THE INFLUENCE OF MOLECULAR CONSTITUTION UPON THE INTERNAL FRICTION OF GASES. BY FREDERICK MALL1NG PEDERSEN, E.E., Sc. D. Instructor in Mathematics in the College of the City of New York. SUBMITTED IX PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF SCIENCE IN THE FACULTY OF SCIENCE, NEW YORK UNIVERSITY, APRIL, 1905. Hew D. VAN XOSTRAND COMPANY, 23 MURRAY AND 27 WARREN STREETS. 1906. THE INFLUENCE OF MOLECULAR CONSTITUTION UPON THE INTERNAL FRICTION OF GASES. BV FREDERICK MALL1NG PEDERSEN, E.E., Sc. D. Instructor in Mathematics in the College of the City of Xeic York. SUBMITTED IX PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF SCIENCE IN THE FACULTY OF SCIENCE, NEW YORK UNIVERSITY, APRIL, 1905. D. VAN NOSTRAND COMPANY, 23 MURRAY AND 27 WARREN STREETS. 1906. Ill TABLE OF CONTENTS. PAGE INTRODUCTION 13 HISTORICAL REVIEW 3-34 Baily 7 Barus 29 Bernoulli 3 Bessel 6 Bestelmeyer 33 Boltzmann 28 Braun & Kurz - 24 Breitenbach 32 Chezy 6 Clausius 9 Couette 28 Coulomb 5 Couplet 4 Crookes 23 De Keen 28 Du Buat 4 Euler 4 Eytehvein 5 Gerstner 5 Girard 6 Girault 10 Graham 8 Green 7 Grossman 24 Guthrie 17 Hagen 7 Helmholtz & Pietrowski 10 Hoffmann 25 Holman 17 Houdaille 31 Jaeger 31 Job 32 Klemencic 24 Kirchhoff 17 Koch 24 Koenig 27 Kundt & Warburg 15 239372 PAGE Kleint 33 Lampe 28 Lampel 28 Lang 13 Ludwig & Stefan 9 Marian 5 Margules. 23 Markowski 33 Maxwell 9 Mathieu 10 Meyer, L 12, 19, 22 Meyer, L. & Schumann *. 20 Meyer, O. E 11, 13, 14, 28 Meyer & Sptingmuhl 14 Naumann 12 Navier 6 Noyes & Goodwin 30 Newton 3 Obermayer 16 Ortloff 30 Perot & Fabry 31 Poiseuille 8 Poisson 7 Prony 5 Puluj 15, 17, 18 Rayleigh 32 Reynolds, F. G 33 Reynolds, O 25 Sabine. . .' 6 Schneebeli 27 Schultze 32 Schumann 26 Stefan 18 Steudel 21 Stewart & Tait 11 Stokes 9 Sutherland 29 Tomlinson 27 Warburg 16 Warburg & Babo 24 Wiedemann, E ^ 17 METHOD EMPLOYED IN EXPERIMENTAL INVESTIGATION 34 GENERAL DESCRIPTION OF APPARATUS 34 APPARATUS No. 1 38 Internal Friction of Air, Apparatus No. 1 39 APPARATUS No. 2 40 Internal Friction of Air, Apparatus No. 2 and No. 3 42 PREPARATION OF ETHERS , 42 Internal Friction of Ethers, Apparatus No. 2 43 PAGE APPARATUS No. 3 45 Internal Friction of Ethers, Apparatus No. 3 46 MOLECULAR VOLUMES 48 MOLECULAR MEAN SPEEDS, FREE PATHS, AND COLLISION FREQUEN- CIES 50, 51 COMPARISON WITH THE RESULTS OF OTHERS 51 SUMMARY OF RESULTS 52 BIBLIOGRAPHY . . 54-59 THE INFLUENCE OF MOLECULAR CONSTITUTION UPON THE INTERNAL FRICTION OF GASES. BY FREDERICK M. PEDERSEN. INTRODUCTION. The effect of molecular structure upon the physical prop- erties of matter has interested many scientists, but much about this question yet remains to be learned. In the hope of making a contribution to the subject the following investi- gation was undertaken. A careful historical research reveals the fact that the in- ternal friction of most of the isomeric ether gases that I have employed has never before been determined. Nevertheless, my object has been, not so much to determine with great accuracy the absolute values of these frictions, as to obtain by an extremely sensitive method accurate comparative re- sults, in the hope of proving that difference in molecular con- stitution is accompanied by measurable difference in friction, and therefore by a difference in the size of the molecule. By the internal friction or viscosity of a gas, is meant the friction between two adjacent layers of the gas, which are moving in parallel directions but not with the same velocity. It is proportional to the relative velocity and the area of con- tact of the two layers. The coefficient of internal friction is the constant factor, depending upon the kind of gas under consideration, which, when multiplied by the difference in velocity, gives the friction per square unit of surface of contact of the two layers. Although there is no sensible cohesion between the par- ticles of a gas, as there is in liquids and solids, the tendency of a moving layer of gas to impart its own velocity to a layer in contact with it, has been satisfactorily explained according to the kinetic theory of gases. As is well known, all the mole- cules of a gas are supposed to be constantly in rectilinear motion, rebounding in straight lines with undiminished ve- locity, after collision with one another or a solid wall. In virtue of its mass and velocity each molecule possesses mo- mentum. When one layer of gas is moving over another, there is a constant passage of molecules from each layer into the other. There is then an interchange of momentum, the result being a tendency to equalize the velocities of the two layers. We see thus that the effect of friction is produced between two layers of gas which are not moving with the same velocity in the same direction. It has been demonstrated 1 that the coefficient of friction of a gas, m G 47TS 2 where m = the mass of a molecule, G = the mean value of the velocity as deduced from the mean kinetic energy, and 5 = the distance between the centres of two molecules at impact, or the diameter of the sphere of action. This formula brings out the remarkable and very important law, that the viscosity of a gas is independent of the pressure at a constant temperature. It also shows that the smaller the sphere of action, the greater will be the coefficient of viscosity. We also see that the viscosity varies directly as the rectilinear velocity of the molecules, i.e., as the square root of the kinetic energy, and therefore as the square root of the absolute tem- perature, since the last two are directly proportional to each other. The first named law has been found true experimentally for all ordinary pressures, but the last law connecting the viscosity and absolute temperature has not been substantiated by experiment. This disagreement between theory and prac- tice I will leave for a fuller discussion later, merely .noting in passing that Maxwell 2 has shown that the assumption in regard to the nature of the impact between two molecules determines the relation between the coefficient of friction and the absolute temperature. That the coefficient of friction of a gas is independent of the pressure seems at first inconceivable, but it can probably 1. O. E. Meyer's Kinetic Theory of Gases, 1899, p. 179. 2. Phil. Mag., 1868 (4), Vol. 35, p. 211. be better understood by a consideration of another formula which has been deduced from the kinetic theory, viz.: T) = 1/3 d G L l where d is the density, G is square root of the mean square of velocity, and L is the mean free path of a molecule. It can be shown that L and d are inversely proportional to each other so that their product is constant, an,d therefore 77 is thus independent of a change in density. Before describing my own experimental investigations I will give a historical review of the work done by previous experimenters in the subject of the viscosity of gases. Methods of research in this line seem to have developed out of or along with investigations in the subject of the internal friction of liquids. I shall therefore frequently have occasion to refer to this last named subject, but I shall make no attempt what- ever to follow out its complete development. HISTORICAL REVIEW. O. E. Meyer 2 points out that Newton 3 made the first at- tempt at a theory of the friction of fluids. His fundamental hypothesis was that the friction between two adjacent layers of a fluid is proportional to the difference of velocity, the area of contact and independent of the pressure. This hypothesis has several times since been enunciated by others, independently of Newton, and used as a basis for a theory of fluid friction. Unfortunately Newton's mathematical development of the theory from this assumption is erroneous. His experiments, like those of many early investigators, seem to have been confined to observations of pendulums. It is interesting to note that on searching for the resistance which air opposes to the movement of a globe, swinging in very small arcs, he used a formula composed of three terms: one containing the square, the second the 3/2 power and the third the first power of the velocity. 4 Later, 1719, in calculating the resistance which is offered to spheres falling slowly in air or water, he reduced the formula to two terms: one varying as the square of the velocity, and the other constant. Bernoulli 5 pointed out Newton's error in his paper before 1 Meyer's Kinetic Theory of Gases, 1899, p. 1 78. 2. Crelle's Journal fur Mathematik, 1861, Bd. 59, p. 22'J 3. Philosophiae naturalis principiamathematica, 1687, Lib. II., Sec. IX. 4. IVincipia Lib. II., Prop. XL. 5. Opera Ornnia. Lausanne et Genevae 1742. Tornus III. Xouvelles pense>s snr le systeme de M. Descartes, XIX. -XXIII. the French Academy in 1730, explaining the orbits of the planets and the precession of the equinoxes, according to the Cartesian vortex hypothesis. On submitting to calculation the experi- ments with pendulums made by Newton, Bernoulli supposes only two terms to represent the resistance; one varying as the square of the velocity, the other constant. He points out that the theory does not agree with the experiments, but states that we cannot conclude anything from that because of the difficulty and delicacy of the observations necessary. 1 Couplet, 2 in 1732, called attention to the fact that the rules then in use for determining the flow of water through pipes were utterly useless, as the error amounted to nearly 100%. He made some experiments on a large scale at Versailles, and found that the water delivered was frequently 1/20 or 1/30 what was promised by his calculations. Nothing was known about internal friction, the effect of bends, diameter of pipe, etc. Couplet apparently made no attempt to discover the true laws of "flow and seemed to scarcely believe in the possi- bility of discovering them. Euler 3 in 1756 fell into the error of supposing that the fric- tion of a liquid is independent of the velocity and propor- tional to the hydrostatic pressure. Du Buat 4 30 years later made a series of quite elaborate experiments with pendulums consisting of spheres of lead, wood and paper, which he oscillated in water and air. He noted that similar laws seemed to hold in both media. He noticed that both liquid and air were dragged along by the spheres. He found the resistance to the motion of the spheres was proportional to their surfaces, and that the resistances of air and water are as their densities. He calculated the resistance of the air to Newton's falling spheres, and found that up to a velocity of 23 feet per second the resistance is as the square of the velocity. At higher velocities he showed it was greater owing to a constant vacuum back of the ball. He pointed out a double correction which must be applied to a pendulum in reducing the results to a vacuum. 1 for its apparent loss in weight owing to the buoyancy of the air. 1. Me"moires de Petersbourg. Tome III. and V. 2. M&noires de 1'Academie, 1732. 3. Tentomen theoriae de frictione fluidorum. Novi Petropolitani, Tomus VI., 1756; 7 Pag. 338. 4. Principes d'Hydrauliques, 1786, Vol. 2, Part 3; Sec. 2, p. 279. and 2 for the increase in the moment of inertia due to the air clinging to it. A full discussion and analysis of his work with pendulums was given by Professor Stokes 1 in 1850. Some years previous to Du Buat's work Mairan 2 made quite an important advance in the experimental study of pendulums by the invention of the method of coincident observations of two pendulums, a method made use of by many later in- vestigators. Gerstner, 3 through experiments made in Prague in 1796, discovered the very great effect which temperature has upon the mobility of water. He found that in many cases a rise of 20 or 30 degrees in temperature doubled the quantity of liquid delivered through narrow tubes, thus showing a very marked decrease in the internal friction of the water. Coulomb 4 in 1801 published an account of experiments undertaken to determine the cohesion of fluids and the laws of their resistance when in very slow motion. He claims that the expression for the resistance of a fluid has two terms, one proportional to the square of the velocity, and the other to its first power, and that if there is a constant term, it is so small in all fluids of small cohesion, that it is almost im- possible to appreciate it. Instead of using the pendulum method, he noted the diminution in amplitude of oscillation of horizontal disks, oscillating in their own plane under the torsion of a brass wire. He was the first one to adopt this important method which has since been employed by many others. Both Prony 5 in 1804, and Eytelwein 8 ten years later, developed theories of the flow of water through pipes, but with the temperature left entirely out of consideration. Prony found the loss of head per unit length is very nearly proportional to the square of the mean velocity of the water. He proposed the formula: 1/4DJ =av+ftv i where D = diameter of the pipe, / = the fall per metre, v = velocity, a =0.0000173314, ft = 0.0000348259. 1. Cambridge Phil. Trans., 1856, Vol. 9, Part 2, p. 8. 2. Historic de 1'Acad. de Paris, 1735, p. 166. 3. Gilbert's Annalen, Bd. 5, 1800, p. 160. Neu.Abh.derkon.B6hm. Gesell. der Wiss., Bd. 3, Prag. 1798. 4. Mem. de 1'Institut Nat. des Sciences et Arts. Annee 9, Tome III., p. 246. 5. Recherches Physico-mathem. sur la The*orie des Eaux Courantes, Paris, 1804. 6. Abh. d. Berl. Akad., 1814 and 1815. Girard 1 in 1815 investigated the flow of liquids through capillary tubes of copper with special reference to the effect of temperature. He found that at 86 C. four times as much water was delivered as at C. He noted that when the cap- illary reached a certain length the term which is proportional to the square of the velocity disappears from the formula for the uniform motion of liquids. In fine tubes he found the loss of head per unit length very nearly proportional to the ve- locity of flow. He refers to the work of Chezy 2 in 1775 as being the first investigation of the flow of water in aqueducts. In 1823 Navier 3 deduced the differential equations of the motion of a viscous medium from considerations which are analogous to those he employed for the derivation of the differ- ential equations of the phenomena of elasticity. He treats of the movement of a fluid in a straight pipe of circular cross section. In a later paper he discusses the discharge of air from pipes and orifices, and endeavors to take into account the effect of bends. The velocity of flow seems to have been his chief concern. The smallest pipe he used had a diameter of 1.579 cm. In 1826 Bessel 4 published the result of his classical investi- gation of the length of the simple second's pendulum. His method consisted in the comparison of two pendulums whose difference in length was exactly one toise or 1.92 metres. He took account of the effect of the resistance of the air and showed that the old correction for reduction to a vacuum was in error and determined its correction factor, viz. : k = 1.946 for a sphere about 5 cm. in diameter. Sabine 5 made the interesting experiment in 1829 of swinging a pendulum in air under a pressure of about half an atmosphere, and in hydrogen at atmospheric pressure, and found that hydrogen had a much greater resistance in proportion to its density than had air. He correctly ascribed the cause to an inherent property of elastic fluids, independent of their density, analogous to that of viscosity of liquids. He also investigated the correction for reduction to a vacuum. His pendulum 1. Me"m. de 1'Institut, Classe Sc. and Math., 1813-15, p. 248; 1816, p.187. 2. Mdm. manuscript de 1'Ecole des Pons et Chausse"es, 1775. 3. Me"m. de 1'Acad. Roy. des Sciences, 1823, Tome 6, p. 389, 1830, Tome 9, p. 311. 4. Abb. Herl. Akad. Math. Klasse, 1826, p. 1. r> Phil. Trans., 1829, p. 207 and 331; 1831, p. 470. made 10.36 more oscillations in a given time in a vacuum than in air, whereas calculations showed that it ought to make only 6.26 more, hence the correction for reduction to a vacuum was in error. In 1829 JL Poisson 1 derived, by a method similar to Navier's, the differential equations of equilibrium and motion of fluids. Two years later he wrote a paper on the motion of a pendulum in a resisting medium, in which he discusses the correction for reduction to a vacuum and infinitely small arcs, and reviews the work of Du Buat, Bessel and Sabine. In 1832 Baily 2 made a very thorough and systematic study of the correction of a pendulum for the reduction to a vacuum. He oscillated 86 different pendulums with bobs of various mate- rials, sizes and shapes in air and then in a vacuum. In the case of a spherical bob two inches in diameter he found the old correction should be multiplied by 1.748 which agrees quite well with Bessel's value of k but not so well with Du Buat's, 1.585. He found the correction varies with the shape but not with the specific gravity of the pendulum. Green, 3 continuing researches on the vibration of pendulums in fluid media, claimed that in the case of a sphere the density of the pendulum should, in the calculations, be augmented by one half the density of the surrounding fluid, while its weight is diminished by the weight of the volume of the fluid it displaces. Hagen, 4 returning in 1839 to the investigation of the flow of water in small cylindrical tubes, tried to improve on Prony's and Eytelwein's equations by taking account of the tempera- ture. He criticized Du Buat for not recording the temperature in any of his experiments in spite of the fact that he knew warm water is more fluid than cold water. Hagen found the rapidity of flow increases with the temperature to a maximum, then diminishes for a rise of 10 or 20 degrees, and then begins to increase again. With large tubes and high speeds both turning points fell below the freezing point. With very narrow tubes and low speeds they both fell above the boiling point. 1. Journal de 1'Ecole Polytech., 1831, Tome 13, p. 139. Connaissance des Terns, 1834, Appendix. Mem. de 1'Acad., 1832, Tome 2, p. 521. 2. Phil. Trans., 1832, Part 1, p. 399. 3. Trans. Royal Soc. Edin., Vol. 13, 1836, p. 54. 4. Pogg. Ann., 1839, Bd. 46, p. 423. Abh. d. Berl. Akad., 1854, p. 17. In the following year Poiseuille 1 published the results of his epoch-making experimental research on the movement of liquids in tubes of very small diameter. He used .glass capil- laries from 0.013 mm. to 0.652 mm. in diameter, and from 2 to 800 mm. long. They were slightly elliptical in cross sec- tion, so he took the mean diameter. The liquid, water or alcohol, was contained in a bladder and pressed out through the cap- illary. He investigated the effect on the quantity of liquid discharged, of the pressure, length of tube and the tempera- ture. He found the volume of liquid discharged varies directly .as the pressure, inversely as the length of tube, directly as the fourth power of the diameter, and with the temperature according to an empirical law. He found the value of a certain factor to be constant for all sizes of tubes. This factor for water was 1836.7 at C. and 2495.9 at 10 C. What Poiseuille had done for liquids, Graham 2 did for gases a few years later. He first investigated the effusion of gases into a vacuum through a very fine aperture in a thin plate, and found the velocity of effusion varies inversely as the square root of the density of the gas. He determined the effusion coefficients of a number of gases referred to air and oxygen as unity. On the other hand, on causing gases to transpire through capillary tubes which were long in com- parison with their diameter, he discovered the transpiration coefficients to be independent of the density of the gases. He either let the gas flow from a gas holder through the capillary into a partial or complete vacuum, or from a com- pression cylinder into the air. In both cases the pressures were not constant during the experiment. He found the velocities of transpiration to vary directly as the pressure, and inversely as the length of tube. A rise in temperature he found decreased the velocity. The transpiration rate was independent of the material of the capillaries, which were of glass or copper. The presence of moisture in the air had but a small effect upon its transpiration. The velocities with which the different gases passed through 1. Soc. Philomath, 1838, p. 77, Comptes Rendus, 1840, Vol. II., pp. 961, 1041. 1841, Vol. 12, p. 112; 1842, Vol. 15, p. 1167. Ann. de Chim. et de Phys. 1843 (3), Vol. 7, p. 50. Me"m. de Savans etrangers, 1846, Vol. 9, p. 433. 2. Phil. Trans., 1846, Vol 136, p. 573. Phil. Trans., 1849, Vol. 139, p. 349. 9 the various capillaries, bore a constant relation to each other. The compounds of methyl had a less velocity than the corre- sponding compounds of ethyl, but a constant relation appeared between them. In 1849 Stokes 1 published an elaborate methematical treat- ment of the theories of the internal friction of fluids in motion, and derived the same complete equations of motion as Navier and Poisson by a different method. The following year he read a paper " On the Effect of the Internal Friction of Fluids on the Motion of Pendulums " in which he gave a very full discussion of the work of Du Buat, Bessel and Baily already referred to. He was well aware of the fact that the air sticks fast to a pendulum, which therefore in its oscillations causes a friction of air on air. He was in error, however, in thinking that what we now know as the coefficient of friction was de- pendent upon the density. From his own experiments and those of others, he deduced values of what he calls the " index of friction," which is our coefficient divided by the density. He found his results agreed well with Baily's but not with Bessel's. Baily's experiments show 7? for air to be = .000104 in c.g.s. units. Stokes also deduced the index of friction of water from Coulomb's observations on the decrement of the arc of rotary oscillation of horizontal disks. Ludwig and Stefan 2 in 1858 called attention to the formation of eddies when a tube discharges liquid into one of larger bore, thus causing an increase in resistance. A similar action was later shown to take place in a gas under the same circumstances. The same year Clausius 3 published his important paper on the kinetic theory of gases which did so much to establish that theory. In 1860 Maxwell 4 entered this field of investigation and made some important contributions to the kinetic theory of gases, proving mathematically that T? must be independent of the density. Six years later appeared his Bakerian Lecture in which he gave an account of his experiments on the 1. Trans. Camb. Phil. Society, 1849 (3), Vol. 8, p. 287. Trans. Camb. Phil. Society, 1856, Vol. 9, p. 8. Phil. Mag., 1851 (4), Vol. 1, p. 337. 2. Sitzber. Wien. Akad., 1858, Vol. 32, p. 25. 3. Pogg. Ann., 1858, Bd. 105, p. 239. 4. Phil. Mag., 1860 (4), Vol. 19, p. 31; 1869 (4), Vol. 35, p. 209, 211. Phil. Trans., 1866, Vol. 150 (1), p. 249. Collected Papers, Vol. II. Proc Royal Society, 1866. Vol. 15, p. 14. 10 viscosity of air and other gases. His method was a modifica- tion of Coulomb's, in which he used several oscillating disks with stationary plates between them. The only uncertainty in his mathematical treatment of the theory of his apparatus is due to the action which takes place at the edges of the oscil- lating plates which is not fully understood. His value for q for air at 18 C. is .0002. He found from his experiments that the coefficient of fric- tion was directly proportional to the absolute temperature instead of to the square root of the latter. He then modified his kinetic theory of gases in order to make the theory agree with his observations, by adopting the hypothesis of a repulsive force between the molecules, which varies inversely as the fifth power of the distance between them, this being the only law of force which causes T? to increase as the first power of the absolute temperature. His observations concerning the increase of TJ were undoubtedly in error as other investigators do not agree with him. Helmholtz and Pietrowski 1 in 1860 tried an interesting modification of Coulomb's method which consisted in observing the logarithmic decrement of the rotary oscillations of a spher- ical vessel containing the liquid under investigation. Helm- holtz worked out the mathematical theory of this method while Pietrowski performed the experiments. With the interior surface of the vessel highly polished and gilded a slipping of the fluid along the side of the vessel was thought to be observed. The results are to Poiseuille's as 4:5. Probable sources of error mentioned are vibrations and changes of temperature. Girault 2 in his pendulum experiments of that year brought out the fact very clearly that most of the resistance to a pen- dulum's motion resides at the bob and not at the point of suspension. In 1863 Mathieu 3 deduced Poiseuille's law for capillaries of elliptical cross section. Poiseuille found Pressure X Diameter 4 length where Q is the quantity of liquid discharged in a unit time 1. Wien. Ber. Mathtn. Naturw., 1860, Vol. 40, p. 607 2. Mdm. de 1'Acad. de Caen, 1860. 3. Compt. Rend., 1863, Tome 57, p. 320. 11 and K is a constant depending upon the nature of the liquid. Mathieu supposes the velocity of the liquid in contact with the capillary wall to be zero, because K is independent of the material of the capillary, and because of observations on blood in the capillaries of a frog's foot. For round capillaries he gives . Pressure R 4 the above equation the form Q =- . -where N is a 8 i\ length constant depending upon the liquid. For elliptical capillaries Pa 3 b* Q = i \T i t 2 . L-> where a and b are the semi-major and minor 4 .V / (a 2 -f o 2 ) axes of the ellipse. In 1865 Stewart and Tait 1 made some interesting observa- tions on the heating of a disk rotating rapidly in vacuo. They found the heating independent of the rarefaction which seemed to show the friction independent of the density or pressure. O. E. Meyer, 2 who probably did more than any other one man to advance our knowledge of the viscosity of gases, began investigations in this subject about this time. A few years before he had begun the study of the viscosity of liquids, but he now turned his attention to gases. He first adopted the oscillation method of Coulomb and found a value of T? for air at 18 C. of 0.00036 which is much too large. Later he pursued his investigations with pendulums. His object was to determine what effect pressure and temperature have on the coefficient of friction of air. He found that the rarefied air in a supposed vacuum has a noteworthy effect on a pen- dulum. He therefore questioned the reliability of Baily's and Sabine's results. Bessel's work he considered much better. From it he calculated rj for air = .000275. His own observations gave y = .00027 to .00052. By varying the pressure and temperature he concluded that with diminishing pressure rj decreases, but less rapidly than the density, and with rise of temperature T) increases an insignificant amount, whereas Maxwell claimed it rises 2% for each 10 C. which is not so far from the truth. The following year in order to check his own results, O. E. Meyer 3 calculated coefficients of friction from Graham's ob- 1. Proc. Royal Society, 1865, Vol. 14, p. 339. Phil. Mag. (4), Vol. 30, p. 314. 2. Crelle's Journal fur Math., 1861, Bd. 59, p. 229. Pogg. Ann., 1861, Bd. 113, p. 55. 1865, Bd. 125, pp. 177, 401, 564. 3. Pogg. Ann., 1866, Vol. 127, pp. 253. 353. 12 servations, although he believed that for absolute measure- ments Coulomb's method was superior to Graham's, because in the former external friction is eliminated, it being settled that the gas adheres to the oscillating disks. For the trans- piration method he derived the formula : in which / is the time; p l is the pressure of the gas entering the capillary; p 2 is the pressure of the gas leaving the cap- illary; V\ is the volume of the transpired gas measured at the pressure p v \ R is the radius of the capillary; L is its length; is the coefficient of slip of the gas on the walls of the cap- illary. The term 4 ^ is so small compared with 1 that most investigators until very recently have neglected it. He found T? for air at about 15 C. = 0.000178 to 0.000206 from Graham's experiments. He was fully convinced by this time that r? is almost, if not absolutely, independent of the pressure, but he was not certain of the law in accordance with which T? increases with the temperature. In 1867 Lothar Meyer 1 published his calculations of the molecular volumes as derived from his brother's determina- tions of f. He derived the formula: in which v = molecular volume; C = a then not obtainable constant which is the same for all gases at the same tempera- ture; m = molecular weight of the gas. He showed that the ratio of the molecular volumes of two substances is about the same as the ratio of their molecular volumes as determined by Kopp 2 from their boiling points. Naumann, 3 working on the same lines that year, derived the formula ^= ^-O which r and r 1 are in the radii of spheres T \ T) \ m i of action, and m and m i are the molecular weights of two sub- 1. Ann. d. Chem. u. Phar., 1867, Suppl. Bd., 5, p. 129. 2. Ann. d. Chem. u. Phar., 1855, Bd. 96, pp. 1, 153, 303. 3. Ann. d. Chem. u. Phar., 1867, Suppl. Bd. 5, p. 252. 13 stances. He showed the methyl ether molecule to be 9 times the volume of a hydrogen molecule. In 1871 O. E. Meyer 1 returned to the observations of pen- dulums and made some careful experiments with pendulums varying in length from 14.5528 to 4.6868 metres, which were set up in the stairway of the University of Breslau. He found i) for air at 18 C. to be 0.000216 to 0.000233. Recognizing that Maxwell's modification of Coulomtys method was superior to his own, which gave only the \ r/ instead of T?, and was less sensitive than Maxwell's owing to the absence of stationary plates between the oscillating ones, Meyer' 2 proceeded now to repeat Maxwell's experiments. In order to avoid irregularities in the internal friction of a single wire he adopted bifilar suspension. He found T? for air at 18 C. = 0.000190 to 0.000197 which he considered good agreement with Maxwell's value 0.0002. In accordance with these more correct values of the coefficient of viscosity of air he revised his list of y for other gases and gave to methyl ether the value of 0.000107 at 15 C. During that same year Lang 3 published the results of some experiments on air and other gases, using the transpiration method. His apparatus was extremely simple, the gas being sucked through a capillary by a falling column of water. His capillaries w r ere of glass, one of round and another of elliptical cross section. The round capillary gave ry for air at 15 0.000168 to 0.000178; the elliptical one gave at 9 0.000143 to 0.000148. The discrepancy between the results of the two capillaries Meyer suggested might be partly owing to the fact that perhaps the cross section of the elliptical one was not a true ellipse. Having already shown theoretically that the volume of gas delivered by a capillary in a unit of time is proportional to the fourth power of its radius, inversely proportional to its length, and directly proportional to the difference in pressure between its two ends, provided the gas is measured at a pressure which is the arithmetical mean betw r een the two terminal pressures, O. E. Meyer 4 proceeded in 1873 to prove experimen- 1. Pogg. Ann., 1871, Vol. 142, p. 481. 2. Pogg. Ann., 1871, Vol. 143, p. 14. 3. Wien. Ber. Mathem. Naturw, 1871, Vol. 63, 2 Abt., p. 604. Wien. Ber. Mathem. Naturw, 1872, Vol. 64, 2 Abt., p. 487. Pogg. Ann., 1872, Vol. 145, p. 290. Pogg. Ann., 1873, Vol. 148, p. 550. 4 Pogg. Ann., 1873, Vol. 148, pp. 1, 203. 14 tally this law, whose similarity to Poiseuille's law for liquids is at once evident. He employed several capillaries varying in cross section from .0008 to .0016 sq. cm. Working under the assumption that the air adheres so firmly to the walls of the capillary that there is no slipping, he obtained the fol- lowing values of y for air: 0.000168 at C., 0.000184 at 14.4 C., 0.000197 at 21.1C. The above experiments were made by causing the air to trans- pire from one copper vessel through the capillary into another similar vessel, allowing the pressure but not the volume of the air to change. He next repeated the experiments by allowing the volume but not the pressure to change by employing con- stant water suction. He found T? at 10 C. to be .000192- .000199 and at 100 C. r? = .000211 - .000218. His results do not agree with Maxwell's in showing T? to vary as the absolute temperature, neither do the} 7 agree with theory in proving T) to vary as the square root of the absolute temperature. His results are midway between Maxwell's and the theory, viz., r) varies as the 3/4 power of the absolute temperature. He proposed the formula: 7? - >? (1+ 0.0024 T) in which T? O = 0.000171 and T = temperature in C. Maxwell's formula was i? = T? O (1+0.003665 T). Using a still smaller capillary whose cross section was .00015 sq. cm. through which he sucked the air by means of a mercury pump he found >? at 22 C. to be .000187 and at 100 C. .000223 .000227. He then modified his formula to read y = 0.000174 (1+0.003 T). Repeating Maxwell's oscillation experiments once more with bifilar suspension Meyer found y = .000190 (1+0.0025 T) which is in fairly good agreement with his transpiration results, so that Meyer concludes Maxwell's results are in error owing to the heating of the suspension wire and the poor position of the thermometer with reference to the oscillating disks. Meyer accounts for the difference between his own results and theory by the fact that heat increases not only the rectilinear velocity of the molecules but also the internal movements of the atoms which build up the molecules. A little later Meyer and Springmuhl 1 using the same apparatus obtained the coefficients of friction of 19 different gases, some 1. Pogg. Ann., 1873, Vol. 148, p. 526. 15 of which had been investigated by Graham, with whom they agreed within 2 to 20%. Their values for air and methyl ether were .000190 and .000102 at 15 C. No difference in y was observed when transpiration took place into a vessel filled with the same kind of gas or with a different gas, showing that there was no diffusion back through the capillary. This proves that a gas has the same pressure upon another kind of gas as upon the same kind as itself. The following year Puluj 1 investigated the coefficient of in- ternal friction of air as a function of the temperature, using a transpiration apparatus like Professor Lang's. He main- tains that the loss of heat due to the expansion of the air in the capillary is compensated for by the heat which the air receives from the walls of the capillary. He found y for air at 15 C. = 0.00018526. For a temperature range from 13.4 C. to 27.2 he found y to vary according to the formula y = r? (1+0.003665 r)- G52776 0-020893. After enlarging the temperature range from 1.5 C. to 91.2 he found the exponent of the above parenthesis changed to 0.590609 0.009510. Kundt and Warburg 2 published in 1875 an important in- vestigation which proves that the law that y is independent of the pressure does not hold for extremely small pressures. They used Coulomb's method, employing only one oscillating glass disk with bifilar suspension, For air at 15 C. they found y = 0.000189 and for pure aqueous vapor at 15 C. and a tension of about 16 mm. y = .000099. They found that with air the logarithmic decrement began to diminish at a pressure of 2| mm. of mercury, and fell off very perceptibly at a pressure of ^ mm. This diminution was ascribed to a slipping of the air on the surface of the oscil- lating disk. From their work they concluded that the coefficient of slip for a gas on a solid partition has sensibly a determined value dependent upon the nature of the gas, so long as the latter is present in layers thicker than 14 times the mean length of molecular path, and it is inversely proportional to the pres- sure. The absolute value, which is the coefficient of internal 1. Wien. Ber. Mathm. Naturw., 1874, Vol. 69 (2), p. 287. Wien. Ber. Mathm. Naturw., 1874, Vol. 70 (2), p. 243. 2. Monatsber. d. Berl. Akad., 1875, p. 160. Pogg. Ann., 1875, Vol. 150, pp. 337, 525. Phil. Mag., 1875 (4), Vol. 50, p. 53. 16 friction divided by the coefficient of external friction of the 760 gas, they found from experiment to te for air 0.0001- whereas 760 theory showed it ought to be 0.000058 where p = pressure expressed in mm. of mercury. Warburg 1 , pursuing the same subject further the following year, proceeded to find the coefficient of slip for air by the transpiration method. He pointed out that because the radius of a capillary is smaller than we can make the distance between oscillating plates, the effect of slipping in transpiration experi- ments is noticeable at higher pressures than in the method of oscillations. Using a pressure of 38 mm. of mercury and a capillary of 0.15 mm. radius he found r? for air was 5% smaller than it ought to be, owing to slipping. The temperature exponent from to 100 C. he found to be 0.77 which is in good agreement with O. E. Meyer. His temperature exponent for hydrogen is 0.63. He gives the formula: - = 1 H 5- where = coefficient of slip and R Observed y R = radius of the capillary. Obermayer's 2 results agree with those of O. E. Meyer in showing that f) for air increases as the 3/4 power of the absolute tem- perature. He used the transpiration method and pointed out two possible sources of error inherent in this method: 1st. the clinging of a layer of air to the inside wall of the capillary which diminishes the cross section more at low than at high temperatures; 2nd. the change in temperature which the gas undergoes in the capillary as it expands from the entering to the terminal pressure. He carried the temperature as high as 270 C. and criticized Puluj for his small temperature range. His absolute values of T) are lower than those of Meyer. He points out that the so-called permanent gases are distinguished by a temperature 1. Pogg. Ann., 1876, Vol. 159, p. 379. 2. Wien. Ber. Mathem. Naturw., 1875, Vol. 71 (2), p. 281. Wien. Her. Mathem. Naturw., 1876, Vol. 73 (2), p. 433. Carls Rep., 1876 (2), Vol. 12, pp. 13, 456. Carls Rep., 1877, Vol. 13, p. 130. Phil. Mag., 1886, Vol. 21. 17 exponent of 3/4, whereas the coercible gases have this exponent almost unity. E. Wiedemann 1 , assuming as did several predecessors that the variation of T) with the temperature can be expressed by the formula TJ = T? O (1-f a T) n , proceeded to determine the exponent n, which I call the temperature exponent, for several gases. He adopted the transpiration method, driving the gas through the capillary by the pressure produced by mercury flowing from a reservoir. His capillary was surrounded by cold water, steam or aniline vapor, so that his temperature range was from to 184.5 C. He assumed that the absolute value of 7) for air had already been determined with sufficient accuracy by Meyer, Maxwell, Kundt, Warburg and others, so that he gave only comparative results. He found the temperature exponent, n, to vary with different gases but in different ways. With most gases he tried, it decreased with higher tempera- tures. Thus for air from to 100 C. it was .7333 and from 100 to 184.5 it was only .6701. For CO it was .6949 from to 184.5 C. Puluj 2 with a modified Coulomb apparatus consisting of an oscillating disk of thin mirror glass between two fixed disks of thick plate glass, found the temperature exponent for air = 0.72196 0.01825 which agrees pretty well with Meyer and Obermayer but not with his own previous results. Kirchhoff 3 in 1877 gave a complete mathematical deduction of the formulae to be used in calculating the coefficient of fric- tion of a gas from the logarithmic decrement of a sphere oscil- lating about a diameter and for an ellipsoid of revolution ro- tating about an axis of symmetry. Holman, 4 in a series of very careful experiments by the transpiration method, found that y for air increases as the 0.77 power of the absolute temperature. Later experiments at higher temperatures showed that the increase of T? falls off in rate as the temperature rises, except possibly in the case of hydrogen, which he therefore regards as oar most perfect gas. Guthrie 5 in order to determine whether there were any 1. Archiv. d. Sc. Phys. et Xat. de Geneve, 1870, Vol. 56, p. 277 Fortschr. d. Phys., 1876, Vol. 32, p. 200. 2.^Wien. Ber. Mathem. Naturw., 1876, Vol. 73 (2). p. 589. :.{. \\Iechanik, 1877, 4 Aufl., 26 Vorl., p. 383. 4. Proc. Am. Acad., Boston, 1877, Vol. 12, p. 41; 1886, Vol. 21. p. 1. Phil. Mag., 1877 (5), Vol. 3, p. 81; 1886, Vol. 21, p. 199. 5. Phil. Mag., 1878 (5), Vol. 5, p. 433. disturbances, such as eddies, in the flow of a gas at the begin- ning and end of a capillary, took the transpiration time of a given volume of air through a capillary, that he afterward cut up into a number of small pieces which he joined together by guttapercha tubing. With the capillary in this condition he found the same transpiration time as before, which seemed to prove the absence of disturbances at the ends of the cap- illary. A later investigator, Hoffmann, does not agree with him. Puluj 1 in 1878 attacked the problem of the internal friction of vapors, using the modified Coulomb-Maxwell apparatus already mentioned, with unifilar suspension. From experi- ments with ether he concluded that its TJ is independent of the pressure. He found T? at 10 = 0.0000716 and at 37.1 C." = 0.0000792. Its variation with the temperature he gives by the equation 7? T = 0.0000689 (1 + 0.0041575 T) ' 94 The coefficient of expansion of ether = .0041575 he calculated from Herwig's 2 data. From this equation he concluded that for ether vapor and probably all vapors ^ is proportional to the absolute temperature. He also experimented on the vapors of alcohol, chloroform, benzol and aceton. Taking the mole- cular volume of hydrogen as 1, he found those for ether and alcohol 101.1 and 52.9 respectively, which is in fair agreement with Kopp's 3 molecular volumes deduced from the boiling points of the liquids. Puluj agreed with Stefan 4 in thinking that each molecule of a gas is surrounded by an envelope of ether, and in explain- ing the increase of internal friction with rise of temperature, by saying that with increased velocity the molecules on impact penetrate further into one another. His experiments with air at very low pressures led to the result, that while the pressure diminishes from 754 to 0.03 mm., the coefficient of friction only decreases to about one half its value, which proves what a great number of molecules remain in quite high vacuum. Puluj 5 next took up the problem of determining the co- efficient of friction of a mixture of the two gases CO 2 and H, using 1. Wien. Ber., 1878, Vol. 78 (2), p. 279; Carls Rep., 1878, Vol. 14, p. 573; Phil. Mag., 1878 (5), Vol. 6, p. 157. 2. Pogg. Ann., 1869, Bd. 137, p. 595. 3. Ann. Chem. Phar., 1855, Bd. 96, pp. 1, 153, 303. 4. Wien. Ber., 1862 (2), Vol. 46, pp. 8, 495; 1872 (2), Vol. 65, p. 360. 5. Carls Rep., 1879, Vol. 15, p. 578 and 633. 19 the same apparatus as for vapors. He drew the conclusions: (1) The coefficient of friction of a mixture of CO 2 and H is not larger (smaller) than the coefficient of that constituent which has the larger (smaller) coefficient; (2) gases with larger mole- cular weights have in a mixture of equal proportions a greater influence on the value of the coefficient of friction of the mixture. He proved that Maxwell's formula for the y of a mixture gives too small a value, and proposed another formula which gives ry too large, but agrees with experiment in showing the curious fact that with a small percentage of the lighter gas >? increases. In 1879 Lothar Meyer 1 began the study of the internal fric- tion of vapors by the transpiration method. The capillary, which was nearly 1 metres in length, was coiled into a helix and fastened in the upper part of a boiling flask. The lower end of the capillary passed through the neck of the flask into a condenser. The substance to be examined was boiled at a regulated pressure, the vapor evolved surrounding the cap- illary and raising it to the same temperature. A part of the vapor, which was of course saturated, entered the upper end of the capillary and passed through it into an air-free cooled space, where it was condensed and measured as a liquid. The volume of the vapor transpired was calculated from the amount of liquid collected in the condenser. This method is open to criticism for several reasons, one of which is the coiling of the capillary into a small helix, thus possibly deforming its bore and causing additional resistance by the curved path of the gas. The vapor at the boiling point of the liquid can also hardly be regarded as a true gas, and should have been tested at several degrees above boiling. The expansion of the saturated vapor as it passed through the capillary probably also changed its state. Furthermore the great length of time, several hours, needed for transpiration made it difficult to keep conditions constant. A preliminary test with air gave y at room temperature = 0.000188 which is higher than most other observers have found by the transpiration method. He found y for benzole about 50% higher than Puluj had found, that its increase with rise of temperature is more rapid than for so-called permanent gases. He discovered that with a difference in pressure of 1. Wien. Ann., 1879, Bd., 7 p. 497. 20 14 cm. of mercury between the two ends of the capillary his apparatus gave y considerably too small. On calculating molecular volumes he found that they are larger, the lower the temperature. Two years later were published the results of observations made by L. Meyer and Schumann 1 on a very large number of substances, using the transpiration apparatus just described. Two capillaries were employed, one 1427 mm. long and 0.31 mm. in diameter; the other 1404 mm. long and 0.3328 mm. in diam- eter. The second capillary gave values 3% higher than the first. Below is the table of their results for Esters. TABLE V. ESTERS C n H 2n O 2 7) X 10 6 . Acid Radical Alcohol Radical Methyl Ethyl Propyl Isobutyl Amyl n = 2 3 4 5 6 Formic Acid 173 156 159 172 160 n = 3 4 5 6 7 Acetic Acid 152 152 160 155 n = 4 5 6 7 8 Propionic Acid 150 158 153 164 158 n = 5 6 7 8 9 Normal Butyric Acid 159 160 164 167 155 Iso-Butyric Acid 152 151 153 158 155 w = 6 7 8 9 10 Valerianic Acid 163 165 167 154 They conclude that all esters at their boiling points and at the same pressure transpire very nearly equal volumes of vapor, which however because the boiling points are different, do not contain the same number of molecules. They found rj for the corresponding acids considerably smaller than for their esters. They believed that the differences of y are partly too small and partly too irregular to draw any sure conclusion of the dependence of r? on the molecular constitu- tion. They point out, however, some differences which seem quite regular. The esters of acetic, propionic and isobutyric acids show almost always a smaller r? than those of formic, normal butyric 1. Wied. Ann., 1881, Bd. 13, p. 1. 21 and valerianic acids. This difference is especially noticeable in the two butyric acids. The influence of the alcohol radical cannot be seen so clearly. Among the isomeric esters, those of formic, acetic, propionic and isobutyric acids have the greater TI with the greater alcohol radical. The isomeric esters of normal butyric and valerianic acids do not follow this rule but have the same friction. The number of carbon atoms in a molecule also seems to have a certain influence on i). The esters for which n = 2, 5, 7 or 8 show a larger T? than those with 3, 4 or 9 carbon atoms. The cause of these differences they were unable to explain. On calculating the relative molecular volumes they found them only about half those given by Kopp's rule. L. Meyer tried to account for this by advancing the hypothesis that by Kopp's method the empty space is included which is open to the atoms for their motion, while from the coefficient of internal friction only the volume of the gas particles themselves is determined. Steudel 1 continued the work of L. Meyer and Schumann, using the same apparatus with the second capillary, the first having been broken. He investigated several homologous lines of organic compounds, viz., alcohols up to four atoms of carbon per molecule, and their halogen derivatives, also some substitution products of ethane and methane. He found the transpiration time increased with the mole- cular weight. Of isomeric compounds at the boiling points, the normal, i.e., those that boil at the highest temperature trans- pire the slowest and the tertiary the fastest, with the exception of isopropyl alcohol which transpires noticeably slower than the normal. Unsymmetrical low boiling compounds have a smaller transpiration time than the symmetrical. The only exception he found to the rule that the transpiration time in- creases with the molecular weight is methyliodide whose time was 1037 minutes, which is almost the same as 1056 minutes taken by isobutyliodide. He points out that the coefficients of friction of each line of homologous compounds are nearly alike or only slightly different. The values for the primary alcohols vary from .000135 to .000143; for three of them they are almost exactly alike. The isopropyl gave a considerably larger figure, also 1. Wied. Ann. 1882, Bd. 16, p. 369. 22 TABLE OF rj X 10 6 . Radical Alcohol Chloride Bromide Iodide Methyl 135 116* 245 Ethyl 142 105* 183 216 Normal Propyl 142 146 184 210 Isopropyl 162 148 176 201 Normal Butyl 143 149 202 Isobutyl 144 150 179 204 Tertiary Butyl 160 150 * Calculated from Graham's results. the tertiary butyl. The chlorides seem to all have the same friction. The greatest differences exist in the iodide column. He found a considerable difference in TJ with a change in pressure. With a greater pressure than 10 to 15 cm. of mercury the formula seemed to lose its validity for vapors. This agrees with what L. Meyer had noticed with benzole. Steudel also gives tables of molecular velocities, mean free paths, combined cross section of molecules and cross-sections of an equal number of molecules. He points out that the cross sections of molecules of isomeric compounds are not the same. For the butyl compounds the areas for the normal substances are the largest, the tertiary the smallest, with the iso com- pounds between. Propyl and isopropyl alcohol, as also both chlorides show the same relation. On the other hand, isopropyl- bromide has a greater molecular area than the normal com- pound. In the substitution products of ethane the sym- metrical ones have a larger area than the unsymmetrical. Steudel also calculated the relative molecular volumes from SO 2 according to the method of L. Meyer, with whom he agrees in finding the volumes about one half as large as those cal- culated according to Kopp's rule. The ratios of molecular volumes of the different substances, however, are almost the same as the ratios of the volumes according to Kopp. L. Meyer, 1 summing up Steudel's and his own previous results, gives the following: TABLE OF )?X10 6 Alcohols C n H 2n+2 O 142 average Chlorides C n H 2n+1 Cl 150 Esters C n H 2n O 2 155 Bromides C^H^+jBr 182 Iodides C n H 2n+1 1 210 "j| 1. Pog. Ann., 1882, Vol. 16, p. 394. 23 Cases in which n = 1 and some few unaccountable varia- tions are omitted from the above. When n = 1 the variations are great but he could see no law of the influence of molecular constitution on the friction. On the other hand the influence of the nature of the atom is very clearly seen; friction of iodine > bromine > chlorine. The molecular volumes do not agree with those L. Meyer 1 calculated from Graham's results. He points out that this one fact seems certain, viz. : the molecules of a tertiary butyl compound are smaller than those of a secondary which are smaller than those of a primary. This is in agreement with the universal view taken of the concatenation of these com- pounds. Those of the tertiary are grouped around a single atom of carbon, hence are more spherical in shape. Propyl and isopropyl compounds on the other hand do not show the same regularity. The alcohols and iodides deviate in opposite directions, which is unexplained. He further points out that the sphere of action of a liquid molecule increases with the temperature, while that of a gaseous molecule diminishes with rise of temperature. In 1881 Crookes 2 published an interesting account of ex- periments on the viscosity of gases at high exhaustions. His method consisted in observing the logarithmic decrement of a plate of mica enclosed in a glass bulb, the axis of suspension being in the plane of the mica. He found that with air, nitro- gen, oxygen and carbon monoxide y diminishes slightly as the pressure falls from 760 to 3 mm., after which it decreases very rapidly. Hydrogen showed no change in r) from 760 to 3 mm. pressure. When the pressure was \ mm. the mean free path of a molecule became comparable with the dimensions of the glass bulb, and the ultra gaseous state of matter, as Crookes names it, was assumed. Margules 3 suggested a new experimental method of finding the internal friction of a fluid or gas. He proposed two coaxial cylinders, one rotating at a constant speed, the other having its turning moment measured directly. He also pointed out that in Coulomb's method the molecules near the rotating plate do not move in circles but in zigzags, the centrifugal 1. Liebig's Ann., 1867, Suppl. Bd. 5, p. 129. 2. Phil. Trans., 1881, Vol. 172, p. 387. 3. Wien. Ber. Mathem. Naturw., 1881, Bd. 83 (2), p. 588. 24 force carrying them outward from the axis. This action Maxwell had apparently neglected. Instead of an oscillating system of plates Braun and Kurz 1 employed a sphere, the mathematical theory of which had been worked out by Kirchhoff. They found y for air at room temperature = 0.000184 .000010. Warburg and Babo 2 investigated the relation between vis- cosity and density of fluid, especially gaseous substances. They employed the capillary method, and experimented on liquid CO 2 and gaseous CO 2 above the critical temperature, and at pressures between 30 and 120 atmospheres. They found that the viscosity of liquid CO 2 increases with the den- sity, as does also that of gaseous CO 2 at very high pressures. The fact that investigators have shown that the viscosity of a gas is not independent of the pressure, when the latter is either very low or very high, does not destroy the validity of the law for ordinary pressures. Grossman, 3 noticing that the transpiration and oscillation methods are not in perfect agreement in their values of >?, undertook to discover the cause of the difference. He chose Coulomb's method for his work as being the simplest. He proposes corrections within certain limits and deduces a for- mula which he thinks applies accurately within those limits. Klemencic 4 in 1881 published a complete mathematical treatment of the damping of the oscillations of a solid body in liquids and gases. He discusses the cases of spheres and cylinders oscillating and swinging in various manners. In order to help settle the question of why the coefficient of friction of a gas increases with the temperature Koch 5 under- took the study of the coefficient of friction of mercury vapor and its dependence on the temperature. Stefan's hypothesis of an ether envelope of the molecule has already been men- tioned. O. E. Meyer's explanation that at higher temperatures the bonds between the atoms are loosened so that the mole- cules on collision penetrate further into each other, thus dimin- 1. Carl Rep., 1882, Bd. 18, pp. 569, 665, 697; 1883, Bd. 19, pp. 343, 605. 2. Wied. Ann., 1882, Bd. 17, p. 390. .",. Wied. Ann., 1882, Bd. 16, p. 619. 4. Wien. Ber. Mathem. Naturw., 1881, Bd. 84 (2), p. 146. Carl's Rep., 1881, Bd. 17, p. 144. Ann. d. Phys. Beiblatter, 1882, Bd. 6, p. 66. 5. Wied. Ann., 1883, Bd. 19, p. 857. 25 ishing the distance between their centres and thus their diam- eters, does not hold in the case of a monatomic gas such as mer- cury. Koch employed two capillaries, one having a radius = 0.004245 cm. and length = 9.875 cm.; and the other a radius = 0.00593 cm. and length = 19.22 cm. He found TJ = 0.000494 at C. and y = 0.000643 at 98 C. With a third capillary whose radius = .008238 cm. and length = 18 cm. he found y 4% larger than the above, hence he con- cludes Poiseuille's law does not hold for a capillary of this size and neglects this result. Assuming the usual equation 7? = JJ Q (\ + at) n he found the exponent n to be 1.6 for mercury from which it appears that the diameter of a mercury molecule is about proportional to j^. where T = absolute temperature. At C. he found the molecular volume of mercury to be 12.9 times, and at 300 only 4.39 times that of hydrogen. In 1884 O. Reynolds 1 read an interesting and suggestive paper on the two manners of motion of water, the steady or direct, and the sinuous or eddying. He experimented on water flowing through glass tubes and made the character of motion visible by means of color bands in the water. In order that motions in tubes of different sizes can be compared, the velocities must be inversely as the tube diameters. The critical velocity at which sinuous motion begins increases with the viscosity of the fluid. If water be flowing in a bent channel in steady streams, the question as to whether it will remain steady or not turns on the variation in the velocity from the inside to the outside of the stream. He enumerates viscosity, converging solid boundaries, curvature with the velocity greatest on the outside as conducive to direct or steady motion, whereas diverging solid boundaries and curvature with the velocity greatest on the inside, tend to sinuous or unsteady flow. It is possible that similar actions take place in the flow of a gas through a capillary, and constitute an objection to a coiled capillary. Hoffmann 2 worked with gases along somewhat the same lines that Reynolds did with liquids. Poiseuille's law takes 1. Proc. Roy. Inst. Grt. Brit., 1884, Vol. 11, p. 44. 2. Wied. Ann., 1884, Bd. 21, p. 470. 26 for granted that the particles of gas move through a capillary in parallel lines. Hoffmann tried to prove that when this law does not apply it is because the molecules have a sinuous motion through the capillary. He thought that the disturbing cause resides chiefly at the beginning and end of the cap- illary, although Guthrie's results seemed to contradict this. Hoffmann repeated Guthrie's experiments and found a longer transpiration time for the many small pieces than when the capillary was all in one piece. He points out that perhaps Guthrie worked with too low pressures and put the pieces of capillary too close together. In order to study the whirling motion he drove tobacco smoke through tubes (of course not capillary) and found a conical contraction of flow at the end, and further on a conical spreading out in which could be seen powerful eddies. The eddies approached nearer to the end as the velocity was in- creased, until finally the contraction vanished and the eddies were right at the end of the tube. These observations agree with those of Sondhaus. 1 As a result of all his work Hoffmann concludes that when Poiseuille's law does not hold for a tube, the chief reason is the phenomena at the end and especially at the beginning of the capillary. As it was impossible to determine by the method employed in 1881 by L. Meyer and Schumann, the dependence of the friction of vapors on the temperature, Schumann 2 proceeded to make this investigation by the oscillation method. His apparatus was similar to Maxwell's, but he found that Maxwell's formula did not give concordant results, so he adopted an empirical formula which gave values in good agreement, but generally smaller than those of other observers excepting Obermayer. His y for air at 20 C. = .000178 and at C. .000168. His results are in fair accord with those of the trans- piration method at ordinary temperatures. At high tempera- tures he claims that capillaries give values which are too small due to the adsorption of the gas by the capillary walls. To express the increase of y with rise of temperature he pro- poses a new formula: 1. Pogg. Ann., 1852, Bd. 85, pp. 58. 2. Ann. d. Phys., 1884, Bd. 23, p. 353. 27 in which a = coefficient of expansion of the gas, and y = co- efficient of diminution of the radius of the sphere of action of the molecule. For air a = 0.003665, y = 0.000802. For CO, a =0.003701, y =0.000889. For all vapors a is .004, but for benzole y = 0.00185 while for other vapors it is 0.00164. He found rj entirely independent of the pressure except when the vapor was saturated. On calculating the molecular volumes he found them to be from 44 to 72% of the values calculated according to Kopp's rule. Schneebeli 1 in 1885 attempted to get with the greatest accu- racy the absolute value of the coefficient of friction of air and its dependence on the temperature. He used five different capillaries and obtained the following values of y x 10 7 at C. : 1712, 1690, 1698, 1703, 1734, corrected for the vapor tension of water. For its variation with the temperature he proposed the formula y t = T? O (1 +0.0027 /) His values agree quite well with those which Obermayer obtained ten years before. Koenig 2 studied the influence of magnetization and electrifi- cation on the coefficient of friction and could detect none. He made an important contribution to the subject of internal friction by deducing a correction for Coulomb's method which brought its results down into agreement with those of the capillary method. Tomlinson 3 made a very careful study of the coefficient of viscosity of air and its change with the temperature. He observed the logarithmic decrement of the torsional vibrations of cylinders and spheres. He found T? at C. = .00017155 and thought Holman's formula correct, j] t = i? (1 +0.002751 t - 0.00000034 t 2 ). This result is about 9% lower than Maxwell's, which Stokes explains on the supposition that Maxwell's disks were not exactly level. Tomlinson also studied the effect of aqueous vapor in the air. He says that air at 15 C. and 760 mm. pressure when saturated with aqueous vapor is only .2% more viscous than dry air. It is only when under a pressure less than 350 mm. that the aqueous vapor begins to show an ap- preciable effect; but when the rarefaction is great, moist air becomes considerably less viscous than dry air. Crookes had 1. Archiv. de Geneve, 1885, Vol. 14 (3), p. 197. 2. Wied. Ann., 1885, Bd. 25, p. 618; 1887, Bd. 32, p. 193. 3. Phil. Trans., 1886, Vol. 177 (2), p. 767. 28 likewise said that at 15 C. and pressures 760 to 350 mm. the presence of aqueous vapor has little or no effect on the internal friction of air. Lampel, 1 after reviewing the mathematical treatment of the torsional oscillations of a sphere with air resistance given by Lampe, 2 Boltzmann 4 , Kirchhoff and Klemencic, proceeded to determine which of these men came nearest the truth. His experiments showed that Boltzmann's formula is the best. In 1887 O. E. Meyer 3 revised his mathematical treatment of Coulomb's method, accepting the correction which Koenig had been fortunate enough to devise for the disturbance which takes place at the edges of the cylindrical surface of the disks. He showed how this correction improved his previous values for j). Because the French physicist, Hirn, rejected the kinetic theory of gases for the reason that he could find no increase in the coefficient of friction of air with increase of temperature De Heen 5 undertook an investigation of this subject. His method consisted in letting a brass piston descend under gravity in a tube and drive air through a capillary stopcock at the bottom. His pressures varied from 10 to 2280 mm. of mercury. He found y at low pressures smaller and at high pressures larger than at ordinary pressures. He maintains that below 80 mm. pressure r? varies as the square root of the absolute temperature in accordance with the kinetic theory of gases. After passing 80 mm. pressure ^ increases more rapidly with the temperature than at lower pressures, the* variability of y attaining a maximum at 300 C. He thinks that the disagree- ment between theory and experiment may be due to the fact that the mean free path of a molecule is possibly not a straight line in gases which are under the relatively high pressure of .the atmosphere. Couette 6 , criticizing Coulomb's method as giving only an ap- 1. Wien. Ber., 1886, Bd. 93 (2), p. 291. 2. Programm des Stadt. Gym. zu Danzig, 1866. 3. Ann .d. Phys., 1887, Bd. 32, p. 642. Sitz. d. Munch. Akad., 1887, Bd. 17, p. 343. 4. Wien. Ber., 1881, Bd. 84 (2), p. 40. 5. Bull, de 1'Acad. de Belgique, 1888 (3), Vol. 16, p. 195. 6. Comp. Rend., 1888, Vol. 107, p. 388. Jour, de Phys., 1890 (2), Vol. 9, p. 414. Ann. de Chim. et de Phys., 1890 (6), Vol. 21, p. 433. 29 proximation, even with very slow oscillations, and Poiseuille's method, as being true only for very slow rates of flow, adopted the method suggested by Margules in 1881. He used two copper cylinders, the inner suspended by a torsion thread and the other one coaxial with it, revolving with a constant velocity. The outer cylinder tends to set the inner one in motion, which is kept in its primitive position by turning the torsion head through a measured angle. He found the angle of torsion divided by the revolutions per minute to be almost constant, increasing only very slightly with the speed. His value of T? for air at 18 C. - .0001847. Barus 1 made a careful investigation of the viscosity of gases at high temperatures. His capillary was of platinum, radius at C. = 0.0079, coiled into a helix. The range of tem- perature was from 5 to 1400 C. He found that the mean increase of gaseous viscosity was proportional to the \ power of the absolute temperature and not in accordance with Schumann's or Holman's formula. He suggested that if the law connecting T? and the temperature were rigorously known his apparatus could be used as a pyro- meter. He pointed out that below 100 air is not rigorously a perfect gas, because below that temperature its temperature exponent increases to 0.73. Above 100 C. it is 0.67 the same as for hydrogen whose exponent' does not change from to 1400 C. Obermayer showed that as the state of vapor is ap- proached the temperature exponent increases, being nearly 1 for many vapors. Barus's absolute values of y are not good, that for air at C. being 0.0002472 to 0.0002508, which is much too high. Sutherland 2 showed that the discrepancy between theory and experimental results concerning the. increase of y with the temperature, disappears if, in the theory account is taken of the forces of attraction between molecules when the latter approach each other. These attractive or cohesive forces tend to increase the number of collisions between the molecules, and thus have the effect of apparently increasing the diameter 1. Amer. Journ. Sci., 1888 (3), Vol. 35, p. 407. Bull, of U. S. Geol. Survey, No. 54, Washington, 1889, p. 239- Wied. Ann., 1889, Bd. 36^ p. 358. Phil. Mag., 1890, Vol. 29, p. 337. 2. Phil. Mag., 1893 (5), Vol. 36, p. 507. 30 of the sphere of action. With increased molecular velocity, due to rise of temperature, the cohesive forces have less chance to act than at lower molecular velocity, i.e., at lower tem- perature. Therefore the sphere of action will be increased less at high than at low temperatures, which produces the same result as if it diminished at higher temperatures. Sutherland proposed the following formula which agrees well with the ob- servations of Holman and Barus: / T U ' 273 f) = T) J I in which T is the absolute temperature, and C is the cohesion constant depending upon the nature of the gas, which for air = 113. Ortloff 1 made a careful study of the friction of the three gases C 2 H 6 , C 2 H 4 and C 2 H 2 by the transpiration method. He found that the difference in cross section of the molecules C 2 H 6 and C 2 H 4 is less 'than that between C 2 H 4 and C 2 H 2 , the relative values being 342, 298 and 228. O. E. Meyer 2 claims that the cross section of a molecule = sum of the cross sections of its atoms. Ortloff found that his molecular cross section was less, his molecular diameter less, and his molecular volume greater than those calculated according to Meyer, with the exception of the C 2 H 2 volume. He concludes that his gas atoms cannot be disposed in a plane. Meyer's supposition that the molecular volume = sum of the atomic volumes presupposes that the atoms are grouped in a sphere. Ortloff 's experiments seemed to show that of his three gases this is true only in the case of C 2 H 2 . Noyes and Goodwin 3 investigated the viscosity of the vapor of mercury because the latter is a monatomic element. They used the transpiration method and experimented also on hydrogen and carbon dioxide. They found that the cross section of a mercury molecule is 2.48 times as large as that 1. Inaug. Diss. Jena, 1895. 2. Kinetic Theory of Gases, 1899, Chapter X. 3. Physical Review, 1896, Vol. 4, p. 207. Zeitsch. Physik. Chem., 1896, Vol. 21, pp. 671-679. 31 of hydrogen at 300 C. They concluded that atoms and mole- cules are of the same order of magnitude, and that the spaces between the atoms within the molecule, if any exist, are not large in comparison with those occupied by the atoms themselves, and that therefore the coefficient of friction is not adapted for distinguishing between monatomic and polyatomic mole- cules. They believed that this explains the fact that the molecular cross section of most comparatively simple molecules is approximately equal to the sum of the atomic cross section as had been pointed out by O. E. Meyer. Their work is open to criticism because a joint in their apparatus could not be made air tight. Houdaille 1 measured the coefficient of friction of air and vapor of water by the transpiration method. At 76 cm. pres- sure he found i? = 0.000186 for air, and for water vapor = 0.0000975. At a pressure of between 1 and 3 cm. he found f) for air the same, while r) for water vapor changed to 0.0000885. He found fair agreement between the calculated and observed values of the diffusion coefficient for water vapor. Perot and Fabry 2 brought out a new kind of absolute electro- meter intended for the measurement of small differences of potential. They observed that it attained its position of equilibrium very slowly when the distance between the plates was small, owing to the viscosity of the layer of air which separates them. Thus- they obtained a new method for deter- mining the viscosity of air, which they found to be 0.000173 at 13 C. Jaeger 3 gave a careful mathematical analysis of the influence of molecular volume on the internal friction of gases taking account of association and expansion of molecules. Instead of the usual formula r t = -- d G L he proposed r t = d G L in which d = density, G = velocity of mean square, L = mean free path of a molecule. He also gave the formula 9 = i) 9 (1+4/3 A) 2 . n whkh A ^ l + D ^ + and p = b_ A 2t V of the molecular and specific volumes. 1. Fortschr. d. Phys., 1896, 52, Jahr. I., p. 442. 2. Compt. Rend., 1897, Vol. 124, p. 28. Ann. de Chim. et de Phys., 1898 (7), Vol. 13, p. 275. 3. Wien. Ber. Mathem. Naturw., 1899 (2), Vol. 108, p. 447. 1900 (2), Vol. 109, p. 74. 32 Breitenbach, 1 using the transpiration method, experimentally determined the coefficients of friction of air, ethylene, carbon dioxide, methyl chloride, and hydrogen. He found that T? varies according to a power of the absolute temperature whose exponent for different gases varies between 0.6 and 1.0. He inferred that the sphere of action of a molecule diminishes with increase of temperature. For the same gas this exponent decreases with increasing temperature. Also a lowering with lower temperatures was noticed. In gaseous mixtures TJ does not vary as the composition, and Puluj's formula is only ap- proximately correct. The difference between the results of the oscillation and transpiration methods at high temperatures cannot be explained by an increase of the slipping of the gas along the capillary walls. This slipping he found could in general be neglected. Two years later Breitenbach 2 compared his work with Suther- land's formula for the variation of r? with the temperature, and found this formula gave very excellent agreement with his experiments. The value of the cohesion constant for air he found to be 119.4 instead of 113 according to Sutherland. Rayleigh 3 on the supposition that jj t = T? O (1 +a t) n found the exponent n for dry air = 0.754, for oxygen = 0.782, for hydrogen 0.681, for argon (impure) 0.801, for pure argon = 0.815. Later, using Sutherland's formula, he found the cohesion constant to be the same for hydrogen and helium, viz.: 72.2, and this value and those for air, oxygen and argon agree well with the values calculated by Sutherland from Obermayer's observations. Schultze 4 , like Rayleigh, used the transpiration method for investigating the internal friction of argon and its change with the temperature. He found its friction at C. to be 2104 X 10- 7 and that it varied with the temperature in accordance with Sutherland's formula, the cohesion constant being 169.9. His argon contained J%. of nitrogen. Job 5 called attention to a new method of measuring the re- sistance offered by a capillary tube to the flow of gases. A voltameter is provided with a capillary outlet; the pressure 1. Wied. Ann., 1899, Bd. 67, p. 803. 2. Drude's Ann., 1901, Bd. 5, p. 166. 3. Proc. Roy. Soc., 1900, Vol. 66, p. 68; Vol. 67, p. 137. 4. Drude's Ann., 1901, Bd. 5, p. 140. 5. Bull. Soc. Franc. Phys., 1901, Vol. 157, p. 2. 33 produced in the voltameter for a given current, measures the resistance of the capillary. The author suggests the applica- cation of this method to various experiments in connection with the flow of gases. F. G. Reynolds, 1 using spheres and cylinders in torsional oscillation, determined the viscosity coefficient of air, and investigated the effect upon it of Rontgen rays. His value of y at 21 C. = 187 X 10- 6 . The results of his experiments with Rontgen rays seem to show that their effect is scarcely, if at all perceptible. Bestelmeyer, 2 using transpiration apparatus very similar to Holman's, investigated the change in the coefficient of internal friction of nitrogen for a temperature range of 192 C. to + 300 C. and found that Sutherland's formula applied with satisfactory accuracy except at 192 where it was 2% in error, perhaps owing to change in the law at this low tempera- ture. The cohesion constant was found to be 110.6. Markowski, 3 employing the same transpiration apparatus used by Schultze in 1901, studied the internal frictions of oxygen, hydrogen, chemical and atmospheric nitrogen, and their change with the temperature. He corrected all his results for slip, using coefficients of slip determined by Breitenbach Kundt and Warburg, for air, oxygen and hydrogen. For nitrogen he used the molecular free path, which is theoretically nearly equal to the coefficient of slip. 4 His corrected value of TI for' air at 15.9 C. - 1814 xlO- 7 , and at 99.62= 2212 XlO- 7 . The correction for slip amounted to J%. He states that Graham's results are preferable to those of Obermayer. Sutherland's formula he found to give excellent results from to 183 C. He also tried the older exponential formula and found the ex- ponent diminishes with rise of temperature as had been noticed by several other observers. Puluj's formula for the friction of a gaseous mixture he found applies well to atmospheric nitrogen as a mixture of chemically pure nitrogen and argon. Kleint, 5 continuing the work of Schultze and Markowski 1. Phys. Review, 1904, Vol. 18, p. 419; Vol. 19, p. 37. 2. Inaug. Diss. Munich, 1902. Ann. d. Phys., 1904, Bd. 13, p. 944. 3. Inaug. Diss. Halle, 1903. Ann. d. Phys., 1904, Vol. 14, p. 742. 4. O. E. Meyer's Kinetic Theory of Gases, 1899, p. 211. o. Inaug. Diss. Halle, 1904. 34 with the same piece of apparatus investigated the friction of mixtures of oxygen, hydrogen and nitrogen. He found Puluj's formula to give only approximately the coefficient of friction of a mixture of gases. He observed that small percentages of oxygen and nitrogen raise the friction of hydrogen markedly, while hydrogen only begins to make itself felt upon the others when it is 5% of the mixture. Sutherland's formula for the increase of y with the temperature he found to be correct. METHOD EMPL.OYED IN EXPERIMENTAL INVESTIGATION. The preceding historical review shows that the methods employed in the past for determining the coefficient of the internal friction of gases, may be divided into two general classes: those in which the movement of a solid body in the gas is observed, and those in which the time of passage of the gas through a capillary tube is noted. There are many modifi- cations of both general methods, especially of the first named one which is the earlier historically. Owing to the difficulty of getting capillaries of perfectly uniform bore, and of deter- mining with the greatest accuracy the shape and size of their cross section, and also owing to the possible formation of eddies at the beginning and end of the flow, and to the possible slipping of the gas on the capillary walls, probably the oscillation method is the better for absolute measurements, provided a solid body is employed of such a shape that the mathematical treatment is rigorously correct. It seems certain however that the transpiration method has been growing in favor of late years, and that it is more convenient for comparative measurements than the other method. As already stated my object was more to get comparative than absolute values of the greatest accu- racy, hence the transpiration method was decided upon for this experimental investigation. GENERAL DESCRIPTION OF APPARATUS. The form of apparatus and the subject of this research were kindly suggested to me by Professor Morris Loeb, under whose guidance these experiments have been made. The apparatus consists essentially of a U shaped tube (see Fig. 1), one limb of which is capillary, while the other is not, but serves as a cylinder of known capacity, down which is forced by gravity a piston consisting of a column of mercury, which drives the gas under it up through the capillary limb. The capillary, A t T M, B FIG. 1. View perpendicular to the plane of the two limbs v FIG. 2. View in the plane of the limbs. f o. FIG. 3. Steam Jacket. 36 does not begin at the bend of the U, but some distance above it. The distance between the limbs is only 3 cm., so that the apparatus can be placed inside of a glass tube 5 cm. in diameter which serves as a steam jacket (see Fig. 3), the steam being introduced through a side tube, S, near the top and passing out freely at the bottom, 0. A cork, C, through which both limbs of the apparatus pass is on a level with the top of the capillary, and serves to close the steam jacket at the top. The mercury limb above the cork is provided with a special stop-cock, H, bored out to the same diameter as the tube below it. Above the stop-cock is a continuation of the tube ter- minating in a small funnel, F, for convenience in introducing ether and mercury. An ordinary thistle funnel with horizontal bottom will not answer; the bottom of the funnel must be inclined about 45 to the axis of the mercury tube, otherwise the mercury will not so readily descend in one unbroken column when released by turning the stop-cock. Two marks, M 1 and M 2 are etched on the mercury tube a convenient distance apart, the lower one, M 2 , being near the bend at the bottom, the volume of the tube between the marks being accurately deter- mined by calibration with mercury. The bore of this tube being slightly conical instead of truly cylindrical, its calibration was also carried far enough above the upper mark, M\, to cover the space passed over by the mercury column during an ex- periment, so that the average height of the mercury column during an experiment could be calculated. The method of making an observation on ether gas is essen- tially as follows: The steam jacket being cold and filled with air the apparatus is placed in it. The stop-cock is opened and about 1 cc. of ether is poured down the mercury tube. By means of an aspirator the ether is sucked up the other limb close to the base of the capillary. The small hermetically closed tube seen at V in Figs. 1 and 2 serves as a reservoir for ether at the base of the capillary. Care was always taken to fill this tube completely with ether so that it could not act as an air pocket. Steam is then led into the jacket and of course vaporizes the ether, driving surplus liquid violently out through the top of the mercury tube. The stop-cock is then closed and after equilibrium has been estab- lished, a weighed amount of mercury is introduced into the funnel and allowed to flow down until it is arrested by the stop-cock, which is not quite air tight, because no grease can 37 be used in it for fear of soiling the mercury. The stop-cock is then suddenly turned and the mercury descends in a solid column. When its lower meniscus passes the upper mark on the tube a stop watch is started, which is stopped as the same meniscus passes the lower mark. The barometer is read when the mercury has covered half the measured distance. A ther- mometer, T, Fig. 1, hanging from the cork at the top of the steam jacket indicates the temperature of the steam', and serves as a plumb line. The readings of this thermometer were corrected by comparison with a standard thermometer. The coefficient of friction of the gas is calculated by the following formula given by O. E. Meyer: 1 xgdr 1GL V in which d = density of mercury at C g = acceleration due to gravitation r = radius of capillary in cm. L = length of capillary in cm. P = pressure of gas on entering the capillary. p = height of barometer at C t = time in seconds V = c.c. of gas transpired. No allowance need be made for the expansion of the mercury at 100 C., for what the column gains in height is compensated for by loss in density. No allowance has been made for the expansion of the capillary as it was assumed the expansion of the mercury tube, hence increase of the volume of gas trans- pired, would counteract this. The advantages of this apparatus over that devised by L. Meyer are quite obvious and numerous. In the first place it is much more simple and available. The capillary is straight instead of coiled into a helix. The friction of the vapors is taken at a temperature so high above the boiling point that they behave more like true gases than at their boiling points, where they are in a condition of unstable equilibrium. The length of time required for an experiment is short enough for all conditions to be kept constant, and yet long enough, so that with a stop watch reading to one fifth of a second, the time can be determined to within less than one part in a thousand. Pogg. Ann., 1866. Bd. 127, p. 269. 38 APPARATUS No. 1. The first piece of apparatus was constructed more to test the practicability of the method than for getting accurate results. It having been determined by experiment, that the largest diameter of tube in which a mercury column would hold together, against a cushion of air, was about .35 cm., this apparatus was constructed with a mercury tube whose mean diameter, where it was traversed by the mercury column was .35104 cm., as shown by the fact that 76.0405 grams of mercury occupied 58.05 cm. at 20 C. The distance between the two etched marks on the tube was 50 cm. This distance was occu- pied by 65.5558 grams of mercury at 20 C. The density of mercury at this temperature being 13.5463 the volume between the two marks was 4.8394 cubic centimetres. The capillary was 34.35 cm. in length which was only about half the length of the mercury tube forming the other side of the U, in other words the capillary formed the upper half of one side of the U. Its bore was conical as shown by the fact that 9.50 cm. of mercury at the small end occupied 9.05 cm. at the larger end which was made the bottom because O. Rey- nolds 1 has shown that converging walls tend to a steady instead of a whirling flow of water and presumably also of gas. The above column of mercury was passed slowly from one end of the capillary to the other and changed its length gradually and steadily, thus showing the absence of abnormal contrac- tions or expansions of the bore. The bore of the capillary was determined from a sample 5.75 cm. long which had been cut from the smaller end. This sample was weighed several times empty and when containing different columns of mercury and the radius of the bore determined in the well known way. A section was also examined under a microscope with a micro- meter eye piece, and found to be so nearly a true circle that it was taken for such. The microscope reading agreed well with the mercury determinations of the radius. After taking the average of the mercury and microscope readings and after allowing for the taper of the capillary its mean radius was found to be .0103908 cm. To facilitate the washing of the apparatus the small vent, V, in Figs. 1 and 2, was left about \ cm. below the base of the capillary. This vent tube which was only 1 cm. long was 1. Proc. Royal Inst. Grt. Brit., 1884, Vol. 11, p. 44. 39 hermetically sealed off after the apparatus was washed and ready for use. The apparatus, after it was received from the glass blower, was washed throughout with a solution of potassium permanganate made alkaline with caustic potash, then with a solution of the same substance made acid with dilute sulphuric acid. Later a half and half solution of potassium bichromate and strong sulphuric acid was substituted for the above. Some- times also strong nitric acid, followed by dilute nitric acid was used. Then followed many washings with distilled water. The apparatus was then dried, by sucking through it by means of an aspirator, hot air which was filtered through cotton and dried by calcium chloride. A column of calcium chloride was also placed in the connection between the aspirator and the apparatus. The drying was not hastened by the use of alcohol or ether. The mercury was cleansed by shaking with dilute nitric acid, passing through a fine pin hole and drying in a porce- lain dish at 110 C. TABLE OF THE INTERNAL FRICTION OF AIR. APPARATUS No. 1. Ht. of Press, of Temp, driving Barom. entering Time in T^XlO 7 Average in C Col. atOC Gas seconds in cm. 15 7.97 76.03 83.09 76.04 1872 14.9 7.97 76.03 83.09 75.0 1862 14.9 7.97 76.03 83.09 74.8 1857 1863. 8 2. 57 14.9 7.966 76.03 83.09 75.2 1866 14.9 7.966 76.60 83.64 75.2 1860 15.4 7.966 76.60 83.64 75.4 1865 20.4 4.29 75.38 78.87 156.8 1967 19723.5 20.2 4.29 75.38 78.87' 157.6 1977 21.4 9.48 75.38 84.01 64.4 1935 19322.0 21.4 9.48 75.38 84.01 64.2 1929 99.9 9.48 75.38 84.01 75.6 2272. 5 99.9 9.48 75.38 84.01 76.0 2284. 5 100.3 7.98 76.96 84.00 92.0 2238. 2 100.3 7.98 76.96 84.00 89.8 2185. 2296. 5 15 8 100.3 7.97 76.60 83.64 94.0 2305. 100.3 7.97 76.60 83.64 96.4 2384. 5 100.3 7.97 76.60 83.64 92.7 2293. 100.3 7.97 76.60 83.64 94.0 2325. 40 The results of experiments on air with this apparatus are given in the preceding table. It will be noticed they are all rather high, but prove at any rate that the method and apparatus are practicable. The probable errors were calculated by the usual formula 0.6745 J ^ dlfp The error for readings at 100 C. \ n(n~l] is large, probably owing to an error of the stop watch, whose hand showed a tendency to fly forward when stopping near 90 seconds. In one case the hand flew forward 10 seconds. The watch was of course repaired as soon as this defect was noticed. In those cases where the mercury was allowed to run back and was used over again the readings were weighted less than independent readings, three readings being con- verted into two by taking their means, and four into three in the same way. It is rather difficult to tell what is the true coefficient of the internal friction of air. Landolt and Boernstein 1 give r) for air at 15 C. 1784 X 10 7 . Markowski 2 at 16 gives 1814 X 10 7 , Kleint 3 at 14.1 to 14.5 C. gives 1808 X 10 7 . According to Landolt and Boernstein then, my readings are 4J% too high at 15, while according to Markowski & Kleint they are about 3% too high. F. G. Reynolds 4 gives for air at 20.7 .000187 which is 3J% lower than my result of .0001932 at 21.4 C. It was found by repeated trials with di-ethyl ether, that the' surface tension of the mercury was so much reduced by contact with the ether gas, that the mercury would not hold together in one column in apparatus No. 1. It was determined by experiment that the mercury piston could not be used for ether gas in a tube whose diameter was much larger than 2 mm. A second piece of apparatus was accordingly constructed with a mercury tube of about this size, and a finer capillary, so that the transpiration time would be increased and greater accuracy be secured. APPARATUS No. 2. The capillary selected for this when examined with a simple microscope was at first thought to be circular in cross section, 1. Phys. Chem. Tabellen, Landolt & Boernstein, 1893. 2. Inaug. Diss. Halle, 1903. 3. Inaug. Diss. Halle, 1904. 4. Physical Rev., 1904, Vol. 18, p. 419; Vol. 19, p. 37. 41 but the use of a high power microscope with, micrometer eye piece showed it to be very elliptical, the ratio of the axes being almost exactly as 3 is to 1. The area of its cross section was determined by mercury several times and these values averaged with the microscope reading with which they agreed well. The capillary which was 84 cm. long tapered in its bore, 2.5 cm. of mercury at the large end becoming 2.685 cm. at the small end. The sample whose bore was determined was taken from the small end. After allowing for the taper of the bore the average semi-major axis was found to be .006057 cm. and the average semi -minor axis .002016 cm. The capillary was placed with its larger end downward and reached to within 15 cm. of the bend in the U. The mean diameter of the mercury tube in that part traversed by the mercury piston during an experiment was .2012 cm. as shown by the fact that 31.3 grams of mercury occupied a length of 72.6 cm. at 16 C. The marks on the mercury tube were made 50 cm. apart, the volume between these marks being found by means of mercury to be 1.5707 c.c. Because the capillary was elliptical the formula used for calculating the internal friction of gas from this apparatus is TT dg a 3 b 3 P 2 -p 2 " 8 L V a 2 + b 2 P where a = semi major axis of ellipse of capillary b = semi minor axis of ellipse of capillary d = density of mercury at C g = acceleration due to gravitation L = length of capillary V -= vol. of gas transpired P = pressure of entering gas p = pressure of leaving gas = barom. at C. t = time in seconds. The results with air with this apparatus are shown in the next table. It will be noticed that the values of TJ are somewhat lower than for apparatus No. 1, and therefore nearer the correct values. The different lengths of the capillary given in the first column are due to the fact that on several occasions the upper end of 42 the capillary became stopped with dust from the atmosphere and had to be cut off. When not in use the capillaries were kept capped with rubber, the funnels filled with cotton and closed by corks in which were inserted tubes of chloride of calcium. TABLE OF THE INTERNAL FRICTION OF AIR. APPARATUS No. 2. Ht. of Pres. of Length Temp, driving Barom. entering Time in f)X 10 7 Average of cap. in C Col. at C Gas seconds >?X10 7 81.8 12.5 23. 17 75. 74 97. 92 2604 1864 .2 84. 18.5 23. 17 76. 19 98. 30 2721. 1897 .4 84. 19.9 23. 15 76. 26 98. 41 2715. 1892 .0 ) 1897.33.5 84. 19.9 23. 42 76. 41 98. 82 2700. 1902 .5f 84. 20.9 23. 15 76. 26 98. 41 2730. 1902 .2 84. 26.9 23. 16 76. 15 98. 31 2810. 1957 .1 84. 29.1 23. 17 75. 90 98. 08 2830. 1973 .6 84. 100. 23. 17 75. 91 98. 09 3201. 2233 .0 ) 81.8 99.9 23. 14 75. 70 98. 85 3130. 2237 81.8 100.2 23. 14 76. 51 98. 65 3155. 2257 .0 ) TABLE OF THE INTERNAL FRICTION OF AIR. APPARATUS No. 3. 73.25 10. 18 .746 76.78 94 .56 509. 8 1843.8 73.25 13. 3 18 .75 76.78 94 .56 516. 1866.9 73.25 14. 4 18 .74 76.80 94 .58 518. 2 1870.0 73.25 19. 4 18 .75 76.00 93 .79 524. 6 1894.0 73.25 20. 8 18 .74 75.99 93 .78 531. 2 1917.2 70. 21. 18 .75 76.99 94 .77 509. 2 1923.6 1909. 9 7. 2 70. 20. 8 18 .75 76.95 94 .74 500. 1889.0 70. 100. 18 .74 75.76 94 .50 592. 8 2244.0 70. 100. 1 18 .74 76.19 94 .93 599. 2267.0 2246. 37. 2 70. 100. 3 18 .75 77.13 94 .92 588. 8 2228.0 PREPARATION OF ETHERS. The di-ethyl ether used in the following experiments was sulphuric ether, U. S. P. purified by two washings with con- centrated sulphuric acid, C. P., each of these washings being followed by one with distilled water. It was then shaken with mercury and dried with sodium wire from which it was dis- tilled into glass tubes which were afterward hermetically sealed. 43 TABLE OP INTERNAL FRICTION OF ETHER GASES. APPARATUS No. 2. Ht. of Press, of Length Kind Temp, driving Barom. entering Time in Capillary of Gas. of Gas Col. at C Gas seconds atO C Average 81.8 Methyl- 99. 9 23.16 75.68 97 .85 1502.0 1074 .5 81.8 Ethyl 100. 23.20 76.09 98 .30 1537.8 1103 81.8 Ether 99. 8 23.19 75.87 98 .07 1471.0 1053 .9 81.8 99. 8 23.16 75.44 97 .61 1558.0 1114 .2 1092.3-^-6.4 81.8 100. 23.06 76.28 98.35 1550.2 1105 .4 81.8 100. 23.16 76.15 98 .32 1540.0 1102.8 84. Methyl- 100. 23.90 76.28 99 .18 1380.0 990 9 84. Propyl 100. 23.22 76.23 98 .46 1409.6 985 .4 84. Ether 100. 23.28 76.18 98 .47 1402.2 982 .5 84. 100. 1 23.14 76.46 98 .60 1461.8 1018 .9 1004.3-^5.5 82.5 99. e 23.13 75.31 97 .46 1440.4 1021 .2 81.8 99. 8 23.11 74.24 96 .39 1439.0 1027 .0 81.8 Methyl- 100. 8 23.15 77.15 99 .30 1481.4 1061 .1 81.8 Isopro- 100. a 23.14 77.10 99 .24 1526.0 1081 .1 81.8 pyl 100. 23.13 76.01 98 .24 1459.6 1047 .3 1054.8-J-5.0 81.8 Ether 100. 23.12 76.00 98 .13 1455.0 1039 .9 81.8 100. 23.14 76.00 98 .15 1460.8 1044 .9 84. Di-Eth- 100. i 23.42 76.39 98 .81 1410.0 993 .4 84. yl Eth- 100. 23.42 76.08 98 .51 1404.0 990 .6 84. er 100. 23.16 76.15 98 .32 1416.6 986 .8 84. 100. 23.17 76.10 98 .28 1417.6 988 .6 999. -(-3.3 84. 100. i 23.16 76.54 98 .71 1437.6 1000 .6 82.5 99. 9 23.16 75.71 97 .89 1434.6 1017 .8 82.5 99. 8 23.21 75.50 97 .73 1428.0 1015 .2 84. Ethyl- 100. 23.17 76.06 98 .24 1285.0 896.2 84. Propyl 100. 23.17 75.89 98 .07 1296.4 904 .6 84. Ether 99. D 23.16 75.70 97 .87 1281.2 892 .7 81.5 99. I 23.15 75.58 97 .75 1295.8 930 .5 91o.2-J.-5. 3 81.5 99. 8 23.14 75.57 97 .73 1298.8 932.3 81.5 99. 8 23.15 75.57 97 .74 1300.8 934 .0 The ethyl-propyl ether, methyl-propyl ether and methyl-ethyl ether used in this investigation were prepared by me in the research laboratory of New York University under the per- sonal supervision of Professor Loeb, according to the continuous etherification method described by Norton and Prescott in the Am. Chem. Journal, 1884, Vol. VI., p. 241. They were care- fully dried with sodium and kept in sealed tubes. The other substances, viz.: di-methyl ether, ethyl alcohol, methyl-isopropyl ether, ethyl-isopropyl ether, di-propyl ether, isopropyl-propyl ether and di-isopropyl, I owe entirely to the great courtesy of Professor Loeb, as I had no part at all in their preparation. As far as is known this is the first time that 44 methyl-isopropyl ether has ever been made, while ethyl-iso- propyl ether and isopropyl-propyl ether have probably been made only once before. The internal frictions of the ethers used in apparatus No. 2 are given in the foregoing table on page 43. The ethers are arranged in this table in the order of theii molecular weights; that of methyl-ethyl ether being 60.064, that of methyl-propyl, methyl-isopropyl, and di-ethyl ether being alike 74.08; while that of ethyl propyl is 88.096. It will be noticed the smaller the molecular weight the greater is the internal friction, which agrees well with the kinetic theory of gases, according to which the friction increases with dimin- ished size of molecule. The most noteworthy fact shown in this table is that the three isomeric ethers, di-ethyl, methyl-propyl and methyl- isopropyl have not the same internal friction. Di-ethyl ether and methyl-propyl practically agree, while methyl-isopropyl ether has a friction 5% higher. According to the universally accepted view of the concatenation of the molecules of these substances, the first two are simple chain compounds, while the last is a chain compound with two branches: Di-ethyl ether H 3 C C O C CH 3 H 2 H 2 Methyl-propyl ether H 3 C O C C CH 3 H 2 H 2 Methyl-isopropyl ether H 3 C O C<^ H H s It is easily conceivable how the last arrangement results in a compacter, smaller molecule, and hence shows a higher in- ternal friction. Although the amount of liquid used varied from ^ cc. to 1 cc. as a rule, this small amount seemed sufficient on vaporizing to drive all the air out of the apparatus, as filling the appar- atus entirely with ether from the base of the capillary to the stop-cock did not give results different from those obtained using 1 cc. of ether. The weight of mercury used in apparatus No. 1 and No. 2 was usually about 10 grams. That in apparatus No. 3 shortly to be described was about 5 grams. The allowance to be made for the friction of the mercury against the walls of the tube was determined in the following 45 manner. Three readings with air were taken with three different columns of mercury, under constant temperature conditions It being known that the internal friction of a gas is independent of the pressure, the values of y from these three readings were equated to each other in pairs, giving, after cancelling out constants of the apparatus, equations of the form P2 c 2_p2 P 2 c 2_p2 PC P,c tl in which c is a constant reduction factor due to the friction of the mercury against the walls of the tube. The effect of cap- illarity is probably negligible, because the upper and lower surfaces of the mercury are convex in opposite directions. In solving the equations for c good agreement was found, the average c for apparatus No. 1 being .9892. For apparatus No. 2 c = .9895 and for apparatus No. 3 c = .9894, all at 100 C. The pressure of the gas on entering the capillary was calculated by adding the height of the mercury column at C. to the barometer at C. and multiplying by the reduction factor given above. The mercury and barometer column were re- duced to C. by the aid of a table given on page 248 of Kohl- rausch's Kleiner Leitfaden der Praktischen Physik. APPARATUS No. 3, In order to have a second piece of apparatus available for experiments with ether, so that two experiments could be made at the same tmie, a third piece of apparatus was constructed. The mercury tube of this apparatus was made a little smaller than that of apparatus No. 2, as some difficulty had been met in the mercury column failing to hold together properly. The capillary of this apparatus was also elliptical and slightly conical. The average semi -major axis as determined by the microscope and mercury was .006176 cm. while the minor axis was .002837. The length of the capillary was 73.25 cm. at first, but was reduced to 70 cm. when the apparatus was repaired after a breakage. The distance between the marks on the mercury tube was 38.45 cm. and the volume of gas transpired was .742 c.c. since 10.0634 grams of mercury occupied the distance between the marks at 16 C. The average diameter of the mercury tube where the mercury traversed it during an experiment was .158112 cm. A table of the results obtained with this apparatus for air 46 is given on page 71 underneath those for air with apparatus No. 2. It will be noticed that the two pieces of apparatus agree very closely in results. TABLE OF INTERNAL FRICTION OF ETHER GASES. APPARATUS No. 3. Ht. of Press of Length Capillary Kind of Gas Temp, driving of Gas Col. atOC Barom. at C entering Time in Gas seconds B X 1 7 Average 70. Di-meth 100.3 18.75 77.24 95. 03 315.6 1191. 8 70. yl Ether 100.3 18.74 77.32 95.10 314.8 1188.6 1190.2-j-l. 70. Ethyl 100. 3 18.74 76.98 94. 67 303.6 1145. 6 70. Alcohol 100. 3 18.76 76.99 94 .70 308.4 1164 .4 1155. -J-6.3 73.25 Di-ethyl 100. 1 18.78 76.43 94 .26 287.4. 1041 .8 73.25 Ether 100. 18.60 76.15 93 .81 287.4 1023.2 73.25 100. 18.79 76.03 93. 88 270.4 978. 4 1001. -[-8.15 70. 99. 9 18.76 75.80 93.61 254.8 963 .7 70. 100. 1 18.56 76.46 94 .07 266.5 998 .1 73.25 Ethyl- 99. 9 18.76 75.79 93 61 253 . 2 914 .8 73.25 Propyl 99. 9 18.26 75.74 93. 06 267.2 939.3 73.25 Ether 100. 18.05 76.39 93. 50 262.4 921. 7 918.4-J-4.2 73.25 100. 3 18.28 76.83 94 .16 256.6 902 .7 73.25 100 4 18.05 77.02 94 .14 263.2 913 .7 70. Ethyl- 100. 2 17.74 76.46 93, 26 267.7 946 ,4 70. Isopro- 100. 2 18.76 76.41 94 .22 255.4 966 .4 960.7-J-5. 70. pyl Ether 100 . 2 18.56 76.34 93 .95 258.8 969 .3 70. Di-Pro- 100. 18.75 75.95 93.73 222 2 839. 5 70. pyl 100. 18.76 76.02 93 .74 222.0 839 .3 837. 5 -J- 1.26 70. Ether 100. 3 18.75 77.14 94 93 220.8 832 .7 70. Isopro- 100. 1 18.73 76.19 93. 97 230.4 870.2 873.4-1-2.2 70. pyl-pro- 100. 1 18.76 76.22 94.03 231.6 876 .6 pyl Ether 70. Di-iso- 100. 3 18.74 77.14 94 .92 230.2 869 .2 70. propyl 100. 3 18.74 77.14 94 .92 230.8 871.5 70. Ether 100. 3 18.76 77.09 94 .89 239.0 902 .8 70. 100. 3 18.74 77.13 94. 91 236.0 890.7 894. 3 -j- 5. 95 70. 100. 1 18.73 76.23 94 .01 241.0 910 .5 70. 100. 1 18.75 76.23 94, ,03 243.6 921 2 The results obtained for ethers with this apparatus are given in the above table. The first point worthy of note is that ethyl alcohol which is metameric with di-methyl ether has about 3% lower coefficient of friction, showing it to have the larger molecule, The values for di-ethyl ether, and ethyl-propyl ether agree well with those given on page 43 by apparatus No. 2. 47 Ethyl-isopropyl ether has a friction 4.6% higher than that of ethyl propyl ether, hence has a smaller molecule. Ethyl-propyl ether Ethyl-isopropyl ether H 3 C C O C< 3 H 2 H CH a Isopropyl -propyl ether has a friction 4.3% higher than that of di-propyl ether, while di-isopropyl ether has a friction 6.8% higher than di-propyl. This is in accordance with our view of the concatenation of the atoms in these molecules. Di- propyl ether is a straight,. chain compound whereas iso-propyl propyl has two branches and di-isopropyl has four branches. Di-propyl ether Isopropyl-propyl ether u 3 n> C h 3<- H H 2 H 2 H 3 Di-isopropyl ether **,C > C O C < 3 H H 3 As already noticed the difference between methyl-propyl and methyl -isopropyl is 5% due to 2 branches; the difference between ethyl-propyl and ethyl-isopropyl due to two branches is 4.6%; the difference between di-propyl and isopropyl-propyl is 4 . 3 % due to the same number of branches. The diminishing effect of the 2 CH 3 branches is probably due to the fact that the molecules of the substances lower on the lists are larger, hence the branches have a smaller relative effect. It would seem that di-isopropyl ether ought to be 8.6% higher than di- propyl whereas it is only 6 . 8% . This discrepancy I am unable to explain. Di-methyl ether, being a gas at ordinary temperatures, had to be handled differently from the other ethers, all of which were put into the apparatus as liquids. The low boiling ethers, methyl-ethyl, and methyl-isopropyl were experimented upon in a room whose temperature was at or very near C. The di -methyl ether gas was kept in sealed glass bulbs of 250 c.c capacity. The lower end of a bulb was connected by rubber tubing to a reservoir of mercury while the upper end was connected by a rubber tube to the funnel at the top of the apparatus. 48 After the aspirator had created a partial vacuum in the apparatus by sucking the air out through the capillary, the tips of the bulb were broken off inside the rubber tubes and mercury allowed to flow into the bulb gradually from the bottom, driving the ether gas before it into the ap- paratus, the action being assisted all the while by the suction of the aspirator. As 250 c.c. was many times the cubic capacity of the apparatus, by the time the mercury had entirely filled the ether bulb, it was judged the apparatus would be filled with pure ether, unmixed with any air. This method of getting the ether gas into the apparatus proved a failure several times for various reasons, but the very close agreement of the two read- ings which were at length obtained points to their correctness. MOLECULAR VOLUMES. L. Meyer 1 gives the following formula for calculating approxi- mate relative molecular volumes, derived from the molecular volume of 5 2 as determined by Andreef at -8 C V == 3.10" 6 j M (1 +at) I* j in which M = molecular weight, t = temperature centigrade T? = coefficient of friction. The values obtained for the substances under investigation are given on the next page. According to Kopp 1 the molecular volume of a liquid composed of carbon, hydrogen and oxygen can be found by substituting in its formula 11 for each atom of carbon, 5.5 for each hydrogen and 7.8 for each oxygen. The values thus calculated are given in the last column. The agree- ment between the molecular volumes calculated in these two ways is quite good, much better than that obtained by L. Meyer and Schumann (p. 21). As both apparatus No. 2 and No. 3 gave values for air about 5% higher than those given by Landolt and Boernstein as prob- ably the most correct values, it is probable that the values of >? for the ethers are correspondingly too high. Accordingly in the second table of molecular volumes the values of IQ were reduced by multiplying by 2113/2244, thus calibrating with air, and the molecular volumes recalculated. A still better 1. Wied. Ann., 1881, Bd. 13, p. 17. 1. Ann. Chem. Phar., 1855, Bd. 96, pp. 1, 153, 303. 49 agreement with Kopp's values is shown, the disagreement being greatest when the boiling point of the ether approaches the tem- perature at which i) was determined. FIRST TABLE OF MOLECULAR VOLUMES. Material Boiling Point Formula Cent. r/XlO 7 Molec. V. Molec. V. Found Calculated according to Kopp Di-methyl ether C 2 H 6 -23.65 1190.2 51.62 I 62 .8 Ethyl alcohol CTT 2 6 78. 1155. 55.25 s Methyl-ethyl ether CTT 3 8 10. -13. 1092. 3 71.65 84 .8 Di-ethyl ether CTT 4 10 34.6-35. 1000. 95.73 ) Methyl-propyl ether CTT jXi 10 40-45. 1004. 3 95.16 106 .S Methyl-isopropyl ether CTT 4 10 7. 1054. s 88.37 ) Ethyl-propyl ether C 5 H 12 66. -68- 910. 8 124.19 I 1 9S Ethyl-isopropyl ether CTT -IT. 12 54. -57. 960. 7 115.77 LAO Di-propyl ether C 6 H 84.5-86.5 837. 5 158.90 Isopropyl-propyl-ether CTT piJ. uO 76.5-77. 873. 4 148.86 [ 150 .8 Di-isopropyl ether CTT 8 14 68.5-69. 894. 3 144.06 ) SECOND TABLE OF MOLECULAR VOLUMES. Material Boiling Point in Formula deg. Cent. 7?X10 7 Molec. V. Molec. V. Calculated Found according to Kopp Di-methyl ether Ethyl alcohol Methyl-ethyl ether cX C 3 H 8 O -23.65 1121 78. 1088 10. -13. 1029 .0 .0 .0 56. 59. 81. 46) 05) 12 62, 84. ,8 8 Di-ethyl ether C 4 H 10 O 36. ,6-35. 942 .0 104. 71 ? Methyl-propyl ether C 4H 10 40. -45. 946 .0 104. 20 [ 106. 8 Methyl-isopropyl ether C *H 10 7. 993 .6 96. 66 ) Ethyl-propyl ether C 5 H 12 66. -68. 863 .6 135. 84 | 1 28 Ethyl-isopropyl ether C 5 H 12 54, ,-57. 905 .1 126. 61 f 1 <6o < Di-propyl ether C 6 H 14 84 ,5-86. 5 788 .9 173. 08^ Isopropyl-propyl-ether C 6 H 14 76. 5-77. 822 .7 163. 20 > 150. 8 Di-isopropyl ether C 6 H 14 68. 5-69. 842 .4 157. 5lJ In this second table r) has been reduced by calibration with air. MOLECULAR MEAN SPEEDS, FREE PATHS AND COLLISION FREQUENCIES. In Meyer's Kinetic Theory of Gases on page 219 is given the formula y =0 . 30967 p LSI whence T . 0.30967 ptt in which i) = coefficient of internal friction of the gas p = density of the gas Q =mean molecular velocity L =mean molecular free path. On page 55 of the same work we find ^f- where p pressure in absolute measure. On page 195 we see that the p = npQ 2 whence o Q Collision Frequency = -=- The values given in the table on this page were calcu- lated according to the above formulae, at a pressure of 76 cm. of mercury and a temperature of 100C., the density being cal- culated from the formula (formula weight) (0.000089) It will be noticed that the values are all of the proper order of magnitude, but do not rise and fall in a periodic way while the FIRST TABLE OF MEAN MOLECULAR SPEEDS, FREE PATHS AND COLLISION FREQUENCIES. Mean Molecular Collision Molecular Molecular Free PathFrequency Material We ight 7?X10 7 speed cm. cm.XlO 10 xio- 6 per Sec. Di-methyl ether 46 .048 1190. 2 41470 61787 6711. 7 Ethyl alcohol 46 .048 1155. 41470 59970 6915. Methyl-ethyl ether 60 .064 1092. 3 36310 49660 7312. Di-ethyl ether 74 .08 1000. 32695 40895 7994. 9 Methyl-propyl ether 74 .08 1004. 3 32695 41111 7952. 7 Methyl-isopropyl ether 74 .08 1054. 8 32695 43180 7572. Ethyl-propyl ether 88.096 916.8 29982 34415 8711. 8 Ethyl-isopropyl ether 88 .096 960. 7 29982 36063 8315. 5 Di-propyl-propyl ether 102 .112 837. 5 27848 29201 9536. Isopropyl-propyl ether 102 .112 873. 4 27848 30454 9144. G Di-isopropyl ether 102 .112 894. 3 27848 31182 8931. 51 SECOND TABLE OF MEAN MOLECULAR SPEEDS, FREE PATHS AND COLLI- SION FREQUENCIES. Di-methyl ether 46. .048 1121 .0 41470 58204 7125. Ethyl alcohol 46 .048 1088 .0 41470 56490 7341. Methyl-ethyl ether 60. 064 1029 .0 36310 46782 7762. Di-ethyl ether 74 .08 942 .0 32695 38570 8477. Methyl-propyl ether 74, ,08 946 .0 32695 38725 8639. 5 Methyl-isopropyl ether 74. 08 993 .6 32695 40682 8036. 7 Ethyl-propyl ether 88. 096 863 .6 29982 32418 9248. 4 Ethyl-isopropyl ether 88. 096 905 .1 29982 33976 8824. 3 Di-propyl propyl ether 102. 112 788 .9 27848 27507 10124. 9 Isopropyl-propyl ether 102. 112 822 .7 27848 28685 9708. 9 Di-isopropyl ether 102. 112 842 .4 27848 29372 9481. 1 In this second table y has been reduced by calibration with air. molecular weight increases, as pointed out for other substances by Meyer on page 196 of his Kinetic Theory of Gases, probably because these ethers are all so closely related. COMPARISON WITH THE RESULTS OF OTHERS. The results of my work cannot be compared satisfactorily and exactly with that of others, because I have determined the friction at a temperature higher than most other observers, and the law of its increase with the temperature is not accurately known in each case. Furthermore, air, di-ethyl ether, dimethyl ether and ethyl alcohol are the only substances which I have employed that others have investigated. Disregarding the results with apparatus No. 1 as being only preliminary, we see that the values of tj for air for the other two pieces of apparatus agree closely and are consistent, though both are about 5% higher than the values which are most probably correct, according to Landolt and Boernstein's tables. That the relative values of y at the different temperatures are correct is shown by calculating y at C. by Sutherland's for- mula (see page 30) and then calculating from tha+ y at 100 C. Using the value of the cohesion constant C=119.4 as deter- mined by Breitenbach we get for apparatus No. 2, >? = .0001793 and T? IOO = .00022367 the latter value agreeing with .0002242 observed within 1/4%. For apparatus No. 3 we get in like manner >? = . 0001795 and ^ 100 = .0002239, the latter value agreeing with .0002246 observed within \%. This agreement is within the limits of error of the experiments. 52 Puluj's formula for the coefficient of friction of di-ethyl ether vapor, y =0.0000689 (1 +.0041575 t) * 94 gives j =0.0000935 at 100 C. which agrees with my corrected value of 0.0000942 within }%. This is good agreement bearing in mind the fact that Puluj's formula was determined from experiments over a small range of temperature, viz.: from to 37 C. The coefficient of friction of di-methyl ether is given on page 192 of Meyer's Kinetic Theory of Gases as 0.000092 at C. No one has determined the law of its increase with the tem- perature. My corrected value is .0001121 at 100 C. which seems reasonable. Steudel (see page 21) gives 0.000142 as the coefficient of friction of ethyl alcohol at 78.4 C., its boiling point. My cor- rected value of .0001088 at 100 C. does not agree well with this. Because his value was determined at the boiling point I think it is open to question. My value for ethyl alcohol agrees much better with Puluj's 1 results which are 0.0000827 at C. and 0.0000885 at 16.8 C. He assumes y is proportional to the absolute temperature which gives y = .0001130 at 100 C., which is nearly 4% higher than my value of .0001088. I think the assumption that ^ increases exactly as the absolute tem- perature is not strictly correct. Concerning the differences which I have found between the normal propyl and the iso-propyl ethers I would point out that Lothar Meyer, Schumann and Steudel (see pages 20 to 23) found similar differences between many, though not all, of the normal propyl and iso-propyl compounds which they examined. They also found that normal butyl, isobutyl and tertiary butyl compounds showed still more regular differences. The weight of evidence gives a larger molecule to the primary, a smaller to the secondary and the smallest to the tertiary com- pound. My values of the molecular speeds, free paths and collision frequencies being for 100 C. of course do not agree with those calculated for C. by others. SUMMARY OF RESULTS. 1. A new and simple apparatus for determining the internal friction of gases and vapors has been developed. 2. The coefficients of internal friction of the following eight ether gases which have hitherto not been experimented with, have been determined with considerable accuracy as follows: 53 Methyl-ethyl ether 0.0001029 at 100 C. Methyl-propyl ether 0.0000946 " Methyl-isopropyl ether .0.00009936 " Ethyl-propyl ether 0.00008636 " Ethyl-isopropyl ether. 0.00009051 " Di-propyl ether 0.00007889 " Iso-propyl-propyl ether .0.00008227 " Di-isopropyl 0.00008424 " 3. The molecular volumes calculated from the friction have T}een shown to agree fairly well with those obtained by Kopp's rule. 4. A marked and unmistakable difference between the nor- mal propyl and isopropyl ethers has been found, proving that the difference in the molecular structure of these ethers has a very noticeable effect upon their internal friction, and there- fore upon the size of their molecules, the molecules having the most numerous branches being smaller than those with fewer or no branches. The object of this research has therefore been accomplished. In conclusion I wish to express my most sincere and heartfelt thanks to Professor Loeb for suggesting, to me both the subject of this research and the form of apparatus, and for his kind interest and help throughout the course of the investiga- tion. In addition I wish to thank him heartily for his very great courtesy in supplying me with the necessary ethers. I also wish to express my thanks to Professors Charles Basker- ville and Charles A. Doremus of the College of the City of New York, for their kindness in permitting me to perform most of the experiments in the chemical laboratory of that institution, in order to save time in going to and from the laboratory of New York University. FREDERICK M. PEDERSEN. New York University, April, 1905. BIBLIOGRAPHY. BAILY. Phil. Trans., 1832, p. 399. BARUS. Am. Journ. Sci., 1888 (3), Vol. 35, p. 407. Bull. U. S. Geol. Survey, No. 54, Washington, 1889, p. 39. Wied. Ann., 1889, Bd. 36, p. 358. Phil. Mag., 1890, Vol. 29, p. 337. BERNOULLI, JOHANN. Opera omnia. Lausannae et Genevae, 1742, Tomus III. Nouvelles pense"es sur le systeme de M. Descartes XIX.-XXIII. BESSEL. Abh. d. Berl. Akad. Math. Klasse, 1826, p. 1. BESTELMEYER. Inaug. Diss. Munich, 1902. Ann. d. Phys., 1904, Vol. 13, p. 944. BOLTZMANN. Wien. Ber. Math. Naturw., 1868, Bd. 58 (2), p. 517. 1872, " 66 (2), p. 324. 1880, " 81 (2), p. 117. 1881, " 84 (2), pp. 40, 1230. 1887, " 95 (2), p. 153. 1888, " 96 (2), p. 891. Wied. Ann., 1897, Bd. 60, p. 399. BRAUN AND KURZ. Carls Rep., 1882, Vol. 18, pp. 569, 665, 697. 1883, Vol. 19, pp. 343, 605. BREITENBACH. Wied. Ann., 1899, Bd. 67, p. 803. Ann. d. Phys., 1901, Bd. 5, p. 166. CAUCHY. Exerc. de Mathe"m., 1828, p. 183. CHALLIS. Phil. Mag., 1833, p. 185. CHEZY. Me*m. Manuscript de 1'Ecole des Fonts et Chausse*e, 1775. CLAUSIUS. Pogg. Ann., 1858, Bd. 105, p. 239. CLEBSH. Crelle's Journ. fur Math., Bd. 52, 1856, p. 119. COUETTE. Compt. Rend., 1888, Vol. 107, p. 388. Journ. de Phys., 1890 (2), Vol. 9, p. 414. Ann. de Chim. et de Phys., 1890 (6), Vol. 21, p. 433. 54 55 COULOMB. Me'm. de 1'Institut National. Year 9 (1801), Tome III., p. 246. COUPLET. Me'm. de 1' Academic, 1732. CROOKES. Phil. Trans., 1881, Vol. 172, p. 387. D'ALEMBERT. Trait^ de I'^quilibre et du mouvement des fluides, nouvelle e"dit. Paris, 1770. D'ARCY. Me'm. de Divers Savans, 1858, Vol. 15, p. 141. DE KEEN. Bull, de 1'Acad. de Belgiques, 1888 (3), Vol. 16, p. 195. Du BUAT. Principes d'Hydraulique, 1786. EULER. Tentamen theoriae de frictione fluidorum. Novi commentarii Ptero- politani, tomus VI., 1756, et 57 Pag. 338. Die Gesteze des Gleich- gewichtes und der Bewegung fliissiger Korper. Translated from the Latin by W. Brandes. Leipzig, 1806. EYTELWEIN. Abh. d. Berl. Akad., 1814 and 1815. GERSTNER. Neu. Abh. der kon. Bohmischen Gesell. der Wiss., Bd. 3, Prag 1798. Gilbert's Annalen, Bd. 5, 1900, p. 160. GlRARD. Me'm. de 1'Institut., Classe Sc. Math., 1813-15, p. 248; 1816, p. 187 GlRAULT. Me'm. de 1'Acad. de Caen, 1860. GIULIO. Memorie di Torino, Ser. 2, tomo 13, 1853. GRAHAM. Phil. Trans., 1846, Vol. 136, pp. 573, 622. 1849, Vol. 139, p. 349. Pogg. Ann., 1866, Bd. 127, pp. 279, 365. GREEN. Transactions of the Royal Society of Edinburgh, 1836, Vol. 13, p. 54. GRONAU. Uber die Bewegung schwingender Korper im widerstehenden Mittel. Danzig, 1850. Program der Johannesschule. GROSSMAN. Inaug. Diss. Breslau, 1880. Wied. Ann., 1882, Bd. 16, p. 619. GROTRIAN. Pogg. Ann., 1876, Vol. 157, pp. 130, 237. 41 1877, Vol. 160, p. 238. Wied. Ann., 1879, Vol. 8, p. 529. GUTHRIE. Phil. Mag., 1878 (5), Vol. 5, p. 433. 56 HAGEN. Pogg. Ann., 1839, Bd. 46, p. 423. Abth. d. Berl. Akad., 1854, p. 17. HAGENBACH. Pogg. Ann., 1860, Bd. 109, pp. 385, 401. HELMHOLTZ AND PIETROWSKI. Wien. Ber. Mathem. Naturw., 1860, Vol. 40, p. 607. HOFFMANN. Wied. Ann., 1884, Bd. 21, p. 470. HOLM AN. Proc. Am. Acad. Boston, 1877, Vol. 12, p. 41. 1886, Vol. 21, p. 1. Phil. Mag., 1877 (5), Vol. 3, p. 81; 1886, Vol. 21, p. 199. HOUDAILLE. Fortschr. d. Phys., 1896, 52 Jahr., I., p. 442. JACOBSON, H. Archiv. fur Anatomic und Physiologic von Reichert und Du Dois 1860 and 1861. JAEGER. Wien. Ber. Mathm. Naturw., 1899 (2), Vol. 108, p. 447. 1900 (2), Vol. 109, p. 74. JOB. Bull. Soc. Franc. Phys., 1901, Vol. 157, p. 2. KLEMENCIC. Carls. Rep., 1881, Bd. 17, p. 144. Wien. Ber., 1881 (2), Bd. 84, p. 146. Beiblatter, 1882, Bd. 6, p. 66. KlRCHHOFF. Mechanik., 1877, 4 Aufl., 26 Vorl., p. 383. KOCH. Wied. Ann., 1883, Bd. 19, p. 857. KOENIG. Wied. Ann., 1885, Bd. 25, p. 618; 1887, Bd. 32, p. 193. Sitz. d. Munch. Akad., 1887, Bd. 17, p. 343. KUNDT AND WARBURG. Monatsber. d. Berl. Akad., 1875, p. 160. Pogg. Ann., 1875, Vol. 155, pp. 337, 525. Phil. Mag., 1875 (4), Vol. 50, p. 53. KLEINT. Inaug. Diss. Halle, 1904. LAMPE. Programm des Stadtischen Gym. zu Danzig, 1866. LAMPEL. Wien. Ber. Mathem. Naturw., 1886, Bd. (2), p. 291. LANG. Wien. Ber. Math. Naturw., 1871, Vol. 63 (2), p. 604. 1872, Vol. 64 (2), p. 487. Pogg. Ann., 1872, Vol. 145. p. 290. 1873, Vol. 148, p. 550. 57 LUDWIG AND STEFAN. Sftzber. Wien. Akad., 1858, Vol. 32, p. 25. MARIAN. Histoire de 1'Acad. de Paris, 1735, p. 166. MARGULES. Wien. Ber. Mathem. Naturw., 1881, Vol. 83 (2), p. 588. MARKOWSKI. Inaug. Diss. Halle, 1903. Ann. d. Phys., 1904, Vol. 14, p. 742. MAXWELL. Phil. Mag., 1860 (4), Vol. 19, p. 31. 1868 (4), Vol. 25, p. 209 and 211. Phil. Trans., 1866, Vol. 156 (1), p. 249. Collected Papers, Vol. II. Proc. Royal Society, 1866, Vol. 15, p. 14. MATHIEU. Compt. Rend., 1863, Tome 57, p. 320. MEYER, LOTHAR. Ann. d. Chem. u. Phar., 1867, Suppl. Bd. 5, p. 129. Wied. Ann., 1879, Bd. 7, p. 497. 1882, Bd. 16, p. 394. MEYER, L. AND SCHUMANN. Wied. Ann., 1881, Bd. 13, p. 1. MEYER, O. E. Dissertation: de mutua duorum fluidorum frictione. Regimonti's Prussorum, 1860. Crelle's Journal fur Mathem., 1861, Bd. 59, p. 229. Pogg. Ann., 1861, Bd. 113, p. 55; 1865, Bd. 125, pp. 177, 401, 564; 1866, Bd. 127, pp. 253, 353; 1871, Bd. 142, p. 513; 1871, Bd. 143, p. 14; 1873, Bd. 148, p. 203. Wied. Ann., 1887, Bd. 32, p. 642; 1891, Bd. 43, p. 1. Sitz. d. Munchen Akad., 1887, Bd. 17, p. 343. MEYER, O. E. AND SPRINGMVHL. Pogg. Ann., 1873, Vol. 148, p. 526. MORITZ. Pogg. Ann., Bd. 70, p. 74. MUTZEL. Wied. Ann v 1891, Bd. 43, p. 15. NAUMANN. Ann. d. Chem. u. Pharm., 1867, Suppl. Bd. 5, p. 252. NAVIER. Me'm. de 1'Acad. des Sciences, 1823, Tome 6, p. 389. 1830, Tome 9, p. 311. NOYES AND GOODWIN. Physical Review, 1896, Vol. 4. Zeitsch. Physik. Chem., 1896, Vol. 21, p. 671. NEUMANN. Einl. in d. theor. Physik., 1883, p. 246. NEWTON. Philosophiae naturalis principia mathematica, 1687, Lib. II., Sect. IX. 58 OBERMAYER. Wien. Ber. Mathem. Naturw., 1875, Vol. 71 (2), p. 281. 1876, Vol. 73 (2), p. 433. Carls. Rep., 1876 (2), Vol. 12, pp. 13, 456. " 1877, .Vol. 13, p. 130. Phil. Mag., 1886, Vol. 21. ORTLOFF. Inaug. Diss. Jena, 1895. PEROT AND FABRY. Compt. Rend., 1897, Vol. 124, p. 28. Ann. de Chim. et de Phys., 1898 (7), Vol. 13, p. 275. POISEUILLE. Soc. Philomath, 1838, p. 77. Compt. rend., 1840, Vol. 11, pp. 961, 1041. 1841, Vol. 12, p. 112. 1842, Vol. 15, p. 1167. Ann. de Chim. et de Phys., 1843 (3), Vol. 7, p. 50. Me*m. de Savans Strangers. 1846, Vol. 9, p. 433. POISSON. Journal de 1'Ecole Polytech., 1831, 20 me cahier, tome 13, p. 139. Connaissance des Terns, 1834. Appendix. M(m. de 1'Acad., Tome 2, 1832, p. 521. PLAN A. Me'm. de 1'Acad. di Torino, T 37, 1835. PRONY. Recherches physico-mathe'matiques sur la the*orie des eaux courantes. Paris, 1804. PULUJ. Wien. Ber. Mathem. Naturw., 1874, Bd. 69 (2), p. 287. 1874, Bd. 70 (2), 243. 1876, Bd. 73 (2), 589. 1878, Bd. 78 (2), p. 279. Carls. Rep., 1878, Bd. 14, p. 573. " 1879, Bd. 15, pp. 427, 578, 633. Phil. Mag., 1878 (5), Vol. 6, p. 157. RAYLEIGH. Proc. Roy. Soc., 1900, Vol. 66, p. 68; Vol. 67, p. 137 RELLSTAB. Inaug, Diss. Bonn., 1868. REYNOLDS, F. G. Phys. Rev., 1904, Vol. 18, p. 419; Vol. 19, p. 37. REYNOLDS, O. Proc. Roy. Inst. Grt. Brit., 1884, Vol. 11, p. 44. Beiblatter, 1886, Bd. 10, p. 217. SABINE. Phil. Trans., 1829, p. 207 and 331; 1831, p. 470. SCHNEEBELI. Arch, de Geneve, 1885, Vol. 14 (3), p. 197. SCHULTZE. Ann. d. Physik., 1801, Bd. V., p. 140. 59 SCHUMACHER. Astronomische Nachtrichten, Bd. 40, 1855. SCHUMANN. Wied. Ann., 1884, Bd. 23, p. 353. STEFAN. Wien. Ber. Mathem. Naturw., 1862 (2), Vol. 46, pp. 8, 495. 1872 (2), Vol. 65, p. 360. STEUDEL. Wien. Ann., 1882, Bd. 16, p. 369. STEWART AND TAIT. Proc. Royal Society, 1865, Vol. 14, p. 339. Phil. Mag. (4), Vol. 30, p. 314. STOKES. Camb. Phil. Trans., 1849, Vol. 8, p. 287. 1850, Vol. 9, p. 8. Phil. Mag., 1851 (4), Vol. 1, p. 337. ST. VENANT, BARRE DE. Compt. Rend., 17, 1843, pp. 1140 and 1240. SUTHERLAND. Proc. Ann., Acad. 1885, p. 13. Phil. Mag., 1893 (5), Vol. 36, p. 507. TOMLINSON. Phil. Trans., 1886, Vol. 177 (2), p. 767. WARBURG. Pogg. Ann., 1876, Bd. 159, p. 399. WARBURG AND BABO. Wied. Ann., 1882, Bd. 17, p. 390. Sitz. d. Berl. Akad., 1882, p. 509. WlEDEMANN, G. Pogg. Ann., 1856, Bd. 99, p. 177. WlEDEMANN, E. Arch. d. Sc. Phys. et Nat. de Geneve, 1876, Vol. 56, p. 277. Fortschr. d. Phys., 1876, Vol. 32, p. 206. ou- i-^-*-"'