zinc placed in contact with o.i formal ZnSC>4 solution must pass into the solution as zinc ions until enough positive electricity accumulates in the layer of solution adjoin- ing the electrode to produce a counter electromotive force equal to that ELECTRODE-POTENTIALS AND LIQUID-POTENTIALS 255 corresponding to the tendency of the zinc to pass into solution. In general, starting with a definite concentration of the ion and definite pressure of the gas (in case a gas is involved) at a definite temperature and with electrically neutral conditions, the reaction will proceed on the surface of the electrode in one direction or the other till enough positive or negative electricity is accumulated on the electrode and withdrawn from the adjoining layer of solution to produce a potential- difference sufficiently great to compensate the inherent tendency of the chemical reaction to take place. The quantity of ion that has to deposit on the electrode or be produced in the solution in order to bring the electrode to the equi- librium potential, which seldom exceeds one or two volts, is however extremely small, owing to the very large value of the faraday in rela- tion to the quantity of electricity which produces considerable elec- trostatic effects. Thus when a large glass flask containing dilute sulfuric acid is coated on the outside with tinfoil and the latter is charged with positive electricity at 1000 volts, and when a capillary mercury electrode is inserted in the acid solution and this electrode is connected to earth, it is found that a bubble of gas consisting of io~ 10 equivalent of hydrogen is set free on the surface of the mercury. 151. The Expression of Electrode-Potentials. Although attempts have been made to determine by various experimental methods the absolute values of electrode-potentials, yet the measurements made by these methods are not nearly so accurate nor reliable as those of the electromotive force of ordinary cells. It is therefore customary to adopt as the value of the electrode-potential of any half-cell (such as Zn(s),ZnCl 2 .iooH 2 O) the electromotive force of a whole cell which consists of the half-cell under consideration combined with a standard half-cell. The electrode-potential of the standard half-cell is thereby arbitrarily assumed to be zero. Various half-cells have been used as such standards of reference; but it has now become a fairly uniform practice to employ the mold hydrogen electrode, which consists of hydrogen gas at a partial pressure of one atmosphere in contact, with the aid of an inert metal electrode (such as platinized platinum), with an aqueous solution i molal in hydrogen-ion. This half-cell is re- presented by the formula H 2 (i atm.), H+(i m.). In this book this half-cell will always be used as the standard of reference; it being further specified that the solution has such a hydrogen-ion concen- tration that it produces the same thermodynamic and mass-action 256 PRODUCTION OF WORK IN VOLTAIC CELLS effects as a solution containing i mol of hydrogen-ion per 1000 grams of water would produce if the hydrogen-ion were a perfect solute at that concentration. (* In other words> the activity as denned in Art. 113, not the concentration, of the hydrogen-ion is to be i molal.) The electrode-potential of any half-cell is therefore equal to the electro- motive force of the whole cell formed by combining it with this stand- ard half -cell. Thus the electrode-potential of the half -cell Zn(s), ZnCl2.iooH 2 O is equal to the electromotive force of the whole cell Zn(s), ZnCl 2 .iooH 2 O || H+(i m.), H 2 (i atm.); and the electrode-po- tential of the half-cell Cl 2 (i atm.), HCl(o.i f .) is equal to the electromo- tive force of the cell Cl 2 (i atm.), HCl(o.i f.) || H+(i m.), H 2 (i atm.). It is, however, further understood that in evaluating this electromotive force the liquid-potential between the two solutions has been sub- tracted from the measured electromotive force of the whole cell a fact which is indicated in the formulation of the cell by inserting parallel lines, instead of a comma, at the liquid junction, as is done in the cells just considered. It follows from these conventions that the electrode-potential has a positive sign when positive electricity tends to flow from the electrode to the solution, and a negative sign when it tends to flow in the reverse direction. In using data from outside sources, it is to be noted that foreign electrochemists employ the opposite convention as to the sign of electrode-potentials. In the case of liquid-potentials the convention here adopted is the same as in the case of whole cells; namely, the liquid-potential is given a positive sign when positive electricity tends to flow from the solution whose symbol is written on the left-hand side to the solution whose symbol is written on the right-hand side; and it is given a negative sign in the reverse case. Prob. 22. Conventions Relating to Electrode-Potentials and Liquid- Potentials. a. Formulate the whole cell whose electromotive force is equal to the electrode-potential of the half-cell Ag(s)+AgCl(s), KCl(o.i f.). b. From the facts that at 25 this electrode-potential is 0.289 volt and that of H 2 (i atm.), HCl(o.i f.) is 0.064 volt, and that the liquid-potential of KCl(o.i f.), HCl(o.i f.) is 0.028 volt, calculate the electromotive force of the cell H 2 (i atm.), HCl(o.i f.), KCl(o.i f.), AgCl(s)+Ag(s). 152. Change of Electrode-Potentials with the Ion-Concentrations, and the Concept of Molal Electrode-Potentials. Just as the total ELECTRODE-POTENTIALS AND LIQUID-POTENTIALS 257 electromotive force of a cell is determined solely by the change in state that takes place in it, so any electrode-potential is determined solely by the change in state that takes place at the electrode. With the aid of this principle, by equating the two general expressions for free-energy decrease derived in Arts. 138 and 145, the change of an electrode-potential with the concentration of the ions or with that of any other gaseous or dissolved substance involved in the electrode reaction can be formulated, as shown by the following problems, provided the concentrations are so small that the dissolved substances behave as perfect solutes. Prob. 23. Change of Electrode-Potential with the Concentration or Pressure of the Substances Involved in the Electrode Reaction. a. For- mulate an electrochemical equation expressing fully the change in state that occurs when two faradays pass from the electrode to the solution in the half-cell Cl 2 (/> atm.), Cl~(c m.), and give an expression for the free-energy decrease in terms of Its electrode-potential E. b. Describe a process of producing this same change in state which involves three steps, of which one is brought about with the aid of half-cell C1 2 (i atm.), Cl~(i m.), whose electrode-potential is E, and formulate expressions for the free-energy decrease that attends each of the three steps of the process and the process as a whole, regarding the substances as perfect solutes, c. Combine these results, in accordance with the law of initial and final states, so as to give an expression for the difference in the electrode-potentials of the two half-cells. The electrode-potential of a half-cell in which all the substances taking part in the electrode reaction are considered to be perfect solutes at a concentration of i molal or perfect gases at a pressure of i atm. is called the molal electrode-potential E of the half-cell. The electrode-potential E of any half-cell in which the electrode reaction is expressed by the equation: a A(at C A ). . +b B(at p*). . +N0 =/.. \ E = E -^ 10 ^-^VT' This equation holds true strictly only for perfecTsolutes and perfect gases, and approximately for solutes or gases only at moderate con- centrations or pressures; and correspondingly, the molal electrode- potential is not that actually observed when the concentrations of the 258 PRODUCTION OF WORK IN VOLTAIC CELLS solutes are i molal, but is the value calculated by the equation for the concentration i molal from the electrode-potentials of half-cells containing the solutes at much smaller concentrations. (*In other words, the molal electrode-potential refers strictly to an activity, not to a concentration, of i molal.) In the problems of this chapter, however, unless otherwise stated, largely ionized substances are as an approximation to be regarded as completely ionized, and their ions are to be treated as perfect solutes up to a concentration of i molal. For convenience in numerical calculations it may be noted that, when ordinary logarithms are used, the coefficient 2.T > oT,RT/Nif pre- ceding the logarithmic term has the value i.984Xio~ 4 r/N volt, or at 25 the value 0.05915^ volt. This last number, approximately 0.059 volt, is evidently the amount by which the electrode-potential at 25 changes when the molality of any univalent ion involved in the electrode reaction is varied tenfold. Relation between Actual and Molal Electrode-Potentials. Prob. 24. Derive by a consideration like that employed in Prob. 23 the general expression given above. Prob. 25. Write the electrochemical reaction that takes place when electricity passes from the electrode to the solution in each of the following half -cells, where the molalities of the solutes are represented (as in mass-action expressions) by their chemical formulas enclosed within parentheses; and formulate the corresponding expression for the relation between the molal electrode-potential and the actual electrode potential at these molalities: a. Zn(s), Zn++ at (Zn++) m. |Cl-at(Cl-)m. * M ' {C1 2 at(Cl 2 )m. M fOH-at (OH-)m. ' \O 2 at/> atm. > ++ at (Fe ++ ) m. ^ +++ at (Fe +++ ) m. e. Hg(l) +H g2 S0 4 (s), S0 4 =at (SO 4 =) m. , ,, f Hg 2 ++ at (Hg 2 ++) m.+Cl- at (Cl~) m. M > \HgCl, at (HgCl,)m. M JMn0 4 = at (Mn0 4 = ) m. g ' ' \MnOr at (MnOr) m. Prob. 26. The potential of a platinum electrode in an acid solution of potassium iodate and iodine is found to depend only on the concen- fFe 'iFe ELECTRODE-POTENTIALS AND LIQUID-POTENTIALS 259 trations of H+, IO 3 ~, and I 2 . a. Write the electrode-reaction attending the passage of electricity from the electrode to the solution, b. For- mulate an expression by which the electrode-potential at 25 for any small concentrations (H+), (IO 3 ~), (I 2 ) could be calculated from the molal electrode-potential. Evaluation of Molal Electrode-Potentials. Prob. 27. Calculate an approximate value, under the assumption of complete ionization, of the molal electrode-potential of Zn(s), Zn +f - / from the fact that the cell Zn(s), ZnCl 2 (o.oi.f.), HC1 (0.02 f.), H 2 (g) ( has at 25 an electromotive force of 0.695 v lt, when the hydrogen gas is slowly passed under a barometric pressure of 754 mm. through the HC1 solution. The vapor-pressure of water at 25 is 24 mm., and the liquic^potential in the cell is estimated to be 0.030 volt. Ans. /0.766J Prvb. 28. Calculate the molal electrode-potential at 25, a, of Ag(s) +AgCl(s), Cl~, and b, of Ag(s), Ag+, from the facts that the cell H 2 (i atm.), HCl(o.i f.), AgCl(s) + Ag(s) has at 25 an electromotive force of 0.3522 volt and the solubility of silver chloride at 25 is i.3oXio- 5 formal. Consider the acid to be completely ionized and its ions to be perfect solutes. Ans. a, 0.234; b, 0.812. *Prob. 29. Compute more exact values of the two molal electrode- potentials calculated in Prob. 28, taking into account the activity- coefficient of the ions in the HC1 solution, as given in the table of Art. 113. Ans. a, 0.223; b, 0.801. *Pr ob. 30. Electrode-Potential of the Normal Calomel Electrode. Calculate the electrode-potential of Hg(l)+Hg 2 Cl 2 (s), KCl(i n.) at 25 from the following data: The molal electrode-potential of Hg(l) + Hg 2 Cl 2 (s), Cl~ has been computed to be 0.270 volt by the method illustrated by Prob. 28; a i n. KC1 solution is 1.033 f- m KC1; ion-activity coefficient of KC1 in this solution is 0.633. Ans. n The difference, -0.281 volt, between the electrode-potentiai-<# the normal calomel electrode, which is the half-cell represented by the formula Hg(l) + Hg 2 Cl 2 (s), KCl(i n.), and that of the molal hydrogen electrode is a quantity of much practical importance, since other electrode-potentials are often measured directly against the normal calomel electrode, instead of against a hydrogen electrode, because of greater experimental convenience and because the liquid-potentials involved are usually much smaller. 153. Values of the Molal Electrode-Potentials. The following table contains some of the more accurately determined values of the molal electrode-potentials at 25 and one atmosphere, referred to that of the molal hydrogen electrode taken as zero. 260 PRODUCTION OF WORK IN VOLTAIC CELLS MOLAL ELECTRODE-POTENTIALS AT 25. Reduced Oxidized Electrode- Reduced Oxidized Electrode- state state potential state state potential Li Li+ 2.958 Cu-hCl- CuCl(s) O.II9 Rb Rb+ 2.924 Ag+Cl- AgCl(s) 0.224 K K+ 2.922 Hg+Cl- ^Hg 2 Cl 2 (s) O.27O Na Na+ 2.713 Cu Cu"^ -0-345 Zn Zn++ 0.758 2OH- ^0 2 +H 2 O -0.399 Fe Fe++ 0.441 Cu Cu+ -0-47 Cd Cd++ 0.398 I- ^I 2 (s) -0.536 Tl T1+ 0.336 Fe++ Fe +++ -0.747 Sn Sn++ 0.136 Hg(l) ^Hg^ -0.799 Pb Pb++ O.I22 Ag Ag + O.SOO H 2 (g) 2H+ O.OOO Br~ Br 2 (i m.) -1.085 ci- ^Cl 2 (g) -1-359 154. Derivation of Related Molal Electrode-Potentials from One Another. Molal electrode-potentials that are related to one another can be calculated in the ways illustrated by the following problems. Prob. 31. Molal Electrode-Potentials Corresponding to Different States of Aggregation. Calculate the molal electrode-potential at 25 a, of C1 2 (i m.), Cl~~, and b, of I 2 (i m.), I~~, from the values given for Cl 2 (g), Cl~, and for I 2 (s), I~, in the table of Art. 153, and from the molalities, 0.062 and 0.0013, of C1 2 and I 2 in water saturated with them at 25 when the pressure is i atm. Ans. a, 1.395; &> 0.621. Prob. 32. Molal Electrode-Potential of a Metal in the Presence of One of Its Slightly Soluble Salts. Calculate the molal electrode- potential at 25 of Hg(l) +Hg 2 SO 4 (s), SO 4 = , from that of Hg(l), Hg 2 ++ given in the table, and from the solubility- product (Hg 2 ++) X (SO 4 ~) of Hg 2 SO 4 which at 25 is 8.2 X lo" 7 . Ans. -0.62 volt. Prob, 33. Molal Electrode-Potentials of an Element that Exists in more than Two States of Oxidation. a. By considering the changes in state that occur on passing electricity at 25 through the voltaic cells Cu,Cu+(i m. m.),Cu, and Cu, m.) derive a relation between the molal electrode-potentials of Cu(s), Cu + ; Cu(s), Cu++; and Cu + , Cu++. b. Calculate the last of these potentials from the first two. Ans. b, 0.22. 155. The Electromotive Force of Cells with Dilute Solutions in Relation to the Molal Electrode-Potentials. The electro- motive force of cells in which the liquid-potentials are negligible or for theoretical purposes are to be disregarded can be calculated from the molal electrode-potentials in the way illustrated by the following problems. ELECTRODE-POTENTIALS AND LIQUID-POTENTIALS 261 Evaluation of Electromotive Forces. Prob. 34. Calculate the electromotive force at 25 of the cell Zn(s), ZnCl 2 (o.ooi f.), AgCl(s)+Ag(s). Prob. 35. Calculate the electromotive force at 25 of the cell H 2 (o.i atm.), HCl(o.i f.), PbCl 2 (s)+Pb(s). The saturated solution of PbCl 2 in water at 25 is 0.039 formal. Prob. 36. Formulate numerical expressions for calculating the electromotive force at 25 of each of the following cells: a, M+CuCl(s), CuCl 2 (o.oi f.), Cl 2 (o.i atm.). The solubility-product of CuCl is i.2Xio~ 6 formal. For the composi- tion of iodide solutions saturated with iodine see Prob. 45, Art. in. 156. The Electromotive Force of Cells with Concentrated Solutions. The electromotive force of any cell in which solutes are present at large concentrations cannot be calculated from the molal electrode- potentials with the aid of the logarithmic concentration formula of 'Art. 152, since this holds true even approximately only when the concentration does not exceed i formal. A knowledge of the change in state taking place in such a cell and of the reactions occurring at its electrodes is, however, of importance, since it shows qualitatively the factors which determine the magnitude of the electromotive force. This is illustrated by the following problems, which relate to certain cells of technical importance. Prob. 37. The "Dry Cell." The Leclanche cell and the common "dry cell," which is a Leclanche cell to which some porous material, such as paper-pulp or sawdust, has been added, consists essentially of a zinc rod (amalgamated to diminish local voltaic action) dipping into a concentrated NH 4 C1 solution containing ZnCl 2 , and a carbon rod coated with MnO 2 dipping into the same solution, a. Formulate the cell and the reaction that takes place at each electrode and the whole reaction in the cell, taking into account the facts that the MnO 2 is reduced to Mn 2 O 3 , and that Zn(OH) 2 is soluble in NH 4 C1 solution with formation of Zn(NH 3 ) 4 Cl 2 . b. Show in what direction the electromotive force would be changed by decreasing the concentration of the NH 4 C1. Prob. 38. The Edison Storage Cell. In the nickel-iron (Edison) storage cell, Fe(s)+Fe(OH) 2 (s), KOH( 2 i%),Ni(OH) 2 (s)+Ni(OH) 3 (s), the main reaction is Fe(s)+ 2 Ni(OH) 3 (s) = Fe(OH) 2 (s)+2Ni(OH) 2 (s) (the degree of hydration of the three oxides being, however, somewhat indefinite), a. Write the reaction occurring at each electrode, b. Show in what direction each of the electrode-potentials, and also the electromotive force of the whole cell, would vary with increase of the KOH concentration. 262 PRODUCTION OF WORK IN VOLTAIC CELLS *Prob. 39. The Clark Standard Cell. One form of the Clark cell, used as a standard cell in electromotive force measurements is at 20 represented by the formula: Zn(s)+ZnSO 4 .7H 2 O(s), ZnSO 4 .i6.8H 2 O, Hg 2 SO 4 (s)+Hg(l). a. Specify the change of state that occurs when two faradays pass through the cell. b. Show that even in an actual cell, with only a finite quantity of solution, no variation of the electro- motive force results when two faradays pass, provided the change in state takes place slowly enough, c. Explain how the electromotive force of the cell would be changed by making the saturated ZnSO 4 solution also i formal in Na2SO 4 . 157. The Nature of Liquid-Potentials and Expressions for their Approximate Evaluation. The electromotive force at the boundary between two solutions arises from the different rates at which the positive and negative ions tend to diffuse from one solution into the other. This principle may be first illustrated by applying it to solutions of the same substance at two different concentrations, such as HC1 (i f.), HCl(o.i f.). In this case both kinds of ions diffuse from the first to the second solution in consequence of their smaller concen- trations in the latter solution; but, owing to their greater mobility, the hydrogen ions tend to pass into the second solution in larger quantity, and they do so to a slight extent, until such an excess of positive electricity is accumulated in the intermediate layers that the unequal diffusion-rates of the two ions are compensated by the effect of the potential-gradient in driving the positive hydrogen ions back- ward and drawing the negative chloride ions forward. The result of the different diffusion-rates of the two ions is in this case evidently an electromotive force directed from the more concentrated to the more dilute HC1 solution. Similar considerations apply to solutions of two different sub- stances at the same concentration, such as HCl(i f.), NaCl(i f.). In this case, the hydrogen ions evidently tend to diffuse from the first solution to the second, and the sodium ions in the reverse direction; but owing to the much greater mobility of the hydrogen ions there is a resultant flow of positive electricity, and correspondingly an electro- motive force, from the HC1 to the NaCl solution. As before, there can be only a very slight accumulation of hydrogen ions in the bound- ary of the second solution because of the counter electromotive force thereby produced. ELECTRODE-POTENTIALS AND LIQUID-POTENTIALS 263 Prob. 40. The Nature and Direction of Liquid- Potentials. a. Ex- plain fully, in accordance with the above statements, the mechan- ism of the process by which a liquid-potential results in the case of KOH(i f.), KCl(i f.). b. Predict the sign of the liquid-potential of NaOH(i f.), NaOH(o.i f.); and of NaCl(i f.), KNO 3 (i f.). Expressions by which liquid-potentials can be approximately evaluated can be derived either by formulating quantitatively the above considerations with the aid of the laws of diffusion, or by apply- ing the free-energy principles in the way shown in Art. 158 to the changes in state taking place at the boundary. By these methods the following expressions for the liquid-potential E L , in terms of the ion-conductances, AC, A C ', A A , etc., and the number of charges or valences, V A and v c , of the cation and anion, have been derived under the assumptions that the substances are completely ionized and that their ions are perfect solutes and have constant mobilities equal to those at zero concentration: For combinations of the ionic types C + A~(at ci), C + A~(atc 2 ); C + 2 A=(at ci), C + 2 A=(at c 2 ); etc: A_C_AA F ._ ^ _ v RT Cl (i) EL ~ A C +AA F 10g ^ For combinations of the types CA, C'A; C 2 A, C' 2 A; CA 2 , C'A 2 ; etc., in which the two substances are at the same concentrations: RT. Ac+A A For combinations of the types CA, C'A; C 2 A, C' 2 A; CA 2 , CA' 2 ; etc., in which the two substances have the same concentrations: RT, Ac+A A . Evaluation of Liquid Potentials. Prob. 41. Calculate the liquid-potential at 18 of the combinations: a, NaCl(o.i.f.), NaCl(o.oi f.); &, K 2 S0 4 (o.i f.), K 2 SO 4 (o.oi f.). Prob. 42. Calculate the liquid-potential at 25 of: o, KOH(o.i f.), KCl(o.i f.) ; 6, HCl(o.oi f.), KCl(o.oi f.) ; c, ZnSO 4 (o.i f.), CuSO 4 (o.i f.). Prob. 43. Calculate the electromotive force at 25 of the cell Ag(s), AgN0 3 (c.i f.), KN0 3 (o.i f.), KCl(o.i f.), KCl(i f.) +Hg 2 Cl 2 (s), For combinations of solutions of two salts having a common ion but different concentrations, such as KNO 3 (o.i f.), KCl(i f.), and 264 PRODUCTION OF WORK IN VOLTAIC CELLS for combinations of solutions of salts without a common-ion, such as KNOs(o.i n.), NaCl(o.oi n.), the calculation of the liquid-potential is more complicated. Combinations of these kinds can, however, be avoided, as illustrated by the cell of Prob. 43, by connecting the two solutions through intermediate ones so as to produce only combina- tions of the two types for which the liquid-potentials can be calculated by the above equations. A liquid-potential can sometimes be experimentally determined by measuring the electromotive force of a cell involving it in which the two electrode-potentials are made substantially equal, as in the cells: Hg(l)+Hg 2 Cl 2 (s), KCl(o.i f.), NaCl(o.i f.), Hg 2 Cl 2 (s)-f-Hg(l). Ag(s), AgNO 3 (o.ooi f.)+KNO 8 (o.i f.), AgNO 3 (o.ooi f.) + NaN0 3 (o.i f.), Ag(s). Prob. 44. a. Show that the electromotive force of the silver cell just formulated is substantially equal to the liquid-potential of KNO 3 (o.i f.), NaNO 3 (o.i f.), by specifying exactly what determines each of the three partial potentials of the cell, b. State what assumption is involved in regarding the two electrode-potentials equal to each other. *158. Derivation of the Liquid-Potential Equations. Just as the electromotive force of a whole cell is determined by the free-energy decrease attending the change in state that takes place in the cell, and just as the electrode-potential is determined by the free-energy decrease attending the change in state that takes place at the electrode, so a liquid-potential is determined by the free-energy decrease attending the change in state that takes place at the boundary between the two solutions. The change in state at such a boundary is of a simple character in the case where the two solutions contain only the same solute at two different concentrations, as in the combination, HCl(o.oi f.), HCl(o.i f.). In this case, in order to derive equation (i) of Art. 157 for the liquid-potential, it is only necessary to consider the number of equivalents of the positive ion-constituent and of the negative ion- constituent that pass through the boundary in the two opposite directions per faraday passed through, to formulate the free-energy decrease attending these transfers, on the one hand, in terms of the electrical work E^ involved when they are brought about by the passage of one faraday, and on the other hand, in terms of the usual logarithmic concentration equations of Art. 138, and to equate these two expressions. This derivation involves the assumptions that the ELECTRODE-POTENTIALS AND LIQUID-POTENTIALS 265 solute is completely ionized, that its ions behave as perfect solutes, and that the ratio of their mobilities, that is, the transference-number, is constant up to the highest concentration involved. Derivation of the Equation for the Liquid-Potential between Solutions with the Same Solute at Different Concentrations. Prob. 45. Derive equation (i) of Art. 157 in the form it assumes for a uniunivalent solute at two different concentrations, such as HCl(at Ci) , HCl(at c 2 ), making the assumptions stated in the text. Prob. 46. Derive equation (i) of Art. 157 in the form it assumes for a unibivalent solute at two different concentrations, such as CaCl 2 (at Ci), CaCl 2 (at c 2 ), making the assumptions stated in the text. It will be evident from these derivations that, provided only that the ratio of the two ion-conductances, and hence the transference-number T C or T A , does not vary between the two concentrations involved, and that the two ions can be considered to have the same activity-co- efficient a. in the same solution, an exact expression for the liquid- potential is given in the case of a uniunivalent solute by the equation: By a similar consideration of the ion-transference that takes place at the boundary of the two solutions and with the aid of the same assumptions, an expression can be derived for the liquid-potential of combinations of the types-KCl, HC1; K 2 SO 4 , Na2SO 4 ;ZnSO 4 , CuSO 4 . Namely, taking as a specific example the combination KCl(at c) t HCl(at c), it will be noted that there must be at the boundary an intermediate portion of the solution in which the two substances are present in varying proportions, the concentration of the potassium chloride decreasing continuously from left to right from c to o, and that of the hydrogen chloride increasing continuously in the same direction from o to c. Considering now within the boundary-portion any two adjoining layers of infinitesimal length in which at equilibrium the prevailing concentrations of the two cation-constituents are C K or CR and c K +dc K or c H +dc Q and their transference-numbers are T K and T H , it is evident, when one faraday passes from the KC1 solu- tion to the HC1 solution, that T K mols of potassium-ion are transferred from the concentration C K to the concentration c K -\-dc K and T H mols of hydrogen-ion from C R to c H +dc s . At the same time T CL equivalents of chloride-ion are transferred in the opposite direction; but this need not be considered, since its concentration is uniform throughout the 266 PRODUCTION OF WORK IN VOLTAIC CELLS whole solution. The free energy decrease dF attending this transfer- ence is therefore given by the equation: A corresponding transference, attended by a corresponding free-energy decrease, takes place between each pair of adjoining infinitesimal layers throughout the whole boundary-portion. The total free-energy decrease, which is equal to the electrical work E L E attending the passage of one f araday, is therefore equal to the integral of this expres- sion, taken for the C K term between the limits C K = C and C K = O and for the CR term between the corresponding limits c s = o and C H = C. That is: Now the transference-number of any ion-constituent, as indicated in Art. 73, is equal to the ratio of the product of its concentration by its conductance to the sum of the corresponding products for all the ion- constituents present. Hence, assuming complete or equal ionization for the two substances in all the solutions and constant ion-mobilities of the three ions up to the concentration c, the following expressions result: Replacing c u in the first of these expressions by c C K , and C K in the second of these expressions by c C H , substituting the resulting values of T K and T H in the above free-energy equation, carrying out the indi- cated integration, and simplifying the result by appropriate transfor- mations, there is obtained the following equation, corresponding to equation (2) of Art. 157. Pr ob. 47. Derivation of the Equation for the Liquid-Potential between Solutions with Two Solutes at the Same Concentrations, Derive the equation just given by the method stated in the text. 159. Determination of Ion-Concentrations and of Equilibrium- Constants by Means of Electromotive Force Measurements. Measurements of electromotive force, interpreted with respect to the electrode-potentials involved, furnish an important means of deter- ELECTRODE-POTENTIALS AND LIQUID-POTENTIALS 267 mining ion-concentrations which are so small that they cannot be readily measured by other methods. The so determined ion-concen- trations may be used for calculating the solubilities of very slightly soluble salts or the ionization-constants or other equilibrium-constants of chemical reactions, as shown in the following problems. Measure- ments of electromotive force similarly interpreted are frequently employed by physiologists for determining the hydrogen-ion concen- tration in blood and other animal fluids, and sometimes by chemical analysts for the titration of acids and bases, especially in cases where the color of the solution makes the use of indicators unreliable. Prob. 48. Determination of Solubility. The electromotive force of the cell Ag(s)+AgI(s), KI(o.i f.), KNO 3 (o.i f.), AgNO,(o.i f.), Ag(s) is 0.814 volt at 25. Calculate the solubility of silver iodide in water at 25. Prob. 49. Determination of the lonization-Constant of Water. Calculate the ionization-constant of water (Art. 104) at 18 from the fact that the electromotive force of the cell H 2 (i atm.), HCl(o.i f.), KCl(o.i f.), KOH(o.i f.), H 2 (i atm.) is -0.653 volt at 18. Prob. 50. Determination of Complex-Constants. The electro- motive force of the cell Ag(s), K+Ag(CN)r(o.oi n.)+KCN(i n.), KCl(i n.), Hg 2 Cl 2 (s) +Hg(l) is oj^ volt at 25. Calculate the dissocia- tion-constant of the complex-ion Ag(CN) 2 ~ at 25. Neglect the liquid- potential, which is small in this case. Ans. i.5Xio~ 21 . Prob. 51. Determination of the Hydrolysis of Salts and the lonization- Constants of Slightly Ionized Acids. When a solution 0.05 formal in Na 2 HPO 4 is saturated with hydrogen at i atm., when a platinum elec- trode is placed in it, and when the half-cell HCl(o.oi f.)+NaCl(o.i f.), H 2 (i atm.), is brought into contact with it, the cell thus formed is found to have at 18 an electromotive force of 0.398 volt. Calculate the hydrolysis of the salt and the ionization-constant of HPO 4 ~~, assum- ing complete ionization of the largely ionized substances, and neglecting the liquid-potential, which is made small by the addition of the sodium chloride. Prob. 52. Determination of Indicator-Constants. When the Na 2 HPO 4 solution of Prob. 51 is made o.oooi formal in phenolphthalein the indicator is found to show a color 13% as intense as that which is produced on adding to the solution an excess of sodium hydroxide. Calculate the ionization-constant of this indicator. Prob. 53. Electrometric Titration of Acids and Bases. Into a solution of an acid to be titrated are introduced the side-arm of a calomel half-cell and a tube carrying a small platinized platinum electrode, around which hydrogen gas at the barometric pressure is kept bubbling. A standard 0.5 n. NaOH solution is added from a burette until the electromotive force between the hydrogen electrode and the 268 PRODUCTION OF WORK IN VOLTAIC CELLS calomel electrode is found to change very rapidly as more alkali is added. Calculate the successive changes that would be observed in the electromotive force, when the alkali added is equivalent to 99.0, 99.5, 99.8, and 100.2% of the acid present, a, in case the acid to be titrated is a o.i normal solution of HC1; b, in case it is a o.i normal solution of an acid of ionization-constant of io~ 5 , referring to the table of Art. 1 08 for the hydrogen-ion concentrations. Neglect the liquid- potential and the change in the volume of the solution produced by the addition of the standard NaOH solution. EQUILIBRIUM OF OXIDATION REACTIONS 269 V. THE EQUILIBRIUM OF OXIDATION REACTIONS IN RELATION TO THE ELECTRODE-POTENTIALS 160. Derivation of the Equilibrium-Constants of Oxidation Re- actions from the Molal Electrode-Potentials. When the concentra- tions of the substances involved in the two electrode reactions of any cell are such that the two electrode-potentials are equal to each other, there is evidently no tendency for the cell to act nor for the chemical change to take place in it. In other words, the concentrations that make the two electrode-potentials equal are concentrations at which the chemical change is in equilibrium. The equilibrium-constant of a chemical change involving solutes or gases at small concentrations or pressures can therefore be evaluated, as in the following problems, by conceiving a cell in which the reaction would take place and calculat- ing from the molal electrode-potentials by the usual logarithmic equation the concentrations of the substances involved hi the electrode reaction that will make the two actual electrode-potentials equal. Evaluation of Equilibrium-Constants by a Consideration of the Equi- librium Conditions of Voltaic Cells. Prob. 54. a. Calculate the concentration of copper-ion at which the reaction Zn(s)+Cu" H " = Cu(s)+Zn" H " is in equilibrium at 25 when the zinc-ion is i molal. b. Calculate the equilibrium-constant at 25. Prob. 55. a. Calculate the concentration of hydrogen-ion at which the reaction Pb(s)+2H + Cl~ = H 2 (g) +Pb' H -Cl- 2 is in equilibrium at 25 when the hydrogen has a pressure of i atm. and the lead-ion is 0.03 molal. b. Formulate an algebraic relation between the equilibrium- constant of the corresponding ion reaction and the molal electrode- potentials, c. Calculate the value of this equilibrium-constant. The foregoing considerations evidently lead to a general expression for the equilibrium-constants of oxidation reactions. Namely, the equilibrium-constant K of the reaction which takes place when N faradays of electricity pass from left to right through a cell whose left-hand and right-hand molal electrode-potentials are E\ and E 2 , respectively, is related to these potentials as follows: l?riog^ = (E 1 -E 2 ) NF. This relation may be directly derived from free-energy considera- tions in the following way. The free-energy decrease attending a chemical reaction when the reacting substances and reaction-products are all at i molal (or if gaseous, at i atmosphere) has been shown in Arts. 140 and 141 to be given by the expression: 270 PRODUCTION OF WORK IN VOLTAIC CELLS The free-energy decrease attending this same change in state, when this is considered to take place in a cell in which all the substances are at i molal (or if gaseous, at i atmosphere), is evidently expressed by the equation: ,~ ~ , -AF = NF (Ej-Eg). By equating these two expressions for the free-energy decrease the general relation given above obviously results. It will be seen from these considerations that electrode-potentials are of great importance in their purely chemical relations. When they are so considered they are appropriately called the reduction- potentials of the corresponding oxidation reactions. Thus, the, elec- trode-potential (0.122 volt) of Pb(s), Pb++ is the reduction-potential of the reaction Pb(s) + 28 = Pb + + . Expressed in chemical terminology, the preceding equation shows how the equilibrium-constant K of a chemical reaction which is the resultant of an oxidation reaction (such as Pb(s) + 20 =Pb++) and of a reduction reaction (such as 2H + =H 2 (g) + 20) is related to their molal reduction-potentials, EI and 2; the symbol N denoting now the number of positive electrons involved in the separate reactions. Evaluation of Equilibrium-Constants by the General Equation. Prob. 56. a. Derive an expression for the equilibrium-constant of the reaction Cu(s) -f-Cu"^ = 2Cu + in terms of the molal reduction-poten- tials, b. Calculate the concentration of cuprous salt resulting when copper is shaken with a o.i f. CuSO 4 solution at 25. c. Calculate the concentration of cupric salt in the equilibrium-mixture produced by shaking a c.i f. CuCl 2 solution with copper at 25, noting that solid CuCl separates, whose solubility-product at 25 is i.2Xio~ 6 molal. Ans. b, 0.0024; c, 0.00172. Prob. 57. a. Formulate an algebraic expression showing how the equilibrium -constant of the reaction Ag(s)+Fe +++ =Ag + +Fe~ H ~, ex- pressed in terms of the equilibrium concentrations, is related to the molal reduction-potentials involved, b. Calculate the equilibrium- constant, c. Find the composition of the equilibrium mixture that results when metallic silver is placed in o.i f. Fe(NO 3 ) 3 solution. Ans. b, 0.14; c, Ag + or Fe" 1 " 4 ", 0.067. Prob. 58. Manganates in aqueous solution undergo partial decom- position into permanganates and manganese dioxide according to the following equation: 3MnO 4 = + 2H 2 = MnO 2 (s) + 2MnO 4 ~+ 4OH~. Calculate the ratio of permanganate to manganate in the equilibrium mixture at 25 when in that mixture the manganate is o.i formal and the free base is i formal. The molal reduction-potentials at 25 are -0.61 volt for the reaction MnO 4 = + 0=Mn0 4 ~, and -0.66 volt for the reaction Mn0 2 (s)+40H+20 =Mn0 4 = +2H 2 O. Ans. 2.21. VOLTAIC ACTION AND POLARIZATION 271 VI. VOLTAIC ACTION, ELECTROLYSIS, AND POLARIZATION 161. Concentration-Changes Attending Voltaic Action and the Resulting Polarization. Throughout the foregoing considerations, as an aid in evaluating the electromotive force, it has been assumed that so large a quantity of solution is present in the cell that only infinitestimal concentration-changes are produced in it by the passage of a finite quantity of electricity. The fact that this is not the case in actual cells must be taken into account, as in the following problem. Prob. 59. Change of Electromotive Force due to Changes in Con- centration. The electromotive force E at 15 of the lead storage-cell varies with the mol-fraction x of the H 2 SO 4 (for values of x up to o.io) according to the equation E = 1.855 +3-8ox - iox 2 . Out of a certain cell which contains 1300 g. of 10 mol-percent H 2 SO 4 a steady current of 5.36 amperes is taken for 10 hours. Calculate the electromotive force of the cell at the beginning and at the end. In practice the electromotive force is often decreased much more than these considerations indicate because the concentration-changes actually occur in the immediate neighborhood of the electrodes and are only gradually distributed by convection or diffusion through the whole body of the solution. Thus in the lead storage-cell water is produced and acid is destroyed by the electrode reaction in the solu- tion impregnating the porous lead-peroxide electrode, and the acid can only be replenished from the main body of the solution by the slow process of diffusion. This phenomenon is one kind of polariza- tion, sometimes called concentration-polarization; the name polarization being used in general to denote the production by the passage of the current of any change in the solution adjoining the electrode or in the surface of the electrode which makes its potential deviate from its normal value. Concentration-polarization is affected by various incidental condi- tions, for example, by agitation of the solution around the electrodes. It is also often greatly influenced by the current-density, by which is meant the current per unit-area of electrode surface, thus the number of amperes per square centimeter or square decimeter of surface of a specified electrode. Prob. 60. Concentration-Polarization and the Factors Affecting It. When a current is taken at 25 out of" a certain cell of the form Zn(s), ZnSO 4 (i f.), CuSO 4 (i f.), Cu(s), the electromotive force soon be- comes fairly constant and remains so for a time at a value 0.06 volt below its normal value, a. Show quantitatively how this might be 272 PRODUCTION OF WORK IN VOLTAIC CELLS accounted for by concentration-polarization, b. Explain how the polarization would be affected by increasing the current; by using smaller electrodes without changing the current, thereby increasing the current-density; and by stirring the solutions, these being separated from each other by a porous cup. 162. Electrolysis in Relation to Minimum Decomposition-Potential. In order to produce electrolysis in any electrolytic cell there must obviously be applied from an external source an electromotive force at least equal to the electromotive force that is produced by the combination of solution and electrodes, considering it as a voltaic cell. The value of the electromotive force that must be applied to compensate the electromotive force of the voltaic cell which develops under the actual conditions, assuming local concentration changes to be equalized, may be called the minimum decomposition-potential. Its value can evidently be calculated in the case of cells with dilute solu- tions by the method considered in Art. 155. It is important to recognize this fact; for the decomposition-potential is sometimes treated as if it were an essentially independent quantity. The actual decomposition-potential, that is, the electromotive force that must be applied to produce continuous electrolysis, may be much greater than the minimum decomposition-potential because of polarization effects, such as are considered in the next article. Prob. 61. Deposition of Metals by Electrolysis. A solution 0.05 formal in H 2 SO 4 and 0.05 formal in CuSO 4 is electrolyzed at 25 between a mercury anode and a platinum cathode. In the mixture assume the salt to be completely ionized and the acid to be 25% ionized into 2H + and SO 4 = and 75% ionized into H + and HSO 4 ~. a. Calculate from the molal electrode-potentials (taking that of Hg(l)+Hg 2 SO 4 (s), SO 4 = to be 0.62 volt) the minimum electromotive force which would have to be applied in order to cause the copper to deposit, b. Find the value to which this electromotive force would have to be increased after 99% of the copper had been precipitated in order that the deposition might continue, assuming the ionizations to be the same as in the original mixture, c. Find the minimum electromotive force at which hydrogen could be continuously set free, assuming that it attains at the cathode an effective pressure of i atm. Prob. 62. Separation of Elements by Electrolysis. A solution o.i formal in KC1, o.i formal in KBr, and o.i formal in KI is placed, together with a platinum electrode, in a porous cup; and this is placed within a larger vessel containing a zinc electrode and a large quantity of o.i formal ZnCl 2 solution. Neglecting the liquid-potential and any polarization effects, calculate the applied electromotive force required at 25, a, to liberate 99.9% of the iodine; b, to set bromine free at a ELECTROLYSIS AND POLARIZATION 273 concentration of o.oooi molal; c,to liberate 99.9% of the bromine (which remains in solution); and d, to liberate chlorine at a concentration of o.oooi molal. 163. Electrolysis in Relation to Polarization. In electrolysis, as in voltaic action, concentration changes are produced in the solutions around the electrodes; and these changes have the effect of increasing the applied electromotive force required. Thus in electrolyzing a copper solution between copper electrodes (as is done in copper plating) only an infinitesimal electromotive force is required to start the deposition; but the solution around the cathode soon becomes less concentrated in copper and the solution around the anode more con- centrated in copper, producing a concentration-cell with an electro- motive force opposite to that applied. This back electromotive force would evidently be diminished by decreasing the current-density, by agitating the whole solution, or by rotating the electrodes. When a gas, such as hydrogen or oxygen, is set free at an electrode, a phenomenon, known as gas-polarization, is observed which does not occur in the deposition of metals. For when a gas is involved in the electrode-reaction its partial pressure determines the electrode- potential. Hence in a cell exposed to the air a slow, continuous electrolysis may take place before the applied electromotive force equals the electromotive force of the cell when the gas-pressure is one atmosphere; for the gas-forming substance is produced on the electrode at a lower pressure and is dissolved in the solution and carried away by convection and diffusion. Moreover, a sudden increase in the rate of electrolysis does not take place when the applied electromotive force is increased beyond that corresponding to a pressure of one atmosphere; for the gas-forming substance then passes into the elec- trode at this higher pressure, producing a supersaturated adsorbed layer on its surface or a supersaturated solid solution within the metal, from which the gas does not escape rapidly enough to reduce the effective pressure to one atmosphere. There is thereby produced a back electromotive force which is equal to the electromotive force of a voltaic cell in which the gas has this higher pressure, and which is larger than the electromotive force of a similar cell in which the gas has the partial pressure that prevails above the solution. The amount by which the back electromotive force exceeds the theoretical electro- motive force at the. prevailing partial pressure is called the polarization or overvoltage. 274 PRODUCTION OF WORK IN VOLTAIC CELLS It will thus be seen that the applied electromotive force may be considered resolved into three parts: one equal to the minimum decomposition-potential or to the theoretical electromotive force of the resulting voltaic cell at the prevailing partial pressures; a second part equal to the polarization; and a third part, that uncompensated by the back electromotive force resulting from these two effects, and therefore available for forcing a current through the cell in accordance with Ohm's law. These considerations are illustrated by Prob. 63. These principles suffice to enable the phenomena of polarization to be treated from experimental and thermodynamic standpoints. The various hypotheses that have been proposed in regard to the molecular mechanism by which gas-polarization results can not be here considered. These hypotheses involve considerations relating to surface-tension, adsorption, formation of unstable compounds, and rate of reaction in solid phases. The back electromotive force produced by the electrolysis may be experimentally determined in the following ways: (1) By the method described in Prob. 63. (2) By rinding the smallest value of the applied electromotive force at which bubbles form slowly but continuously at the electrode surface (this giving a value corresponding practically to a minimum current-density). (3) By applying a definite electromotive force to the cell long enough to charge the electrodes with the decomposition-products and then short-circuiting the electrodes through a high-resistance potentiometer, the applied potential being at the same time removed. From the so-measured back electromotive force the overvoltage is obtained by subtracting the theoretical electromotive force corresponding to the prevailing gas-pressure, which is usually about one atmosphere. Prob. 63. Gas-Polarization. Electromotive forces successively increasing in magnitude were applied at 22 to an electrolytic cell con- sisting of a large, unpolarizable, platinized platinum plate as anode, a small mercury surface as cathode, and a o.i f. H 2 SO 4 solution as electrolyte. Hydrogen gas at i atm. was bubbled steadily through the cell, and a resistance of 100,000 ohms was placed in series with it, the resistance of the cell being negligible in comparison. The current- strengths in millionths of an ampere corresponding to various applied electromotive forces in volts were as follows: Current-strength .... 0.06 0.44 1.20 2.20 3.70 4.82 Applied electromotive force . 0.32 0.48 0.62 0.77 0.95 1.08 ELECTROLYSIS AND POLARIZATION 275 a. Plot these current-strengths as ordinates against the electromotive forces as abscissas, b. Calculate the back electromotive force and polarization corresponding to each of these applied electromotive forces, tabulate these values, and plot the current-strengths against them on the same diagram, c. Calculate the effective pressure of the hydrogen at the electrode corresponding to the smallest of these back electromotive forces, assuming the logarithmic relation valid for perfect gases to hold up to this pressure. The overvoltage is always found to increase with increase of the applied electromotive force and of the current-density; but it varies in a highly specific way with the chemical nature of the gas and with the chemical nature and physical state of the metallic electrode. Thus in certain experiments made with i normal H 2 SO 4 the over- voltage of the hydrogen was found to have the following values: Small Large current-density current-density Platinized platinum . . . o.oo volt 0.07 volt Smooth platinum .... 0.03 0.65 Lead 0.36 1.23 Mercury 0.44 1.30 In certain experiments with 2 normal KOH solution with a moderate current-density the overvoltage of the oxygen was found at the start to be 0.44 volt on platinized platinum, 0.84 volt on smooth platinum, and 0.50 volt on iron; the value increasing to 1.46 on smooth platinum and to 0.59 on iron after two hours' passage of the current. The phenomenon of gas-polarization and the overvoltage attending it are of great significance in technical processes. The overvoltage may greatly diminish the energy-efficiency of the process, the energy- efficiency being the ratio of the minimum electrical energy theoretically required to produce a definite quantity of some product of the elec- trolysis to the energy actually expended. Overvoltage may also make processes practicable which would otherwise not be possible; thus in charging a lead storage-cell hydrogen is not set free at the lead owing to its overvoltage, although the potential of the half-cell H 2 (i atm.), H 2 SO 4 .2oH 2 O is about 0.4 volt less than that of the half- cell Pb+PbSO 4 (s), H 2 SO 4 .2oH 2 0. Prob. 64. Energy-Efficiency and Overvoltage. In a certain com- mercial alkali-chlorine cell sodium hydroxide and chlorine are produced at an iron cathode and graphite anode, respectively, by the continuous electrolysis of a 25% NaCl solution which is slowly flowed through a diaphragm from the anode to the cathode compartment (to prevent the hydroxide-ion from migrating to the anode). In a certain case 4.5 volts 276 PRODUCTION OF WORK IN VOLTAIC CELLS were applied to the cell, whose resistance was 0.00080 ohm, yielding a current of 2000 amperes; and there flowed off each hour from the cathode 27,460 g. of solution containing 10% of NaOH and 13% of NaCl. The electromotive force of the voltaic cell H 2 (i atm.), NaOH(io%) + NaCl(i3%), NaCl(25%), Cl 2 (i atm.) has been independently deter- mined to be 2.3 volts, a. Calculate the current-efficiency in the production of the sodium hydroxide, b. Calculate the energy-efficiency of its production, c. Calculate the voltage used to produce the current and that used to overcome the polarization, and tabulate these together with the minimum decomposition-potential. A ns. b, 47.0%. CHAPTER XII THE EFFECT OF TEMPERATURE ON THE WORK PRO- DUCIBLE BY ISOTHERMAL CHEMICAL CHANGES AND ON THEIR EQUILIBRIUM CONDITIONS I. THE FUNDAMENTAL SECOND-LAW EQUATION 164. The Quantity of Work Producible from a Quantity of Heat that Passes from One Temperature to Another. This chapter is mainly devoted to a consideration of the effect of temperature on the free-energy changes that attend isothermal changes in state. To this effect is closely related, in virtue of the free-energy relations already derived, the effects of temperature on the equilibrium of chemical changes and on the electromotive force of voltaic cells. Before these effects can be properly considered the more general thermodynamic relation referred to in the title of this article must be known. This will now be derived. The second law of thermodynamics has already been expressed (in Art. 135) in fundamental form by the statement that a process whose final result is only a transformation of heat into work is an impossibility; and it was shown that, in accordance with this state- ment, when work is produced in surroundings of constant tempera- ture, there is always a change in the state of the system employed for the transformation. Chapters X and XI have been devoted to a determination of the quantities of work producible by various isothermal changes in state. There will now be considered the other conditions, involving difference in temperature in the surroundings, under which the second law permits that work be produced out of heat by systems which undergo no permanent change in state. When a quantity of heat is transformed into work by a cyclical process (that is, by a process in which the system undergoes a cycle of changes in state), an additional quantity of heat is always taken up from surroundings at a higher temperature and given out to sur- roundings at a lower temperature. That is to say, even when a differ- ence of temperature exists, only a fraction of the heat taken up by the system from the warmer surroundings can be transformed into work. Important questions at once arise as to what determines the fraction that can be so transformed whether it is dependent on the nature of the process employed, and how it varies with the temperatures. 277 278 EFFECT OF TEMPERATURE ON EQUILIBRIUM In order to determine whether the quantity of work that can be produced when any definite quantity of heat is transferred by a cyclical process from a higher to a lower temperature is dependent upon the nature of the system employed for the transformation, or upon the way in which the transformation is carried out, assume that two different reversible cyclical processes, carried out with different systems or in a different way with the same system, could produce two unequal quantities of work by transferring an equal quantity of heat from a higher to a lower temperature. Then cause the process that produces the larger quantity of work W" to take place in such a way that it takes up a quantity of heat Qi at the higher temperature 7\, transfers a part of it Qz to the lower tempera- ture T 2 , and transforms the remainder into work W"; and cause the other process, which in transferring the same quantity of heat Q% from TI to Tz produces the smaller amount of work W, to take place in the reverse direction that is, so that it takes up the heat Qz transferred by the former process to the lower temperature, and raises it to the higher temperature by expending the required amount of work W. It is then evident that the net result of these operations would be the production of a quantity of work W"W from an equivalent quantity of heat without any other change having been brought about either in the systems or in the surroundings. Since this is contrary to the fundamental statement of the second law, the supposition made that the two processes produce unequal quantities of work is unten- able. This important conclusion may be explicitly stated as follows : the quantity of work which can be produced when a definite quantity of heat is transferred from one temperature to another by any process in which the system employed undergoes no permanent change in state is not dependent on the nature of the process. By the conclusion just reached the determination of the relation between the temperatures and the proportion of heat transformable into work is greatly facilitated; for evidently it is now only necessary to determine what that relation is for a single reversible cyclical process. Such a process is considered in the following problem. Prob. i. Derivation of the Second-Law Equation. Consider the following cyclical process carried out reversibly with N mols of a perfect gas contained in a cylinder closed with a weighted, frictionless piston. Start with the gas at the temperature T+dT and volume Vi. (i) Keeping the gas in a large heat-reservoir at T+dT, diminish gradu- ally the weight on the piston and cause the gas to expand till its volume THE SECOND-LAW EQUATION 279 becomes v 2 . (2) Fix the piston so that the volume must remain con- stant, and place the gas in a large heat-reservoir at a temperature T. (3) Keeping the gas in the heat-reservoir at the temperature T, com- press it by releasing the piston and gradually increasing the weight upon it until the volume % of the gas has been restored to its original value VL (4) Fix the piston again so as to keep the volume constant, and place the gas in a heat-reservoir at the temperature T+dT. a. Represent this process by a lettered diagram in which the ordinates denote temperatures and the abscissas volumes, b. Formulate an expression for the quantities of work Wi, Wz, Wz, W^ and for the quantities of heat Qi, Q z , Q 3 , Q*, produced in the surroundings in the separate steps of this process, taking into account the principles con- sidered in Arts. 24, 25, and 26. c. Formulate a relation between the quantity of work 2TF produced in the whole process and the quantity of heat Q imparted to the reservoir at the temperature T. Since it was previously shown that the second law requires that the same quantity of work be produced when a definite quantity of heat is transferred by any reversible cyclical process whatever from one definite temperature to another, it is evident that the equation de- rived in the preceding problem for one such process is an exact expres- sion of the second law for every such process. This equation there- fore expresses one of the fundamental principles of physical science. The equation just derived, which will be called simply the second- law equation, may be written in the form: In this equation S W denotes the algebraic sum of all the quantities of work produced in any reversible cyclical process taking place at two temperatures T and T + dT in which the quantity of heat Q is transferred to the surroundings at the temperature T. The work- quantity 2W is obviously infinitesimal in correspondence with the infinitesimal temperature-difference d T. Prob. 2. General Principles Deduced from the Second-Law Equa- tion. Show that the second-law equation leads to the following conclusions: a. When there is no difference of temperature in the surroundings heat cannot be transformed at all into work by any cyclical process. b. In order to carry heat from a lower to a higher temperature work must be withdrawn from the surroundings, c. The fraction of the heat transformable into work for a given difference in temperature is greater the lower the temperature, d. When the lower temperature is the absolute zero heat can be completely transformed into work. 280 EFFECT OF TEMPERATURE ON EQUILIBRIUM II. THE EFFECT OF TEMPERATURE ON THE EQUILBRIUM OF UNIVARIANT SYSTEMS 165. Effect of Temperature on the Pressure at which the Phases of Univariant Systems are in Equilibrium. An important relation derivable from the second law of thermodynamics is that named in the title of this article. It is expressed by the Clapeyron equation: = dT ~ T Av ' In this equation dp denotes the increase produced by a temperature- increase dT in the pressure at which the phases of a univariant system are in equilibrium at the temperature T, and AH and Av denote the increases in heat-content and in volume which attend the conver- sion at the temperature T and at the equilibrium-pressure p of any definite quantity of substance from the state in which it exists in one phase or set of phases into the state in which it exists in the other phase or set of phases. For example, the temperature-co- efficient dp/dT of the dissociation-pressure p of solid calcium car- bonate at any temperature T can be calculated by substituting in the Clapeyron equation for AH the increases in heat-content and for Av the increase in volume that attend the change in state CaCO 3 (s) = CaO(s) -f- CO 2 (at p) at the temperature T. This equation will now be derived from the second-law equation. Prob. j. Derivation of the Clapeyron Equation. Consider the following cyclical process: Start with some definite quantity of a substance (for example, iH 2 O) existing as a phase (for example, liquid water) which is in equilibrium with a second phase (for example, ice) at the temperature T and pressure p. (i) Cause the substance to pass under these equilibrium conditions from the first phase (liquid water) in which its volume is vi into the second phase (ice) in which its volume is %; (2) heat the substance to T+dT, whereby the pressure becomes p+dp and the volume v 2 +dv 2 ; (3) cause it to pass from the second phase (ice) into the first phase (liquid water) at T-\-dT under the equilibrium pressure p+dp, whereby the volume becomes v\-\-dv\; (4) cool the substance to T, whereby it reverts to its original state. a. Represent this process by a diagram in which the ordinates denote pressures and the abscissas volumes, b. Formulate the quantities of work, Wi, W 2 , W 3 , Wt, produced in the four steps of the process when it is carried out reversibly, summate these quantities of work, cancel the differentials of the second order, substitute the result in the second- law equation, and replace the heat quantity Q in the equation by its equivalent AH. c. Specify the change in state to which the heat- content decrease AH refers. UNIVARIANT SYSTEMS 281 In applications of the Clapeyron equation a definite change in state should first be formulated, and then A# and Az> should be evalu- ated in accordance with it, expressing the heat-quantity A# and the work-quantity dp AD in the same units. When one of the phases in- volved is a gas at small pressure, Av may be determined by neglecting the volume of the solid or liquid phases and expressing the volume of the gaseous phase in terms of its temperature and pressure with the aid of the perfect-gas law. Applications of the equation to systems of this kind (those involving liquid and gaseous phases) were considered in Art. 33. Other applications are illustrated by the following problems. Effect of Pressure on Melting Point. Prob. 4. Derive from the Clapeyron equation a principle expressing the direction of the effect of pressure on the melting-point of solid sub- stances, taking into account the fact that fusion is always attended by an absorption of heat. Prob. 5. Calculate the variation per atmosphere of the melting- point of ice. At o and i atm. its density is 0.91 7 and the heat of fusion of i g. is 79.7 cal. Prob. 6. Effect of Pressure on Transition-Temperature. a. Calcu- late the variation per atmosphere of the transition- temperature (95.5) of rhombic into monoclinic sulfur from the densities of the two forms, which are 2.07 and 1.96 respectively, and from the fact that the transi- tion of i at. wt. of sulfur from the rhombic into the monoclinic form is attended at 95.5 by an absorption of 105 cal. b. State what this shows as to the slope of the line BE in the sulfur diagram of Art. 115; and what conclusion as to the slope of the line CF can be drawn by considering the density of liquid sulfur, which is 1.81 at 115. Prob. 7. Heat of Hydration derived from the Vapor-Pressure of Salt- Hydrates. With the aid of the temperature- vapor-pressure curves of the diagram of Art. 120 derive approximate values of the heat-effect attending the reaction Na2HPO 4 .7H 2 O(s)-f-5H 2 O(g) =Na 2 HPO 4 .i 2H 2 0(s) at 25. In order to integrate the Clapeyron equation exactly, A# and Afl must be expressed as functions either of the equilibrium-temperature or of the equilibrium-pressure. When the system consists of only solid and liquid phases and when only moderate changes of pressure (such as 20 atm.) are involved, AH, Av, and T in the second member of the equation may be considered constant in approximate calculations. When one of the phases consists of a perfect gas, &v may be expressed as a temperature-function by means of the perfect-gas equation. The heat-quantity AH may always be so expressed in terms of the heat- capacities of the substances involved, in the way described in Art. 131. 282 EFFECT OF TEMPERATURE ON EQUILIBRIUM Applications of Integrated Forms of the Clapeyron Equation. Prob. 8. Calculate the melting-point of ice at 20 atm. with the aid of the data of Prob. 5. Prob. 9. State what heat-quantity can be calculated from the dis- sociation-pressures of solid NH 4 SH into NH 3 and H 2 S, which are 500 mm. at 25 and 182 mm. at 10; and calculate its value, assuming it to be constant through the temperature interval involved. *Prob. 10. Calculate the vapor-pressure of solid iodine at 25 from the following data: Its vapor-pressure at 100 is 47.5 mm., its molal heat of vaporization at 100 is 14,600 cal., its atomic heat-capacity is 6.7 between 25 and 100, and the molal heat-capacity of its vapor at constant pressure is 7.8 between those temperatures. *The derivation of the Clapeyron equation shows that it determines the equilibrium conditions of any type of system in which an iso- thermal change in state can take place under a constant equilibrium- pressure, which is determined only by the temperature; for evidently in all such cases, and only in such cases, the expressions for the work- quantities involved in the cyclical process by which the equation was derived will have the same form. An isothermal change can, how- ever, take place without any change in the pressure only when the pressure is determined solely by the temperature and is not changed by any transfer of substance from one phase of the system to another. In other words, the Clapeyron equation is applicable to all systems, and only to systems, which are actually or in effect univariant. Thus it has been shown applicable to one-component systems existing in two phases, like ice and water or water and water-vapor, or to two- component systems existing in three phases, like CaCOs, CaO, and CO 2 . It is also applicable, however, to systems that have been made in effect univariant by the specification that the composition of each of the phases present shall remain constant when the isothermal change takes place and when the temperature of the system is varied. Thus the equation can be applied to a two-component system existing in only two phases, like a solution and its vapor, provided such a change in state be considered that the solution remain constant in composition when some of the solvent vaporizes out of it, as in the case when an infinite quantity of the solution is subjected to the process. Hence the equation can be used to determine the change in the vapor-pressure of a solution with the temperature. Similarly, it can be applied to a two-component system consisting of a solution and a solid phase, like a melted binary alloy in contact with one of the solid UNIVARIANT SYSTEMS 283 metals, so as to determine the effect of pressure on the temperature at which the solid metal will separate from a melt of the specified composition. *Prob. ii. Application of the Clapeyron Equation to Solutions. a. In applying the Clapeyron equation to determine the effect of temper- ature on the vapor-pressure of a 10% NaCl solution, specify the change in state to which the quantities A# and A0 would correspond. b. State what equilibrium could be studied by applying the Clapeyron equation to a system consisting of solid sodium chloride and its satu- rated solution, and specify the change in state to which the quantities Aff and Az> would correspond. 284 EFFECT OF TEMPERATURE ON EQUILIBRIUM III. THE EFFECT OF TEMPERATURE ON THE EQUILIBRIUM OF CHEMICAL REACTIONS IN GENERAL 166. The Effect of Temperature on the Free-Energy Decrease Attending Any Isothermal Change in State. By applying the funda- mental second-law equation to certain reversible cyclical processes there can be derived, as shown in Probs. 12 and 13, the following differential equations expressing the change with the temperature of the decreases AA and AF, denned as in Art. 136, in the work- content and in the free energy of a system attending any isothermal change in its state. d(-AA) AU-AA , , W dT T d(-AF) Aff-AF (2) dT T These equations are general expressions of the second law in forms especially convenient in considerations relating to chemical equi- librium and electromotive force. It is important fully to appreciate the significance of the quantities occurring in them and the case to which each equation is applicable. These are shown by the derivations of the equations (in Probs. 12 and 13) to be as follows. The quantities A A and AF denote the decreases in the work-content and in the free energy of the system which attend any isothermal change in its state at the temperature T', and AU and AH denote the accompanying increases in its energy-content and heat-content. The differential quantities d( AA) and d(AF) signify that when the same change in state takes place at T+dT, instead of at T, it is attended by a work-content decrease of AA+d( AA) and by a free-energy de- crease of -AF+d(-AF), instead of one of -AA and of -AF. The work-content equation is applicable to cases where the initial volume, and also the final volume, of the system is the same at the two tem- peratures; and the free-energy equation is applicable to cases where the initial pressure, and also the final pressure, is the same at the two temperatures. In accordance with the practice adopted throughout the preceding considerations, only the free-energy equation will hereafter be em- ployed. For purposes of integration this equation is more conve- niently written in the following forms: -AFX Ari. CHEMICAL REACTIONS IN GENERAL 285 'T = I T\ JTV This equation in any of its forms will be called the second-law free- energy equation. Derivation of the Work-Content and Free-Energy Equations. Prob. 12. To derive equation (i) given in the above text, consider a reversible cyclical process involving the following steps: (i) Any change in state of any system at the temperature T, by which a quantity of work W is produced; (2) a change in the temperature of the system at constant volume from T to T+dT; (3) a change in the state of the system at the temperature T+dT which is the reverse of the change in state in the first step, this change being attended by a production of a quantity of work (W + dW)', (4) a change in the temperature of the system at constant volume from T+dT to T. a. Represent this pro- cess by a lettered diagram in which the ordinates denote temperatures and the abscissas volumes, b. Find an expression for ZJF for this process, c. By substituting it in the second-law equation and by making other appropriate substitutions, show that equation (i) results. Prob. 13. To derive equation (2) given in the above text, consider a reversible cyclical process just like that described in the preceding problem, except that in steps (2) and (4) the system is kept at constant pressure, instead of at constant volume, a. Represent this process by a lettered diagram in which the ordinates denote temperatures and the abscissas volumes, designating the pressures and volumes of the system at the beginning of each of the four steps by (i) pi and v\\ (2) pz and v 2 ', (3) p 2 and v 2 +dv 2 ', and (4) pi and v\+dv\. b. Find an expres- sion for 2W for this process, c. By substituting this expression in the second-law equation and making other appropriate substitutions based on the definitions of-AF and -AH given in Arts. 136 and 128, show that equation (2) results, d. Show that the free energy equations (2) and (3) are identical by carrying out in the first member of equation (3) the indicated differentiation and by making simple transforma- tions. 167. The Effect of Temperature on the Equilibrium of Chemical Changes Involving Perfect Gases. By substituting in the second-law free-energy equation (3) of Art. 166 the expression derived in Arts. 140 and 142 for the free-energy decrease attending an isothermal chemical change between perfect gases or between solid substances and perfect gases, and noting that the initial and final pressures occurring in this expression must not vary with the temperature if this equation (3) is to be applicable, there is obtained the following equation, com- monly called the van't Hof equation, expressing the effect of tempera- 286 EFFECT OF TEMPERATURE ON EQUILIBRIUM ture on the equilibrium pressures p A , pz, etc., and on the equilibrium- constant K of the chemical reaction: Prob. 14. Derivation of the Van't Hoff Equation. Derive in the way indicated in the text the van't Hoff equation from the general expression given in Art. 140 for the free-energy decrease attending any chemical change between gases. Prob. 15. Qualitative Principle Expressing the Effect of Temperature on Chemical Equilibrium. Derive from the van't Hoff equation a principle showing how the direction in which an equilibrium is dis- placed by increase in temperature is related to the sign of the heat- effect attending the reaction. Applications of the Qualitative Principle. Prob. 16. Show how the equilibrium hi a gaseous mixture of Ck, HC1, O 2 , and H 2 O at 25 would be displaced by increasing the tempera- ture. The heat of formation at 25 of iHCl(g) is 22,000 cal. and that of iH 2 O(g) is 57,800 cal. Prob. 17. The dissociation-pressure of solid CaCO 3 (and of all other solid substances which dissociate into one or more gaseous products) increases with rising temperature. State whether heat is absorbed or evolved when the dissociation takes place at constant temperature. In order to integrate the van't Hoff equation, the heat-content increase MI must be expressed as a function of the temperature. This temperature function is usually derived, in the way shown in Art. 131, from a knowledge of the heat-content increase at some one temperature and of the heat-capacities at constant pressure of the substances involved in the reaction. Thus the function may always be obtained by integrating the following differential equation, in which AC represents the difference between the heat-capacity (equal to CE+CF. .) of the system in its final state and its heat-capacity (equal to C A -f-C B . .) in its initial state: The integration can evidently be carried out when AC is known to be constant or when it can be expressed as a function of the tempera- ture, as illustrated by the following problems. Integration of the Van't Hoff Equation. Prob. 18. a. Derive the expression d(AZ7)=AC.dr in the way described hi Art. 131. b. Integrate it for the case that AC is a function of the form AC=ACo+ar+j3r 2 , where AC , a, and are constants. Prob. ig. Find a numerical expression for AZ7 for the reaction CHEMICAL REACTIONS IN GENERAL 287 2CO(g) +O 2 (g) = 2CO 2 (g) from the heat-capacity data given in Art. 27 and in Prob. 29 of Art. 134 and the heats of formation of iCO(g) and iCO 2 (g) at 20, which are 25,900 and 94,200, respectively. Prob. 20. Integrate the van't Hoff equation between the limits Ti and T 2 , KI and K 2 , for the following cases: a, when AC is zero and therefore A# does not vary with the temperature; b, when AC has a finite value which does not vary with the temperature; c, when AC varies with the temperature in the way considered in Prob. i8b. In numerical applications of the van't Hoff equation, in order to guard against errors in the sign of the heat-content increase and in its value with respect to the multiple chosen, the reaction under con- sideration should first be formulated in a definite chemical equation and then values of K, AH, and AC should be adopted in conformity with it. For the heat-capacities of some important gaseous sub- stances and of elementary solid substances see Arts. 27 and 55. Applications of the Van't Hoff Equation to the Equilibrium of Gas Reactions. Prob. 21. It has been found that when dry air (containing 21.0 mol-percent of oxygen and 78.0 mol-percent of nitrogen) is kept at 1957 till equilibrium is reached, 4.3% of the oxygen present is con- verted into nitric oxide. Calculate the percentage that would be so converted at 3000. The formation of iNO(g) from its elements at 20 is attended by a heat-absorption of 21,600 cal. For the heat- capacities of the three gases see Art. 27. Ans. 18.5%. Prob. 22. Formulate an exact numerical expression by which the dissociation 7 of carbon dioxide into carbon monoxide and oxygen at any temperature T and any total pressure p can be calculated. Its dissociation at 1205 and i atm. is 0.032%. Refer to Prob. 19 for the heat data needed. Prob. 23. When a mixture of 0.49 mol of O 2 and i.oo mol of HC1 is kept at 386 and i atm. in contact with solid cuprous chloride (which acts as a catalyst) till equilibrium is reached, 80% of the HC1 is con- verted into C1 2 and H 2 O. a. Formulate the reaction, and calculate its equilibrium-constant at 386. b. Formulate as a function of the absolute temperature the increase in heat-content attending this re- action, using the heat-content data of Prob. 16 and the heat-capacity data of Art. 27 and of Prob. 17, Art. 131. c. Derive from these results a numerical expression for the equilibrium-constant at 25. Prob. 24. Calculate from the heat data given below and the values of the heat-capacities given in Art. 27: 0, the heat of dis- sociation of iHgO(s) into mercury vapor and oxygen at 357; 6, a temperature-function expressing the heat-content increase attending this dissociation above 357; c, the dissociation-pressure of solid mercuric oxide at 357. The heat of formation of iHgO(s) at 20 is 21,700 cal. 288 EFFECT OF TEMPERATURE ON EQUILIBRIUM The vaporization of iHg(l) at its boiling-point (357) absorbs 14,160 cal. The mean value of the atomic heat-capacity of liquid mercury between 20 and 357 is 6.36 cal. per degree; and the heat-capacity of iHgO(s) may be taken as 10.87 at all temperatures. The dissociation- pressure of mercuric oxide at 390 is 180 mm. Ans. a, 35,500 cal.; c, 70 mm. 168. The Effect of Temperature on the Equilibrium of Chemical Changes Involving Perfect Solutes. By substituting in the second-law free-energy equation of Art. 166 the expression given in Art. 141 for the free-energy decrease attending any chemical change between perfect solutes, and noting that in that expression the last term containing the initial and final molalities does not vary with the temperature when the solution is heated at constant pressure, there again results the van't Hoff equation, now expressed in terms of the equilibrium molalities: This equation is applicable also to chemical changes between solid substances and solutes at small concentrations, since it was shown in Art. 142 that the participation of solid substances in the change does not affect the expression for the free energy. It applies also to changes in which the solvent (thus the water in aqueous solutions) enters into reaction with solutes at small concentrations. From this equation, as from the corresponding one in terms of pressures, can be derived the important qualitative principle that the equlibrium of a chemical change is displaced by increase of tempera- ture in that direction in which the reaction is attended by an increase in heat-content (or by an absorption of heat). The integration of the equation evidently involves the expression of the heat-content as a function of the temperature. Since in the case of solutions the temperature interval often is not large, the in- crease in heat-content can frequently be regarded as constant; and it is to be so regarded in the following problems unless otherwise stated. When this is not admissible its variation with the temperature must be known. This must usually be derived from direct determinations of the heat-effects attending the reaction at two or more different temperatures; for there is likely to be a large error involved in cal- culating it from the partial heat-capacities of the solutes, since they form only a small part of the total heat-capacity of the solution. CHEMICAL REACTIONS IN GENERAL 289 Prob. 25. Application of the Qualitative Principle. State the conclusions in regard to the heat of solution that can be drawn from the facts that solubility of all gaseous substances is decreased, and the solu- bility of most solid substances is increased, by an increase of tempera- ture. Prob. 26. Effect of Temperature on the lonization of Water. The neutralization of largely ionized univalent acids and bases in fairly dilute solution has been found by direct measurements at 2, 10, 18, 26, and 34 to evolve 14,750 52^ cal. at the centigrade temperature /. The value of the ionization-constant of water at 25 is i .00X10-". Calculate its value at o. Ans. o.i Xio- 14 . Prob. 27. Effect of Temperature on the Hydrolysis of Salts. Calcu- late the hydrolysis of o.i f. NH 4 CN at o from its hydrolysis (47%) at 25 and from the fact that on mixing at 25 a solution containing o.2NH| and loco g. of water with one containing 0.2HCN and 1000 g. of water and bringing the mixture back to 25 there is a heat evolution of 152 cal. Prob. 28. Effect of Temperature on Solubility. The solubility of silver chloride in water is i.io X io- 5 formal at 20 and 15.2 X io~ 6 formal at 100. a. State just what heat-quantity can be computed from these data, and calculate its value, b. State by what thermochemical measurement it could be determined experimentally. 290 EFFECT OF TEMPERATURE ON EQUILIBRIUM IV. THE EFFECT OF TEMPERATURE ON THE ELECTROMOTIVE FORCE OF VOLTAIC CELLS 169. The Effect of Temperature on the Electromotive Force of Voltaic Cells. By substituting in the second-law free-energy equa- tions of Art 166 the quantity ENF shown in Art 145 to represent the free-energy decrease attending any change in state taking place reversi- bly hi a voltaic cell there result the following expressions, known as the Gibbs-Helmholtz equation, for the effect of temperature on the electromotive force of voltaic cells: fa In this equation AT denotes the increase in heat-content which attends the change in state that takes place when N faradays of elec- tricity flow through a cell of electromotive force E containing infinite quantities of the constituent substances. Effect of Temperature on Electromotive Force. Prob. 29. Show from the Gibbs-Helmholtz equation under what conditions the electromotive force of a cell, a, is independent of the tem- perature; b, is proportional to the absolute temperature; c, increases with rising temperature; d, decreases with rising temperature. Prob. jo. a. Calculate the temperature-coefficient at 25 of the electromotive force of the cell H 2 (i atm.), H 2 SO 4 (o.oi f.), O 2 (i atm.), which was found in Prob. 20, Art. 149, to be 1.228 volts, b. Calculate the electromotive force of this cell at o, assuming that the heat-effect at- tending the change in state does not vary appreciably between o and 25. Prob. 31. Calculate the temperature-coefficient at 25 of the electro- motive force (1.262 volts) of the cell H 2 (i atm.), HC1(4 f.), Cl 2 (i atm.) from the heat-content (-22,000 cal.) of iHCl(g) and its heat of solu- tion (16,000 cal.) in 4 f. HC1 solution. Prob. 32. Calculate the electromotive force at 40 of the storage-cell Pb(s)+PbS0 4 (s), H 2 S0 4 .ioH 2 O, PbS0 4 (s)+PbO 2 (s), from its electro- motive force (2.096 volts) at o and the heat data needed. The heat- contents at 18 of iPbSO 4 (s), iPbO 2 (s), and iH 2 SO 4 (l) are -216,210 cal., 62,900 cal., and 192,900 cal., respectively. The heats of solution at 18 of iH 2 SO 4 (l) and of iH 2 O(l) in H 2 SO 4 .ioH 2 O are 12,820 cal. and 230 cal., respectively. Prob. 33. Calculation of Heat of Reaction. The electromotive force of the cell Hg(l) +Hg 2 Cl 2 (s), KCl(o.oi f.) +KNO 3 (i f ),KNO 3 (i f.) +KOH(o.oi f.), H g2 0(s)+Hg(l) at o is 0.1483 volt and at 18.5 is 0.1636 volt. a. Formulate the change in state that takes place in the cell when two faradays pass through it. b. Calculate the attendant increase in its heat-content. ELECTROMOTIVE FORCE OF VOLTAIC CELLS 291 *It will have been noted that the calculation of the increase in heat- content attending chemical changes in voltaic cells involves the heat- effect produced by the addition of definite quantities of the solute or solvent to an infinite quantity of the solution at a specified concen- tration. This kind of heat-effect often has to be considered also in evaluating the increase in heat-content used in calculating the change with the temperature of the equilibrium conditions of chemical sys- tems, as in the application of the Clapeyron equation to solutions in Art. 165. It is called (because of the relation derived in Prob. 356) the partial heat of solution of the substance. It can be derived from calorimetrically measured heats of dilution, like those considered in Art. 132, in the ways illustrated by the following problems. * Determination of Partial Heats of Solution. *Prob. 34. The measured heat-effect Q attending the mixing of iH 2 SO 4 (l) with iVH 2 O4(l) at 18 is expressed by the empirical equation, Q = i7&6oN/(N + i.8o) for values of N not exceeding 20. Calculate the heat-effect attending the addition to an infinite quantity of H 2 SO 4 .ioH 2 O, a, of iH 2 O(l); b, of iH 2 SO 4 (l). Am. a, 231; b, 12820. *Prob. 35. a. Determine with the aid of a plot of the heats of dilu- tion given in the table of Art. 132 the heat -effect attending the addition of iH 2 0(l) to an infinite quantity of ZnCl 2 . iooH 2 O. b. Derive a relation betweenthe heat-effect at tending the dissolving of iZnCl 2 (s)iniooH 2 0(l), that attending the addition of iooH 2 O(l) to an infinite quantity of a solution of the composition iZnC! 2 .iooH 2 O, and that attending the addition of iZnC! 2 (s) to an infinite quantity of such a solution. c. Calculate the value of the last named heat-effect, d. Calculate the heat-content of iZnC! 2 in iZnC! 2 .iooH 2 0, taking that of iZnC! 2 (s) equal to 97,30003!. *Prob. 36. The E/ect of Temperature on Electrode-Potentials. a. Specify precisely the change in state whose heat-effect would be used in calculating the temperature-coefficient of the molal electrode- potential of Zn(s), Zn 4 ^. b. Calculate a value of this temperature- coefficient from the heat values found in Prob. 35, the heat-content (-22,000 cal.) of iHCl(g), and its heat of dilution as given in Art. 132, considering for this purpose solutions with iooH 2 O per iZnC! 2 , or per iHCl as infinitely dilute. CHAPTER XIII *SYSTEMIZATION OF FREE-ENERGY VALUES 170. Importance of Systemizing Free-Energy Values. Since a knowledge of the changes in free energy attending chemical changes makes it possible to predict the direction in which they can take place and the conditions under which they are in equilibrium, the working- out of a complete system of free-energy values becomes an experi- mental problem of great chemical importance. The more general principles by which changes in free energy can be determined have already been fully considered in the preceding chapters. The pur- poses of this chapter are the direct application of these principles to the determination of free-energy values, and the presentation of certain conventions by which free-energy values can be system- atized. These conventions will be fir-st considered. 171. Expression of Free-Energy Changes by Equations. - Since the free energy of a substance is a quantity which, like its heat-content, is fully determined by its state, changes in free energy can be repre- sented by equations corresponding in every respect to those used for expressing changes in heat-content; conventions being adopted for showing the state of the substance identical with those described in Art. 129. It will suffice to supplement the specifications of that article by the following statements. Free-energy values are commonly expressed in calories on account of their close relations with heat-content data, which are commonly so expressed. The pressure at which each substance exists must always be definite, since the free-energy is in general in a high degree dependent on the pressure. When not specified, the pressure is understood to be one atmosphere. In correspondence with the convention as to the arbitrary zero- point of the heat-content scale, the free energy at any definite tem- perature of any definite quantity of a substance in any definite state is equal to the increase in free energy that attends the formation of the substance in that state out of the pure elementary substances at the same temperature, and at a pressure of one atmosphere, each elemen- tary substance being in the state of aggregation that is the stable one 292 SYSTEMIZATION OF FREE-ENERGY VALUES 293 at this temperature and pressure. Thus the free energy of iHI(i atm.) at 25 refers to its formation from gaseous hydrogen and solid iodine at 25 and one atmosphere; but at 400 it refers to its forma- tion from gaseous hydrogen and gaseous iodine at 400 and one atmosphere. The free energy of a substance in a solution is understood to mean the increase in free energy attending the formation of the pure substance out of the elementary substances plus that attending the introduction of it into an infinite quantity of the solution under consideration. This free energy is indicated by attaching to the chemical formula a paren- thesis showing the concentration or composition of the solution; for example, iKCl(at o.i f.), iH 2 SO 4 (in H 2 SO 4 .ioH 2 O). The free energy of water in a solution is often conveniently referred to that of pure water (instead of to that of hydrogen and oxygen) as zero; and when so referred the symbol Aq (instead of H 2 O) is used; for example, 5Aq (in H 2 SO 4 .ioH 2 O). The free energy of an ion in aqueous solution is conveniently de- fined, in correspondence with the definition of electrode-potentials given in Art. 151, as the increase in free energy that attends the forma- tion of the ion in solution at the specified concentration out of the elementary substance at one atmosphere, hydrogen-ion at i molal being formed at the same time out of hydrogen gas at one atmosphere. Thus the free energy of Pb++ (o.i m.) is equal to the increase in free energy that attends the reaction Pb(s) + 2H + (i m.)=Pb ++ (o.i m.) +H 2 (i atm.) ; the free energy of H + (i m.) as well as that of H 2 (i atm.) being thereby arbitrarily assumed to be zero. When necessary, equations expressing changes in free energy may be distinguished from those expressing changes in heat-content by prefixing to them the symbols (F) and (H), respectively; and the absolute temperature may be shown by attaching to this letter a subscript . For example : (F 298 ) H 2 (g) + JOsfe) = H 2 0(l) + 56,620 cal. (# 298 ) H 2 (g) + i0 2 (g) = H 2 0(l) + 68,400 cal. (F 298 ) H 2 0(l) = - 56,620 cal. (# 298 ) H 2 0(l) = - 68,400 cal. Since the free energy of a substance at various small pressures or concentrations can be calculated, by the logarithmic expressions of Art. 137 and 138, from the free energy at any one definite small pressure or concentration, it suffices to determine and record the free energy 2Q4 SYSTEMIZATION OF FREE-ENERGY VALUES of each gaseous substance at one atmosphere and of each solute in each solvent at i molal. These standard values are, however, com- monly so expressed as to represent the free energy of the substance at this pressure or concentration under the assumption that it behaves as a perfect gas or perfect solute up to this pressure or concentration; that is, they are calculated by the logarithmic equation from those determined at much smaller pressures or concentrations. In the case of solutes so concentrated that they show large deviations from the behavior of perfect solutes the free energies must be deter- mined at a series of concentrations, and these values must be separately recorded. Or, if preferred, the activity-coefficients corresponding to the free energies at various formalities of the substance may be calculated and recorded. 172. Determination of the Free Energies of Substances. With the aid of the principles relating to free energy considered in chapters X, XI, and XII, and the conventions presented in the preceding article, the free energies of substances in their different states can be determined, as illustrated by the following problems. Prob. i. Free Energies of Elementary Substances in Different States. With the aid of the principles of Arts. 137-139, calculate the free energy at 25 of, a, I 2 (g. at i atm.); b, I 2 (at i m. in H 2 O); and c, I 2 (at i m. in CC1 4 ). At 25 the vapor-pressure of pure iodine is 0.305 mm., its solubility in water is 0.00132 molal, and its distribution-ratio between carbon tetrachloride and water is 86. Prob. 2. Free Energy of Solutes at High Concentrations. a. Calculate the free energy of iNH 3 in a 8.6 n. solution in water at 25 from the following data at 25. The free energy of iNH 3 (g) is 4740 cal. The vapor-pressure of NH 3 from a 0.2 n. solution in water is 2.70 mm. The distribution-ratio of NH 3 between CC1 4 and H 2 O is 0.0040 when the NH 3 is 0.2 n. in the water, and 0.0086 when the NH 3 is 8.6 n. in the water, b. State the principles involved, c. State how the same free energy might be obtained from another kind of measure- ment with the 8.6 n. aqueous solution. Free Energies Derived from Electromotive Forces. Prob. j. Calculate the free energy of iHCl(g) at 25 from the facts that at 25 the cell H 2 (i atm.), HC1(4 f.), Cl 2 (i atm.) has an electro- motive force of 1.262 volts, and that the vapor-pressure of the HC1 in its 4 formal solution is 0.0182 mm. Ans. 22,800 cal. Prob. 4. Calculate the free energy of iAgCl(s) at 25 from the molal electrode-potentials and the solubility of AgCl which is i.3oXio~ 6 formal at 25. Ans. 26,200 cal. Prob. 5. Calculate the free energy of iH 2 0(l) at 25 from the SYSTEMIZATION OF FREE-ENERGY VALUES 295 electromotive force (0.926 volts) of the cell H 2 (g), NaOH(o.i f.), HgO(s) H-Hg(l) and from the free energy (-13880 cat.) of iHgO(s) at 25. Ans. 56,600 cal. Prob. 6. Calculate the free energy of iAg 2 O(s) at 25. The electro- motive force of the cell H 2 (g), KOH(o.i f.), Ag 2 O(s) +Ag(s> is 1.172 volt at 25. Ans. 2500 cal. Prob. 7. Formulate the free energy equation which can be derived from the molal electrode-potential of, a, Cl 2 (g), Cl~, and b, O 2 (g), OH~. Find the free energy at 25, c, of Cl~ (i m.), and d, of OH~(i m.).' Ans. c, 31,400 cal.; d, 37,400 cal. Free Energies Derived from Equilibrium Measurements. Prob. 8. Calculate the free energy of iNO(g) at 1957 from the data of Prob. 21, Art, 167. Ans. 13,700 cal. Prob. 9. Calculate the free energy of iHgO(s) at 357 from its dis- sociation pressure at 357 found in Prob. 24, Art. 167. Ans. 5700 cal. 'The Free Energy of a Substance at One Temperature Derived from Us Free Energy at Another Temperature. Prob. 10. With the aid of the free-energy equation of Art. 166 cal- culate the free energy of iNO(g) at 25 from its free energy at 1957 found in Prob. 8 and from the heat data of Prob. 21 of Art. 167. Ans. 2 0,500 cal. Prob. ii. Calculate the free energy of iHgO(s) at 25 from its free energy at 357 found in Prob. 9 and from the heat data of Prob. 24, Art. 167, considering the heat of dissociation of solid mercuric oxide into liquid mercury and oxygen to be a linear function of the tempera- ture between 25 and 357. Ans. 14,100 cal. Prob. 12. Calculate the free energy at 25 of i at. wt. of monoclinic sulfur from the facts that its conversion into rhombic sulfur at its transition-point (95.5) is attended by a heat-evolution of 105 cal. and that the atomic heat-capacities of the monoclinic and rhombic forms are 6.0 and 5.7 cal. per degree. Tabulate this result with that obtained in Prob. 19 of Art. 139. Prob. 13. Derivation of Free Energies from Original Data. By a thorough examination of the chemical literature to find all the data bearing on the problem and by a critical consideration of these data derive the best values of the free energy at 25, and of a function ex- pressing its change with the temperature, for one of the following solid substances (assigned by the instructor): Ag 2 0, HgO, AgCl, PbCl 2 , Hg 2 Cl 2 , HgCl 2 , CuCl, Agl, etc. 173. The Equilibrium-Conditions of Chemical Reactions Derived from the Free Energies of the Substances Involved. When values of the free energies of substances have been once determined by the methods already described, these values can be employed, conversely, for calculating the free-energy decrease attending chemical changes 296 SYSTEMIZATION OF FREE-ENERGY VALUES and their equilibrium-constants, as in the following problems. From this fact arises the great importance, already referred to, of a complete, systematic knowledge of the free energies of substances. The Equilibrium-Constants of Chemical Reactions Calculated from the Free Energies of the Separate Substances. Prob. 14. a. Calculate from the free energy values already con- sidered the equilibrium-constant at 25 of the reaction: Ag 2 O(s) +H 2 O+2C1- = 2 AgCl(s) +2OH- b. State what mixture would finally result if iAg 2 O(s) were treated at 25 with 1 1. of o.i f. Nad solution. Prob. 15. a. Calculate from the free energy values the equilibrium- constant of the reaction 4HCl(g)+O 2 (g) =2Cl 2 (g)+2H 2 O(g) at 25. b. Calculate the mol-percents of chlorine and oxygen in the gas which at i atm. and 25 would escape from a 4 f . HC1 solution if oxygen at i atm. were passed through it in contact with a catalyst so that equi- librium was established. At 25 in 4 f. HC1 solution the vapor-pres- sures of the water and HC1 are 19.6 mm. and 0.0182 mm., respectively. Am. a, 1.3X10"; 6, C1 2 , 7.2%, O 2 , 90%. NOTATION a * ^ activity; area. n number. a, b van der Waals constants. n number of molecules. c molal concentration; mola- n number of molecules in one mol lity. P pressure. d density. r radius. d differential. s solubility. e base of natural logarithms. t time. f S force, gravity-acceleration. t u centigrade temperature, velocity. h height; hydrolysis. V volume. i mol-number. V specific volume. k kinetic-energy constant. V molal volume. k specific reaction-rate. X mol-fraction. I distance; length. x,y,z coordinates. m mass. A atomic weight ; work-content M molecular weight. A,B constants. N number of mols. C heat-capacity. P osmotic pressure. C specific heat-capacity. P probability. C molal heat-capacity. Q heat evolved. E energy. R gas-constant. F free energy. S entropy. H heat-content. T absolute temperature. K equilibrium-constant. U energy-content. K kinetic energy. W work. c normal concentration; nor- L specific conductance. mality. N number of equivalents or E electromotive force. faraday s. E electrode-potential. Q quantity of electricity. E molal electrode-potential. R resistance. F faraday. T transference number. I current-strength. U ion-mobility. L conductance. V potential. a activity-coefficient, coefficients. X A wave-length, equivalent conductance. 7 dissociation or ionization. V frequency. A increment. 7T circumference-diameter ratio. V viscosity. S summation. 297 INDEX In this index formal statements of laws or principles and their formula- tions in equations are itemized under the heading "laws, principles, funda- mental equations." Definitions of "concepts, properties, and fundamental constants," values of the "constants of chemical substances," and defini- tions and values of "units" are entered under these headings, respectively. ACIDS and Bases, dibasic acids, ionization, 157-159. displacement from salts, 159-160. heat of ionization, 220. heat of neutralization, 219-220. ionizatioh, 118, 123, 152. titration, electrometric, 267. titration with indicators, 162-164. Acid salts, ionization, 158-159. Activity and activity-coefficient, applications, 172-173. change with concentration, 171-172. concept, 169, 170. determination, 170-171, 234, 250. relation to electromotive force, 250. free energy, 233-234, 237. solubility, 173. vapor-pressure, 170. Adiabatic expansion, 41, 44. Adsorption in contact catalysis, 142. Alloys, / annealihg or tempering, 197. compound formation, 193-194, 199. cooling-curves, 194, 198, 202. diagrams, 194, 196, 199, 201. eutectic point and composition, 192, 193, 202. freezing-point lowering, 193. solid solutions, 198-200. ternary, 202. Annealing, 197. Arc process of nitrogen fixation, 136, 287. Association, molecular, 63, 71. Atomic and molecular theories, gaseous state, 12, 22-34, 41-46. general discussion, 5, 95. liquid state, 51-54, 64. (See also Ionic theory.) Atomic weights, determination, 6, 17, 18, 42, 94, 95. relation to combining weights, 6, 7, 17, 18. table of values, 8. Avogadro's principle, 12, 24. BASES. See Acids and bases. Bessemer steel process, 15. Boiling-point composition diagrams, 68, 73. concentrated solutions, 67-69, 72- 75- constant, determination, 62. raising, relation to composition, 60-63. vapor-pressure, 59-60. relation to barometric pressure, 48. heat of vaporization, 49, 60, 61. vapor-pressure, 48, 59-61. Boyle's law, derivation, 23. statement, 10. CARRIER catalysis, 140-141. Catalysis and catalysts, carriers, 140-141. contact agents, 141. enzymes, 143. hydrogen-ion and hydroxide-ion, 142. mechanism, 140-142. solvents, 178-179. water, 143. Cells, voltaic. See Voltaic cells. Chemical formulas, 8, 18. principles, 3. substances, classes, 3, 4, 6. relation to Henry's law, 64. relation to Raoult's law, 58. Clapeyron equation, applications, 49, 60, 81, 281-283. approximate, 50. derivation, kinetic, 54. derivation, thermodynamic, 280. formulation, 49, 50, 280. integration, 50, 281-282. Clark standard cell, 262. Cohesion of gases, 27, 28, 46- 299 3 oo INDEX Color-change of indicators, 161. on neutralization, 160. Combining volume of gases, 12. Combining weights, determination, 6, 7. law, 4, 5. Combustion, heat of, 209. Complex-salt formation, 65, 78, in, 167, 169, 203, 267. Components, concept, 174, 181-183. determination, 183. (See Phase equilibria.) Composition by weight, 4-9, 55, 56. Compound formation in solid alloys, 193-194, 199. in solutions, 71, 72. Compression of liquids, 231. Concentrated solutions, boiling-point, 67-68, 73~7S- freezing-point, 82-83. osmotic pressure, 87-88. vapor-pressure, 67-72. Concentration by hot gases, 222. cells, 250-251. polarization, 271, 273. CONCEPTS, PROPERTIES, FUNDAMENTAL CONSTANTS, absolute temperature, u. acceleration of gravity, 13. activity, 169. activity-coefficient, 170. additive property, 93. adiabatic, 41. adsorption, 142. ampere, 100. anion, 100. anode, 98. atmosphere, 13. atom, 5. atomic energy, 43. atomic weight, 6, 8, 17. Avogadro number, 25. bimolecular reactions, 133. bivariant systems, 180. boiling-point, 48. boiling-point constant, 61. boiling-point, molal raising, 62. carrier catalysts, 140. catalysis, 140. cathode, 98. cation, 100. chemical substance, 6. chemistry, 3. cohesion-constant a, 27. CONCEPTS, PROPERTIES, FUNDAMENTAL CONSTANTS (continued) cohesion pressure, 27. combining weight, 4. component, 174, 181-182. compound substance, 4. compression-coefficient, 231. concentrated solutions, 56, 67. concentration, 55, 56. concentration-cell ,250. conductance, actual, 112. conductance, equivalent, 112. conductance-ratio, 117, 119. conductance, specific, 112. cooling-curves, 192. current-density, 271. cycle of changes, 223. cyclical process, 277. decomposition-potential, 272. density, 10. dissociation, 16. dissociation-constant, 148. dissociation-pressure, 165. distribution-ratio, 77. electrode, 98. electrode-potential, 256. electrode-potential, molal, 257. electrolyte, 98. electrolysis, 98. v electron, 101. electronic charge, 101. element, 4. elementary substance, 4. energy, 35. energy-efficiency, 275 enzyme, 143. equation of state, 21. equilibrium, chemical, 145. equilibrium-constant, 146, 148. equivalent weight, 9. eutectic mixture, 192. eutectic point, 192. eutectic ternary, 202. faraday, 100. formality, 55. formula- weight, 8. fractionation, 3. free energy, 227, 292. free path, mean, 30. freezing-point, 80. freezing-point-constant, 81. freezing-point, molal raising, 82. gas-constant, 13. gases, perfect, 10. gas-polarization, 273. heat, 36. heat-capacity, molal, 39. heat-capacity-ratio, 41. teat-capacity, specific, 39. INDEX 301 CONCEPTS, PROPERTIES, FUNDAMENTAL CONSTANTS (continued) . heat-content, 210, 213. heat of dilution, 218. fusion, 217. reaction, 209. solution, 217. partial, 291. transition, 217. ^ vaporization, 217. humidity, 190. hydrate, 65. hydrolysis, 154. hydrolysis-constant, 156. indicator, 161. indicator-constant, 161. ion, 96, 100. ion-conductance, 120. ion-constituents, 100. ionic theory, 96. ionization, 96. ionization-constant, 152. irreversible process, 224. isomorphism, 91. Joule-Thomson effect, 44. kinetic energy, 24. kinetic-energy constant k, 25. kinetic hypotheses, 22. liquid-potential, 254. mixture, 3. mobility of ions, 107. mobility of ion-constituents, 107. molal concentration, 56* properties, 10. quantities, 21. weight, 12. molality, 55. molecular composition, 18. formula, 18. weight, 12. molecule, 5. mol-fraction, 15. mol-number, 96. mol-ratio, 55. normal concentration, 56. normality, 55. osmotic pressure, 84. overvoltage, 273. partial pressure, 14. perpetual motion first kind, 35. second kind, 76, 223. phase, 55. phase rule, 181. physical chemistry, 3. CONCEPTS, PROPERTIES, FUNDAMENTAL CONSTANTS (continued) physics, 3. polarization, 273. porous-plug experiment, 44. pressure, 13, 37. probability-function, 33. pure substance, 3. reaction order, 133. rate, 131. specific rate, 131. uni-, bi-, tri- molecular, 132, 133- reduction-potential, 270. resistance, 112. reversible process, 224. salting-out effect, 66. semipermeable wall, 14. solid solutions, 197. solubility, 64. solubility-constant, 64. solubility-product, 167. solute, 55. solution, 55. solvate, 65. solvent, 55. specific heat, 39. specific quantities, 21. state of system, 35, 174. substance, chemical, 6. substance, pure, 3. system, 35. temperature, absolute, n. temperature, centigrade, n. ternary alloys, 202. thermochemistry, 207. transference, 104. transference-number, 105. transition-temperature, 176, 189. triple-point, 177. uni variant, 180. vapor-pressure, 47. variance, 180. velocity of molecules, mean, 34. mean square, 34. most probable, 34. viscosity, 118. voltaic action, 98. voltaic cell, 244. volume constant b, 28. molal, 21. specific, 21. weight-normality, 55. work, 36. work-content, 227. 3 02 INDEX CONDUCTANCE, electrical, actual, specific, and equivalent, 112. determination, 113. equivalent, at zero concentration, 114-116. general discussion, 112. ion-, determination and values, 1 20. ion-concentration, determination, 121. mixtures, 116. relation to activity, 171-172. concentration, 114-119. ion-mobility, 113-114, 116- 119. ionization, 117-119, 152-154, 171-172. transference, 113-114. viscosity, 118-119. Conduction, electrolytic Faraday's law, 99-100. general discussion, 98, 112. mechanism, 100-102, 106-107, 113-119- metallic, 98,101. Conservation of energy, law, 35, 207. CONSTANTS OF CHEMICAL SUBSTANCES, activity-coefficients, 171. atomic weights, 8. boiling-point constants for sol- vents, 62. composition of air, 15. conductance-ratios for largely ion- ized substances, 124, 171. density of air, 14. mercury, 13. deviation-coefficients for gases, 19. electrode-potentials, molal, 260. freezing-point constant for benzene, 83. water, 82. heat-capacities of elementary substances, 92. elements, 93. gases, 42. solid compounds, 93. water- vapor, 216. heats of vaporization, 217. hydration of ions, in. hydrogen-ion concentration, change during neutralization, 163. indicator-constants, 164. ion-conductances, 120. CONSTANTS OF CHEMICAL SUBSTANCES (continued) ionization-constants of acids and bases, 123, 152, 157. poly basic acids, 157. water, 154. molal weight of air, 14. molecular collisions, 30. diameters, 30. mean free-path, 30. mol-numbers of salts, 96. overvoltages, 275. phase-diagrams of substances, acetic-acid benzene, 191. alloys, 196, 199, 201. organic liquids, 73. salt-hydrates, 188-189, 194- 195- sulfur, 176. pressure-volume product of gases at moderate pressure, 19. at high pressure, 19, 20, 28. solubilities of air-gases, 65. transference- numbers, 109. van der Waals' constants, 28. vapor-pressure of ice, 80. water, 48, 80. viscosities of salt solutions, 119. CONTACT agents, 141. Contact-process for sulfuric acid, 15, 222. Cooling-curves, 192, 194. D ALTON'S partial-pressure law, 15. Deacon chlorine process, 221. Decomposition-potential, 272. Definite proportions, law, 4. Dehydration. See Drying. Diagrams. See Phase diagrams. Diffusion-potentials. See Liquid-potentials. Dilution, heat of, 218. Displacement of acids or bases, determination, 160. principles, 159-160. Dissociation of gases, determination from density, 16, 148-149- mass-action relations, 148-149. pressure, effect of, 149. Dissociation of solids, dissociation-pressure, 165, 187, 203. gas-phase with one component, 165-166, 187-190. gas-phase with two components, 167. INDEX 303 Dissociation of solids (continued) gaseous and liquid phases, 203. heat of, 282. pressure-temperature relations, 165-166, 187-190. Distillation, of solutions, 68, 69, 72-74, 79. steam, 48, 79. Distribution of base between acids, 159-160. kinetic energies, 31-34. molecular velocities, 33, 34, 52, 53. solutes between phases, 63-66, 76-77, 235. Dry cell, 261. Drying of organic liquids, 190. salt hydrates, 190. Dulong and Petit's law, 91. EFFLORESCENCE of salt-hydrates, 190. Electro-analysis, 272. Electrochemical equations, 102-103. Electrode-potentials, concentration effect, 257-259. electromotive force, relation to, 260-261. equilibrium-constants, relation to, 269-270. expression, 255-256. mechanism, 254. molal, 257-260. normal calomel cell, 259. temperature effect, 291. Electrode, molal hydrogen, 255. normal calomel, 259. processes, 98-102, 254-255. Electrolysis, deposition of metals, 272. energy-efficiency, 275. Faraday's law, 99-100. formulation, 102-103. mechanism, 100-102. polarization, 273-276. products, 98-99. separation of elements, 272. Electrometric titration, 266-267. Electromotive force, back, determination, 274. cells involving chemical changes, 252-253. concentrated solutions, 261- 262. physical changes, 249-251. concentration-cells, 250-251. relation to activity, 250. Electromotive force (continued) relation to concentration, 249, 271. electrode-potentials, 254, 260- 261. equilibrium-constant, 252-253. free energy, 247-253, 294-295. temperature, 290-291. Elements and elementary substances, general relations, 4. heat-capacity of gaseous, 41. heat-capacity of solid, 91-92. molecular formulas of gaseous, 19, . 41-44. periodic law, 95. separation by electrolysis, 272. Energy, conservation, 35. content in general, 39-40, 207-208, 209-211. content of imperfect gases, 44-46. ^perfect gases, 40-44. efficiency of electrolysis, 275. general discussion, 35-40. intramolecular, 43. molecular, 42-43. units, 37. Enzymes, 143. Equations, electrochemical, 102-103. free-energy, 292-293. heat-content, 211-214. of state of gases, 13, 19, 21, 27. thermochemical, 211-214. Equilibrium of gas or solute reactions, derivation from reaction-rate, 138. derivation from free-energy, 238- 242. determined thermochemically, 221. general discussion, 145, 174-175. oxidation-reactions, 269-270. relation to electromotive force, 252-253. free-energy, 236, 240, 242, 295-296. pressure, 150. temperature, 285-289. (See also Mass-action law.andPhase equilibria.) Equilibrium of polyphase systems, general considerations, 174-175, 202-203. phase rule, 180-186. relation to activity, 236. pressure and temperature, 280-283. 304 INDEX Equilibrium of polyphase systems (continued) systems with one-component, 176-179. solid solutions, 197-200. three components, 201-203. two components, 187-200. two liquids, 76-79. unstable conditions, 177-179, 195. (See also Phase diagrams, and Phase rule.) Equilibrium-constants, relation to dissociation-constants, 150. electrode-potentials, 269-270. electromotive force, 252-253, 266- 267, 269-270. free-energies, 240, 242, 295-296 ionization-constants, 157. oxidation-reactions, 269-270. reaction-rate, 138. Eutectic composition, 192-193, 202. point, 192, 202. texture, 192. Explosions, pressure and temperature, 221. Extraction by solvents, 77. FARADAY'S law, 99-100. Flames, maximum temperature, 221. Formation, free energy of, 213-215. heat of, 213-215. Free energy, activity, relation to, 237. change attending activity changes, 234. compression of liquids, 231. concentration-changes, 231- expansion of gases, 230. formation of compounds, 294- 295- reactions between gases, 238- 240. reactions between solutes, 240- 241. reactions involving solids, 242. transfer between phases, 234- 237. transfer between solutions, 231-234. transition of unstable phases, 235-236. voltaic action, 247-253. volume changes, 230-231. concept, 226-229. electromotive force, relation, 247- 253. 294-295. Free energy (continued) elementary substances, 292, 294. equations, 231, 234, 239, 241, 242, 284-285, 293. equilibrium, relation to, 236, 240, 242, 295-296. expression by equations, 293. importance, 228-229, 292. ions, 293. of formation, 292-293. temperature, effect of, 284-285, 295. values, determination, 294-296. values, systemization, 292-296. Freezing-point, constants, 81-82. heat of fusion, relation, 81, 82. lowering, composition relations, 81-83. lowering, effect of salts, 96-97. lowering of metals, 193. pressure, effect of, 177, 281. raising by solutes, 198. solid solutions, 198-200. vapor-pressure relations, 80-82. Fusion, heat of, freezing-point relation, 81, 82. heat-of-vaporization relation, 81. GAS-CONSTANT, calculation of, 13, 38. Gases, cohesion, 27, 28, 46. dissociation, 16, 148-149. energy of imperfect, 44-46. energy of perfect, 40-44. kinetic theory, 22-34, 42-46. mass-action relations, 148-151, 165-166. partial pressure, 14-16. reactions, equilibrium, temperature-effect, 285-288. heat of, 209. rate of, 135-136. solubility, 64-66. volume at high pressures, 19, 20. at moderate pressures, 19, 25-28. of perfect, 10-16, 22-25. Gibbs-Helmholtz equation, 290. HEAT-CAPACITY, additive property, 93. general discussion, 39. influence on heat of reaction, 216. of perfect gases, kinetic interpretation, 42-44. physical laws, 40, 41. relation to atomic weight, 42. relation to composition, 41, 42. INDEX 305 HEAT-CAPACITY (continued) of solid substances compound, 93. elementary, 91-92. ratio for gases, 41. Heat-content decrease, attending aqueous-solution reactions, 218-220. combustions, 209. dilution, 218. formation of compounds, 213- 216. gaseous reactions, 209. ionization, 220. , neutralization, 219-2267- solution, 217-218. transition, 218. determination, 208, 214-215, 290. energy-content relation, 210. expression by equations, 211-214, 286. temperature-effect, 215-216, 286. Heat of fusion. See Fusion. Heat of reaction of various types. See Heat-content decrease. Heat of vaporization. See Vaporization. Henry's law, 63-66. Hydrates of salts. See Salt-hydrates. Hydration of ions, in, 121. salts, heat of, 281. solutes, 65. Hydrogen-ion, acid salts, 142, 158. catalytic effect, 142. change on neutralization, 163. concentration in water, 122. determination, 142, 158, 159, 162. Hydrolysis of salts, constant, 156-157. determination, 122, 155, 267. mass-action relations, 156-157. temperature-effect, 289. Hygroscopicity of salts, 190. INDICATORS, constants of, determination, 162, 267. constants, values, 164. principles, 161-164. Ionic theory, explanation of conductance, 113-119. electrode-potentials, 254. electrolysis, 101-103. electromotive force, 254-255. heat of reaction, 220. Ionic theory, explanation of (continued) largely ionized substances, 123- 126, 171. liquid-potentials, 262. mass-action effects, 152-164, 167- 173- molal properties of salts, 96-97. transference, 106-108. voltaic action, 243-244. Ions or ion-constituents, activity, 171-172. complex, 78, in, 167, 169, 267. composition determined, in. concentration, determined, 121- 122, 266-267. concentration in water, 122. conductance, 120-121. electric charge on, 101. electron relations, 101. electrostatic separation, 255. free energy, 293. general discussion, 100. hydra tion, in, 121. ionization, 106, 117-119, 123-126. mobility. See Mobility, ionic. transference, 107. Ionization, acids and bases, 123, 152. acid salts, 157-158. common-ion effect, 152. dibasic acids, 157-159. from conductance, 117-119, 152. from molal properties, 97. general discussion, 96, 106, 123- . 126, 153. heat of, 220. hypotheses, 124-126, 153. mass-action relations, 152-154. salts, 123-126, 153. water, 122, 154, 157. lonization-constants, acids and bases, 152. determination, 152, 154, 157, 159, 267. dibasic acids, 157. Irreversible processes, 224-225. JouLE-Thomson effect, 44. KINETIC derivation of Clapeyron equation, 53-54- energy of molecules, 24,31 -34, 4 2 43 hypotheses, 22. interpretation of vaporization, si- interpretation of vapor-pressure of solutions, 58, 70, 71. 306 INDEX KINETIC (continued) mechanism of reaction-rate, 135, theory of imperfect gases, 25-28, 45-46. theory of perfect gases, 22-25, 42- 44- Kopp's law of heat-capacity, 93. LAWS, PRINCIPLES, FUNDAMENTAL EQUATIONS, Avogadro's, 12, 24. boiling-point raising, 60, 61. Boyle's, 10. Clapeyron equation, approximate, 50. exact, 49, 280. combining volumes, 12. conservation of energy, 35, 207. constant energy-content of gases, 40. Dalton's partial pressure, 15. definite proportions, 4. distribution between phases, 77. Dulong and Petit's, 91. Faraday's, 100. free-energy equations, 231, 234, 239, 241, 242, 284, 285, 293. freezing-point lowering, 81, 82. gases, at moderate pressure, 19. gases, perfect, 10-13. Gibbs-Helmholtz equation, 290. heat-content equation, 212. Henry's, 63. indicator equation, 162. initial and final states, 36. independence of reaction-rates, 137. kinetic equation for distribution of energies, 33. imperfect gases, 27. perfect gases, 22. vapor-pressure, 54. Kopp's heat-capacity, 93. liquid-potential equations, 263. mass-action law of equilibrium, 146, 148. mass-action law of reaction-rate, 131. 134- Maxwell's distribution, 32. Ohm's, 112. osmotic pressure, 87. perfect solutions, 56. periodic, 95. perpetual motion, 35, 76, 223. phase rule, 181. Raoult's, 57, 58, 59. salting-out, 66. solubility of gases, 64. solubility-lowering, 78. solubility-product, 167. LAWS, PRINCIPLES, FUNDAMENTAL EQUATIONS (continued) thermodynamics, first law, 35. thermodynamics, second law, 224. transference, 105. Trouton's rule, 217. van der Waals equation, 21, 27. van't Hoff equation, 286. Lime-burning, 187. Liquid air processes, 45. Liquid-potentials, conventions, 256. derivation of equations, 264-266. equations for, 263. mechanism, 262. Liquids, boiling-point, 48. compression of, 231. heat of vaporization, 49,52-54, 217. mixtures of. See Solutions. partially miscible, 79. pressure, effect on vapor-pressure, 85-86. temperature-change on mixing, 70. vapor-pressure, 47-54, 85-86. volume-change on mixing, 70. MASS-ACTION law of equilibrium, applied to equilibrium in general, 146- 147, 165. imperfect solutes, 169-173. indicators, 161-164. perfect gases, 148-151. perfect solutes, 152-169. solids and gases, 165-166. solids and solutes, 166-169. titration, 162-164. derivation, 138, 240, 241. formulation, 146, 148, 167. phase-rule, contrasted, 203. Mass-action law of reaction-rate, 131- 138. ^ Maxwell's distribution law, 32. Melting-point. See Freezing-point. Metals, deposition by electrolysis, 272. polarization, 275. (See also Elements, and Alloys.) Molal properties of gases, 10-46. ionized substances, 96-97. solutions, 57-90. Molal quantities, 21. Mobility (ionic) conductance, relation to, 113. experimental determination of, 107- 109. INDEX 307 Mobility (ionic) (continued) general discussion, 106-107, 123- 126. transference, relation to, 107-108. viscosity, effect of, 118-119. Molecular formulas, 18. formulas of elementary substances, 19. magnitudes, 30-31. mechanism of electromotive force, 243- mechanism of reactions, 134-1351 140-142. nature of gases, 12, 18, 19, 22-34, 41-44. nature of solutes, 57, 63, 65, 70-72. species. See Chemical substance, weight effect of solvent, 63, 83. of gases, 12, 16, 17. of solutes, 57, 62, 63, 83. (See also Molecules and Kinetic.) Molecules, atomic composition, 19, 41-44- attraction, 26-28, 46, 51, 70, 71. collisions with one another, 30, 3i- diameter, 29-30. distribution of energies, 31-34- distribution of velocities, 33-34- free-path, 30. impacts on unit-surface, 23, 26, 53. kinetic energy, 24, 31-34* 42- mass, 25. number in one mol, 25. rotational energy, 43. velocity, 24, 30, 33, 34, 52. volume, 26-29. Mol-number, 96. Moving-boundary method of transfer- ence, 108. NEUTRALIZATION, heats of, 219-220. indicators, 161-164. Nitrogen-fixation, arc process, 136, 287. OHM'S law, 112. Order of reactions, 133. Osmotic pressure, experimental determination, 84. general discussion, 84. relation to composition, 87-88. hydrostatic pressure, 88. vapor-pressure, 85-87. Overvoltage. See Polarization. Oxidation-reactions equilibrium in relation to electrode- potentials, 269-270. potentials of, 270. PERIODIC law, 95. Perpetual motion, first kind, 35. phase-equilibrium derivations, 76- 78. phase-rule derivation, 183-186. second kind, 76, 223. Phase diagrams, boiling-point composition, 68, 73. pressure-temperature, 176, 188- 189, 196. substances acetic-acid benzene, 191. alloys, 193, ^194, 196, 199-202. organic liquids, 73. salt-hydrates, 188-189, 194- 195- sulfur, 176. temperature-composition, 68, 73, 190-197, 199-202. triangular, 201. vapor-pressure composition, 68, 72, 79, 188. vapor-pressure freezing-point, 80. Phase rule, component-concept, 174, 181-183. derivation deductively, 183-186. derivation inductively, 180-181. mass-action law, contrasted, 203. statement, 181. variance-concept, 180. Phase equilibria. See Equilibrium of polyphase sys- tems. Polarization, concentration-, 271-273. gas-, 273-276. Porous-plug experiments, 44. Pressure of gases, cohesion, 27, 28, 46. height, change with, 86. kinetic interpretation, 22-29. partial, 14-16. volume and. temperature relations, 10-21. wet and dry, 47. Probability of kinetic energies, 3 I ~34- Producer-gas process, 166. RAOULT'S law. 57-59- Rate of reactions, catalytic effects, 140-143- concentration effects, 131-139- dissolving of solid substances, 139- 308 INDEX Rate of reactions (continued) mass-action law, 131-138. mechanism, 134-135, 140-142. order of reactions, 132-134. simultaneous reactions, 136-138. surface effects, 139. temperature effects, 144. transition of solids, 178-179. Reduction-potentials, 270. Reversible processes, 224-226. SALT-hydrates, determination of existing, 189, 195. diagrams, 188-189, 194-195. drying, 190. efBorescenceandhygroscopicity,i9O. heat of hydration, 281. preparation, 195. solubility, 194-196. transition, 189, 190. unstable forms, 195-196. vapor-pressures, 188-189, 196. Salting-out effect, 66, 77. Salts. See Substances, largely ionized. Saturation of solution, rate, 139. Semipermeable walls, 14, 84. Shaking-out by solvents, 77. Solid solutions, description, 197. freezing-point, 1 98-20x5. Solid substances, atomic properties, 91. heat-capacity, 91-94. rate of dissolving, 139. solubility, 166-169, I 73- Solubility of gases, 64-66. . largely ionized substances, 167-169, 173- salts, determined from conductance, 121. electromotive force, 267. salts, effect of common-ion, 168. complex formation, 167, 169. metathesis, 168-169. temperature, 289. solvents in one another, 78, 79. unstable phases, 177, 179. Solution, heat of, 217-218, 220, 291. Solutions (and solutes), boiling-point of concentrated, 67- 69, 73-75- boiling-point of dilute, 59-62. classes, 56, 67. composition, expression of, 55, 56. conductance, 112-122. electrolysis, 98-103. Solutions (and solutes) (continued) freezing-point, 80-83. ionization, 96, 123-126. molal properties, 57-90. osmotic pressure, 84-88. Raoult's law deviations, 70, 71. reactions, equilibrium of. See Equilibrium, rate of, 131-135, 139, 144- solid, 197-200. thermochemistry of, 217-221. transference, 104-111. vapor-pressure of concentrated, 67-72, 79.^ vapor-pressure of dilute, 57-66. Specific heat. See Heat-capacity. Standard cell, Clark, 262. Steam distillation, 48, 79. Storage cells, 245-246, 261. Substances, chemical, classes of, 3, 4, 6. largely ionized, activity, 171-172. concentration determined from conductance, 121. conductance, 115-116, 119. heat of reactions, 219-220. ionization, 96-97, 119, 123- 126, 153, 171-173. solubility, 167-169. slightly ionized, conductance at zero concentra- tion, 115. ionization, 118, 123. (See Acids and bases.) TEMPERATURE-change produced by expansion of gases, 45. flames and explosions, 221. mixing of liquids, 70. Temperature-composition diagrams. See Phase diagrams. Tempering, 197. Thermochemical applications, 221-222. equations, 211-214. principles, 207-216. results, 217-221. Thermodynamics, first-law, 35-37, 207-208. second-law, 76, 223-229, 277-279. Titration of acids and bases, electrometric, 267-268. polybasic acids, 164. principles, 162-164. Transference (electrical), concentration, effect of, 109-110. INDEX 309 Transference (electrical) (continued) conductance, relation to, 113-114. determination of composition of ions, iio-ui. experimental determination, 104- 106, 108. general discussion, 104. ion-mobility, relation to, 106-108. mechanism, 106-107. moving-boundary method, 108-109. numbers, 105-106. Transition catalysis of, 178-179. free energy of, 234-236. heat of, 218, 281. salt-hydrates, 189. temperature, determination, 178. temperature, effect of pressure, 177, 281. unstable phases, 177-178. Trouton's rule, 217. UNITS, definitions, ampere, 100. bar, 13. calorie, 37. coulomb, 100. dyne, 13. erg, 37- faraday, 100. joule, 37. megabar, 13. mho, 112. ohm, 112. mol, 12. reciprocal ohm, 112. Univariant systems, temperature-pres- sure relations, 280. Unstable phases, diagrams, 176-178. discussion, 177-179. free-energy, 235-236. solubility, 177, 179, 195-196- transition, 178-179. vapor-pressure, 177. VAN DER WAALS equation, 21, 27. Van't Hoff equation, 285-289. Vapor-density, determination, 16. Vaporization, heat of, boiling-point relations, 49, 60, 61, 217. kinetic interpretation, 52-54. Trouton's rule, 217. vapor-pressure relation, 48-50, 282. Vapor-pressure composition diagrams, 68, 72, 79, 189. determination, 47, 67. law), Vapor-pressure (continued) general discussion, 47. molecular interpretation, 51-54, 58, 70, 71. of various systems, partially miscible liquids, 79. perfect solutes, 63-66. salt-hydrates, 188-189, 196. solutions, concentrated, 67-72. solutions, dilute, 57-59, 63-66. solvents, 57-59. unstable phases, 177. relation to activity, 170. boiling-point, 48, 59-61. composition (Raoult's 57-59, 67-72, 79- freezing-point, 80-82. heat of vaporization, 49, 54. osmotic pressure, 85, 87. pressure, 85-86. temperature, 48-50. Variance, applications, 180-181^ '203. concept, 1 80. Viscosity, conductance, effedt on, 118-119. general discussiop, 118. ion-mobility, effect on, 118-119. Voltaic cells, changes in state, 243-247, 252. concentration-cells, 250-251. conduction in, 99-102. electrode-potentials, 254-261. electrode products, 98-99. electromotive force, 249-268. formulation, 102-103, 244-247. free energy decrease, 247-253. involving chemical changes, 252-253. concentrated solutions, 261- 262. physical changes, 240-251. liquid-potentials, 262-266. mechanism of action, 243-244. polarization, 273-276. standard cell (Clark), 262. storage cells, 245-246, 261. work produced, 247-248. Volume-change attending compression of liquids, 231. displacement of acids, 160. mixing of liquids, 70. neutralization, 160. WATER as catalyst, 143. gas process, 150-151. ionization, 122, 154, 267, 289. 3 io INDEX Work-content, effect of temperature, 284-285. general discussion, 226-228. Work, produced by changes in general, 223-229. gas reactions, 209, 211. heat-transfer between tempera- tures, 277-279. Work, produced by (continued) reversible changes, 226-228. voltaic cells, 247-253. volume-changes in general, 37, 38. volume-changes of perfect gases, 38. HOME USE CIRCULATION DEPARTMENT MAIN LIBRARY This book is due on the last date stamped below. 1 -month loans may be renewed by calling 642-3405. 6-month loans may be recharged by bringing books to Circulation Desk. Renewals and recharges may be made 4 days prior to due date. ALL BOOKS ARE SUBJECT TO RECALL 7 DAYS AFTER DATE CHECKED OUT. MAR 25 1975 9 6 RETURNED SEP 3 LOAN AHC FEB 26 1981 REC.CIR. SEP 03*80 LD21 A-40m-5,'74 (R8191L) General Library University of California Berkeley YC 91717