University of California At Los Angeles The Library Form L I QC and . 4. tance between the two rings, through which the hy- drometer sinks under the action of the weight added, IT 5.] PKECAUTIONS. 9 is estimated roughly by a small millimetre scale. We now calculate the effect of one centigram in sinking the instrument. This is called the sensitiveness ( 41) of the hydrometer, and is useful in determin- ing the degree of precision with which the adjust- ments of the instrument should be made (see 48). Thus if the effect of one centigram is distinctly per- ceptible, we should try to avoid errors even less than a centigram in magnitude. In using a Nicholson's hydrometer, several precau- tions should be observed. It frequently happens that through friction against the sides of the vessel, or through capillary phenomena where the surface of the water meets the stem, the hydrometer is unaf- fected by any slight change iu the load. To avoid the first difficulty, the instrument should be kept floating in the middle of the jar, by the use of a guide of some sort. Such a guide may be con- veniently constructed of wire, as in Fig. 5. To avoid the uncer- tainty of capillary action, the stem of the hydrometer should be kept wet, by a camel's-hair brush, for at least a centimetre above the water level. FlG - 5 - In water freshly drawn bubbles of air are apt to form, clinging to the sides of the hydrometer. These should be removed by the same brush. The forma- tion of air bubbles may generally be prevented by using either distilled water, or water which has been standing for some time in the room. K) NICHOLSON'S HYDROMETER. [Exp. 2. It is important to keep the upper part of the stem, the pan, and the weights absolutely dry. The guide (Fig. 4) should prevent the hydrometer from sinking completely below the surface. 1 ^f 6. Accurate Adjustment of a Nicholson's Hydrom- eter. __ A mark is made near the middle of the stem of the hydrometer and the load is altered, a centi- gram at a time, until this mark is floated as nearly as possible in line with the surface of the water. If a glass jar is used, it is better to sight this mark by the under surface of the water, as shown in Fig. 6. In the absence of weights smaller than one centigram, we estimate and record fractions of a centigram as follows : when the mark is floated exactly on a level with the surface of the water, the fact is expressed by placing a cipher in the third decimal place (be- longing to the milligrams). If, however, a given weight fails to sink the mark to this level, while the addition of one centigram sends it as much below the surface as it was before above it, half a 1 A sheet of cardboard or metal with a hole in the middle, is rec- ommended by some authorities (see Pickering's Physical Manipu- lation, Article 45) to serve as a guide, and at the same time to prevent the weights from falling into the water. A student relying upon this safeguard is apt, however, not to acquire a sufficient degree of skill to prepare him for the manipulations of a delicate balance. (Exps. 6-14.) 1 7.] EFFECTS OF TEMPERATURE. 11 centigram or 5 milligrams is obviously the weight to be added ; hence the original weight should be followed by a 5 instead of a in the last place. Thus if with 25.99 g. the mark is 2 mm. above the surface of the water, and with 26.00 g. it is 2 mm. below it, the weight sought must be 25.995 g. Again, if the lesser of two weights differing by one centigram is evidently nearer than the other to the weight desired, we substitute a figure 2 or a 3 for the 5 in the last place, or if the greater weight is more accurate, we write a 7 or an 8 in- stead. Any distinct information of this kind should always be recorded when possible, by means of a figure in the last place, even if that figure be ex- tremely doubtful ( 55). Closer estimates will hardly be justified in the case of a Nicholson's hydrometer. ^[ 7. Effect of Temperature on a Nicholson's Hydrom- eter. The temperature of the water in the jar is now taken. The water is then cooled with ice to about 10, and the weight required to balance the hydrometer is determined as before, with a new observation of temperature. Then the jar is filled with tepid water (at about 30) and the experiment is repeated. A comparison of the different results shows how much the buoyancy of water is affected by temperature. For this purpose the observations which we have now obtained at three different tem- peratures are to be represented graphically on co- ordinate paper by three points, A B and (7, as explained in 59, and through these points the 12 NICHOLSON'S HYDROMETER. [Exp. 2. curve A B C is to be drawn with a bent ruler. (See Fig. 7.) The ambitious student may supplement this ex- periment by using water hotter than 30 and colder than 10, also water at intermediate temperatures. He will thus obtain data for plotting a more com- FIG. 7. plete curve than that shown in the figure. This curve, if we neglect the expansion of the metal of which the hydrometer is composed, represents the relative buoyancy, or (see 64) the relative density, of water at different temperatures. 119.] WEIGHING WITH A HYDROMETER. 13 a EXPERIMENT III. WEIGHING WITH A HYDROMETER. ^| 8. Determination of Weight in Air by a Nicholson's Hydrometer. From the results of Experiment 2 it is possible to find (see 59) the weight necessary to sink a hydrometer to a given mark in water of any ordinary temperature. It is obvious that in all determinations with a Nicholson's hydrometer, the tem- perature of the water must be observed at the time of weighing. To find the weight of a body, place it in the upper pan (a, Fig. 8), and with it enough weights from the box to sink it to the same mark as before. Evidently less weight will be required than at the same temperature without the body, and the dif- ference will be equal to the weight of the body in question. ^[ 9. Reasons for Neglecting Corrections for the Buoyancy of Air. Since the air buoys up both the brass weights and the body used, the result of this experiment is what we call the apparent weight of the body in air ( 65). The amount of this buoy- ancy depends (see 68) in one case upon the density of the brass weights, in the other case, upon that of FIG. 14 NICHOLSON'S HYDROMETER. [Exp. 4. the body in question ; hence if these two densities are approximately equal, the air will exert nearly the same force in both cases. The result, obtained as we have seen by difference, will not therefore be affected to an appreciable extent. For the purposes of this and other experiments which follow, we choose ten steel balls, perfectly round and uniform in size, such as are used in the bearings of the front wheel of a bicycle. The density of these balls (7 -8) is not far from that of the brass weights (84), and it will be seen by reference to Table 21 that the correction for the buoyancy of air may be wholly disregarded. EXPERIMENT IV. WEIGHING IN WATER WITH A HYDROMETER. ^j 10. Determination of Specific Gravity by a Nich- olson's Hydrometer. The steel balls used in the last experiment are now to be placed in the lower pan of the hydrometer (L Fig. 9), which is lifted by the stem out of the water for this purpose. The instru- ment is then balanced with weights, and the tem- perature of the water observed as in the last two experiments. In lowering the hydrometer into the jar, care must be taken to remove with a camel's-hair brush all bubbles of air from the steel balls, as well as from the sides of the hydrometer, and also, of course, not to spill any of the balls. In the adjustment of weights SPECIFIC GRAVITY. lo the same precautions must be used as in the last two experiments. We have already obtained the weight of the steel balls in air (^[ 8) ; we find similarly their weight in water from the results of Experiments 2 and 4, and finally their apparent specific gravity (see 66). T[ 11. Use of the Methods of Substitution and Multiplication. It will be noted that in Ex- periment 3 the unknown weight of a body takes the place of a known weight of brass used in Experiment 2 ; the one is in fact substituted for the other. The method of finding the weight of a body by a Nicholson's hy- drometer is therefore essentially a method of substitution ( 43). This statement also applies to the determination of weight in water by the same instrument ; for the weight of a body in water is here substituted for a known weight of brass in air. The errors committed with a Nicholson's hy- drometer depend upon the peculiarities of the instru- ment itself, rather than upon the quantities weighed. We are in fact liable to the same error in weighing one bicycle ball as in weighing ten. The proportion which the error bears to the total quantity weighed is, however, diminished when this quantity is increased. The use of a large number of bicycle balls for the determination of specific gravity in Experiments 3 FIG. ( J. 1G NICHOLSON'S HYDROMETER. [Exp. 4. and 4 is a good example of the accuracy gained by the method of multiplication ( 39). ^[12. Corrections Disregarded in Experiment 4. In the last experiment we disregarded the effects of the buoyancy of air on the steel balls and on the brass weights, because these effects were so nearly equal, both being in air. Here, however, the balls are in water and the weights in air. There is, therefore, nothing to compensate for the buoyancy of air on the brass weights. It is seen by reference to 65 that 7 grams of brass are buoyed up by the air with a force of about 1 milligram ; and as a Nicholson's hydrometer can float only about 4 times 7, or 28 grams, the effect of buoyancy on the weights cannot be greater than 4 milligrams. This error may generally be disregarded in comparison with the errors of observation. The manner of applying a correction for the buoyancy of air is ex- plained in Experiments 8 and 9, also in 65-68. In calculating apparent specific gravity, no correc- tions need be taken into account; but the result should be expressed as the apparent specific gravity of a given body at a given temperature referred to water at a given temperature. The result will be affected somewhat by the density of the air, but hardly to a perceptible extent. The student is ad- vised, as a matter of habit simply, to note the condi- tions of the atmosphere in which his weighings are performed (see Experiment 5). IT 13.] THE BAROMETER. 17 :f EXPERIMENT V. ATMOSPHERIC DENSITY. ^[13. Determination of Barometric Pressure. The three conditions of the atmosphere which affect the results of physical measurement are barometric press- ure, temperature, and humidity. Let us first con- sider how barometric pressure is observed. A very rough but serviceable form of mercurial barometer consists simply of a glass-tube (a 5, Fig. 10), which, having been filled with mercury, 1 is inverted in a cistern of mercury Q (6). The mercury sinks in the closed end of the tube to a level a, above which there will be a nearly perfect vacuum. 2 As there is no pressure at a, to counter- act the atmospheric pressure be- low, the mercury stands in the tube at a level (a) above the level (i) in the cistern. It is found by experiment 3 that the atmospheric /. 1 pressure is transmitted through the cistern of mercury and the open end of the tube to a point, 5, on a level with the surface of the mercury in the cistern. The atmospheric pressure is accordingly determined by 1 The tube and the mercury must be perfectly clean and dry. For cleaning mercury, see Pickering's Physical Manipulation, I. 9. 2 The " Torricellian vacuum." See 62. FIG. 10. 18 ATMOSPHERIC DENSITY. [Exp. 5. the length, a 6, of the column of mercury which it sustains. 1 The distance a b can be measured by means of a graduated wooden rod, by which the tube is supported in a vertical position. The level b is first sighted in the ordinary manner with care to avoid parallax ( 25) ; and the reading thus found is subtracted from that of the level a, obtained in a similar manner (see 32). In the case of a standard mercurial barometer the lower end of the column of mercury should always be looked at first, else a considerable error is likely to arise ( 32) ; for even when the barometer ends in a large cistern of mercury, the level in this cistern must vary somewhat as more or less mercury rises into the tube. In some barometers this rise and fall is compensated by turning a screw (d, Fig. 11). This raises or lowers the mercury in the cistern, and when a certain steel or ivory point (a, Fig. 11), just touches its own reflection, the level of the mer- cury is known to be at the right height. When the lower end of the mercurial column in the tube has been thus ad- justed, the height of the upper end is usually read by a movable sight, pro- vided with a vernier ( 40). The lower edge of the sight is to be set on a level with the highest part of the mercurial column, so as to appear to be tangent to the meniscus or curved surface of the mercury (Fig. 12, a). To avoid par- 1 See 63. 1 14.] CORRECTIONS OF A BAROMETER. 19 allax ( 25), a double sight is frequently used, con- sisting of two edges in the same horizontal plane, one in front of, the other behind the mercurial column. The student should find by direct measurement whether the distance from the zero-point (a, Fig. 11), to the lower edge of the sight (a, Fig. 12) is indicated correctly upon the scale of the barometer. If the reading of the barometer is in inches, it may be reduced to centimetres conveniently by Table 16. FlG - 12 - Aneroid barometers are generally constructed so as to agree very closely with mercurial barometers. They will be found accurate enough for correcting the results of most physical measurements. If an Aneroid barometer is to be used, the student should compare its indication with that of a mercurial baro- meter, determined as explained above. ^[ 14. Corrections of a Barometer. A small quan- tity of air almost always finds its way sooner or later into the space above the mercury in a barometer (a, Fig. 10), where it causes a slight depression of the column. To test a barometer for air, we tilt the tube ab (Fig. 10) into a new position a f 5, being careful to keep the mercury in the cistern at a constant level, b, either by raising the cistern or by adding more mercury to compensate for that which flows into the tube. In the absence of air> the mercury should follow the horizontal line a a', and should completely fill the tube when the inclination is sufficiently in- creased. 20 ATMOSPHERIC DENSITY. [Exp. 5. A simple way of correcting for air in a barometer is to adjust the angle a' b a (Fig. 10) by trial, so that the space above a' is half that above a. By thus reducing the air to half its original volume, the pressure will be doubled; 1 hence a! will be as much below a as a is below its proper level. By measuring the difference between the levels, a and a', we find accordingly the correction for air. A correction of 2 or 3 mm. may be disregarded, as it will probably be offset by other corrections which the accuracy of the instrument will not justify us in considering. In case the correction is much larger than this, the ba- rometer should be refilled with mercury. The filling of a standard barometer should be attempted only by a skilled workman. Unless perfectly free from air, such a barometer is little better than the rough instru- ment shown in Fig. 10. In all exact readings of a barometer, the three following corrections are usually applied : (a) for ex- pansion, (b) for capillary depression, and (c) for the pressure of mercurial vapor. 2 The temperature of the mercury in a barometer is found by a thermometer beside it. Let t be this temperature, reduced if necessary to the Centigrade scale (see Table 39), and let h be the height in centimetres of the mer- curial column ; then the correction for expansion is .00018 ht, which is to be subtracted from the ob- served height. The object of this correction is to find This follows from the law of Boyle and Mariotte ( 79). 2 The reduction of a barometric reading "to the sea level " is not required for the purposes of physical measurement. 1 14.] COEKECTIONS OF A BAROMETER. 21 how high the mercury would stand if its temperature were Centigrade. Since 1 cm. of mercury when heated 1 Centigrade expands by the amount .00018 cm. (see Table 11), h cm. would expand h times as much ; and h cm. heated t would expand Jit times as much, whence we obtain the correction in question. At the ordinary temperature of a room (20), and at the barometric pressure, 75 cm., this correction for expansion would be .00018 X 20 X 75 cm. = 2.7 mm. It is therefore useless to read a barometer (as is often done) to tenths or hundredths of a millimetre, when no correction for temperature is made. The correc- tion given above may be applied to barometers with wooden or glass scales, the expansion of which may be neglected. When, however, the body of the in- strument consists of steel, the coefficient .00017 should be used instead of .00018 ; and if the barome- ter is mounted in brass or white metal, the factor .00016 will be still more accurate. These numbers represent the difference of expansion between the mercury and the scale by which it is measured. For more accurate values see Table 18 a. When the tube of a barometer is less than a centi- metre in diameter, there is found to be a perceptible depression of the mercurial column due to " capillar- ity," or "surface tension," the general nature of which will be investigated farther in Experiment 67. The internal diameter of the tube should be found if possible by measuring a plug which fits it in the part where the column of mercury ends (see a, Fig. 10). A different method of calibration will be considered 22 ATMOSPHERIC DENSITY. [Exp. 5. in Experiment 26. Wheii the internal diameter is known, the correction for capillarity may be found roughly from Table 18 b. Thus for a tube 5 mm. hi diameter, in which the height of the mercury menis- cus is unknown, the capillary depression may be taken as 1.5 mm. In various barometers which are con- structed so that the internal diameter cannot be measured, we generally assume that the instrument- maker has allowed for capillarity in adjusting his scale, and we therefore neglect this correction. It is customary, also, to neglect the effect of capillary phenomena in the cistern of mercury. Owing to the evaporation of mercury into the space above it in the tube of the barometer, that space is never quite empty. The quantity of mer- curial vapor which it contains is found to increase when the temperature increases, and also the press- ure which it exerts. To allow for the slight de- pression of the mercurial column due to this cause, Table 18 c has been constructed from the results of actual observation. Thus for a temperature of 20, we find that the mercurial column is depressed to the extent of 0.02 mm. by the pressure of its own vapor. We have found in a particular case that 2.7 mm. should be subtracted from the observed height of a barometer on account of expansion ; that 1.5 mm. should be added for capillarity and also 0.02 mm. to offset the pressure of mercurial vapor. The resulting correction is 1.18 mm., to be subtracted ; or let us say, 1.2 mm. nearly. The student who employs a mercurial barometer should find in the same way an If 15.] TEMPERATURE AND HUMIDITY. 23 average correction for it. If an Aneroid is used, such a correction is found by comparing one reading at least with the corrected reading of a mercurial ba- rometer. In the course of experiments which follow, readings of the barometer are needed only for slight corrections in the results of physical measurement. By applying to the barometer an average correction, much labor will be saved, and the error introduced will be insignificant. ^| 15. Determination of Atmospheric Temperature and Humidity. The temperature of the air of a room may be determined, with a sufficient degree of ac- curacy for most purposes, by an ordinary mercurial thermometer, the reading of which may be reduced from the Fahrenheit to the Centigrade scale by Table 39. The thermometer should be brought as near as may be practicable to the place where the tempera- ture is required. It should, for instance, be inside of the balance case in very delicate weighings. It must not, however, be exposed to the rays of the sun, nor for any length of time to the heat radiated by a lamp or by the human body. When the greatest accuracy is desired, the bulb of the thermometer should be protected from radiation to or from surrounding objects, by a shield of polished metal. The humidity of the atmosphere is most conven- iently determined by a class of instruments of which the hygrodeik is an example. The indications of these instruments depend upon the cooling produced by evaporation (see 88). It is found that when the bulb of a thermometer is covered with wet wicking 24 ATMOSPHERIC DENSITY. [Exp. 5. FIG. 13. (a, Fig. 13), its reading differs from that of an ordi- nary thermometer (6) by an amount depending upon the dryness of the air. When the air is completely saturated with moisture, as in a dense fog, there is no evaporation from the wet bulb, hence the two thermometers agree; if the air, however is heated, the fog disap- pears, evaporation begins, and the wet-bulb does not rise so high as the diy-bulb thermome- ter. On the other hand, when the air of the room is cooled sufficiently, either fog is formed or dew is precipitated on various objects ; and the two thermometers again agree. The temperature at which this occurs is called the dew-point, and is cal- culated from the readings of the wet and dry-bulb thermometers by reference to Table 15, or by a spe- cial mechanical device, for the operation of which directions are usually furnished by the instrument- maker. ^[16. Observation of the Dew-point. Unless a hy- grodeik is known to give accurate indications, the latter should be confirmed by a direct determination of the dew-point, as follows : a polished metallic vessel is partly filled with water, and as much ice and salt are added as may be necessary to make a film of moisture condense on the surface. The tem- perature at which this first occurs is just below the dew-point. Soon, however, the contents of the ves- T 17.] DEW-POINT. 25 sel become warmer through contact with the air, and the film begins to disappear. The temperature is now a little above the dew-point. By observing carefully a thermometer with which the cold con- tents of the vessel are continually stirred, the dew- point may be determined within two limits, differing by less than one degree. Care must be taken not to breathe on the metallic vessel, since the breath is much damper than the air of the room ; and as there is more or less evaporation from all parts of the human body, even the hand should be kept as far away as possible. ^[ 17. Relation of Relative Humidity to Dew-point. The actual amount of moisture in a given quantity of air has been determined by extracting it through the action of certain hygroscopic substances, such as chloride of calcium, and measuring the gain in their weight. It is found that hot air can hold more moisture without forming fog than cold air. We have a common instance in the air of a room which, though apparently dry while warm, deposits moisture upon the window-panes by which it is cooled. 1 The ratio of the amount of moisture actually held in the air (at a given temperature) to the maximum amount (which can be held at that temperature) is called the relative humidity of the air. The relations be- tween temperature, dew-point, and relative humidity do not follow any simple law ; but if any two of these quantities are given, the third may be found by 1 For a further illustration see list of Experiments in Elementary Physics, published by Harvard University, Exercise 22. ATMOSPHERIC DENSITY. [Exp. 5. referring to Table 15, containing the results of various experiments. It may be noted that the dew-point depends solely upon the amount of moisture in the air; that dry air has a lower dew-point and less relative humidity than moist air at the same temperature, while for a given dew-point the relative humidity increases with a fall of temperature, until fog is finally formed, or decreases as it becomes warmer until the air is prac- tically dry. It should also be noted that dry air is denser than moist air. We must regard the latter as a mixture of air, not with water, but with steam, which is only about two-thirds as heavy as air. Hence in Table 20 the correction for moisture is negative. ^| 18. Determination of Atmospheric Density by means of a Barodeik. From the temperature, pres- sure, and humidity of the atmosphere, the de- termination of which has been explained above, the density of air may be calculated by the data of Tables 19 and 20. Whenever great accuracy is de- sired this calculation FIG. 14. must be performed. For most purposes, however, the density of the at- mosphere may be found from a single observation of a barodeik (Fig. 14), the principle of which is spoken 1f 19.] MANIPULATION OF A BALANCE. 27 of in 71. It is important to compare the indication of the instrument in at least one case with the calcu- lated density of the atmosphere. A reading of the barodeik should accompany every weighing in which more than three figures are to be preserved, except when the pressure, temperature, and dew-point have been determined. EXPERIMENT VI. TESTING A BALANCE. TJ 19. Manipulation of a Balance. The delicacy of a balance depends upon the sharpness of the knife-edges (a and c, Fig. 15) from which the pans are suspended, also upon the sharpness of the central knife-edge (6) upon which the beam (a c) turns. In order that these edges may not become dull, the pans should be supported by some mechanical device at all times except when an observation is actually being taken. It is particularly important that they should be so supported when they are being loaded or un- loaded, or when the balance is liable to be jarred in any other manner. In an ordinary prescription bal- ance (Fig. 15), the pans rest upon the bottom of the case when the instrument is not in use. Such a balance is thrown into operation by turning a milled head outside of the case. The beam is thus raised as slowly as possible, so as not to injure the knife-edges by suddenly throwing weight upon them. It is not necessary in every case to raise the beam as far as it 28 THE BALANCE. [Exp. 6. will go. As soon as the pointer moves decidedly to one side or the other, the beam should be slowly lowered again. In other cases a prolonged observa- tion of the pointer must be made in order to decide in which direction the beam tends to incline. During such observations the beam should be raised to its fullest extent. Whenever accuracy is desired, the FIG. 15. door of the balance case should be closed, in order to cut off currents of air ; in fact, the door should never be opened except when the purposes of manipulation actually require it. This precaution is necessary to protect the instrument from moisture and dust, and is especially important when the air within the bal- ance case is kept artificially dry by chloride of cal- f 19.] MANIPULATION OF A BALANCE. 29 ciurn or other hygroscopic material. The glass case should be cleaned when necessary with a damp cloth, to avoid charging it with electricity. 1 Before weighing with a balance the case should be levelled and firmly supported, the scale-pans should be scrupulously cleaned and returned to their places, and any dust which may have collected on the knife- edges or their bearings should be cautiously removed with a camel's-hair brush. The beam is now thrown into operation by the mechanism already alluded to. If the instrument is correctly adjusted, the pointer attached to the under side of the beam will oscillate slowly and for some time through nearly equal arcs on either side of the central division of a scale (/, Fig. 15) directly behind it. If it tends to one side, that side is the lighter ; and bits of paper or tinfoil should be fastened to the scale-pan until an exact balance is established. 2 In loading the pans, pincers should be used as much as possible. In the case of the smaller weights, especially, contact with the fingers should be avoided. It makes no difference, theoretically, where the loads in the pans are placed ; but many practical difficulties will be avoided by keeping them as nearly as possible in the centre. Both loads should be at the same 1 By nibbing the glass at one side of a balance case with a piece of silk, a considerable error may be introduced into a weigh- ing. The student should be cautioned, in general, against the effect of charges of electricity on delicate instruments. An eyeglass rubbed on the sleeve has been known to cause serious errors in physical measurement. 2 See, however, first footnote, T 26. 30 THE BALANCE. temperature as the air within the balance case ; for though heat weighs nothing, a hot body may be lifted slightly by upward currents of hot air around it. With non-metallic loads we should avoid friction, which, as we have seen, may generate charges of electricity. When magnetic matter (as iron or steel) is to be weighed, all magnets ( 126) should be re- moved from the immediate neighborhood. In an actual weighing, the scale-pans should be prevented from swinging, both on account of currents of air and because of the irregular motion given to the pointer. ^[ 20. Method of "Weighing by Oscillations. The reading of a pointer is usually taken while it is in motion, since much time would be lost in waiting for it to come to rest, and even then friction might stop it somewhat on one side of its true position of equi- librium. While in motion the pointer swings first to one side of its position of equilibrium, then to the other. The furthest point reached in a given swing to the right or to the left is called as the case may be a right-hand or a left-hand turning-point. Owing to friction, each swing is smaller than the one before it ; hence the position of equilibrium is not exactly mid- way between any two successive turning-points. To avoid errors from this source we adopt the following rule : observe any ODD l number of consecutive turning- 1 The object of making an odd number of observations is that the first and last may be on the same side ; for in this case the turning- points on one side are on the whole neither earlier nor later than on the other side, and the gradual diminution of the swing affects each average alike. U 21. J WEIGHING BY OSCILLATIONS. 31 points ; find the average of those on the right and the average of those on the left ; add these averages alge- braically and divide by 2. The result is the point about which the oscillation is taking place, and at which the index tends eventually to come to rest. It is convenient for many reasons to call the middle scale-division number 10, not 0, since otherwise plus and minus signs must be employed. In practice it is sufficient to observe three consecutive turning- points of the index. It is frequently impossible to balance a given load exactly by any combination of weights which we are able to obtain. Let us suppose that with a weight, w, the index tends to rest at a distance from the middle-point equal to x scale-divisions ; while with the smallest possible addition of weight, a, it tends to rest on the other side of the middle-point and at a distance from it equal to y scale divisions. Then the exact weight indicated for the load, /, is (see 41), The quantity x + y is called the sensitiveness of the balance to the weight (#) under the load () ; and as it occurs in all exact estimations of weight by interpola- tion, it may be made properly the subject of further investigation. ^[ 21. Determination of the Sensitiveness of a Bal- ance. To test the sensitiveness of a balance with the pans empty, after carefully adjusting it as sug- gested in ^T 19, we add a small weight, let us say 2 eg. 32 THE BALANCE. [Exp. 6. to the left hand pan. Instead of swinging about the middle scale-division, which we have agreed to call number 10, it will swing about a new point corres- ponding, let us say, to number 12-6 on the scale. This would show that the balance is sensitive to the extent O f 12-6 10, or 2-6 divisions for 2 eg., or 1-3 divisions per eg., when the pans contain little or no load besides their own weight. This fact is recorded by making a cross (as in Fig. 16) on a piece of co-ordinate paper at the right of the number 0, repre- senting the load, and below the number (1-3) representing the sensitiveness in ques- tion. We now place, let us say, 20 grams in each pan, and find as before the sensitiveness per centi- gram. It will not necessarily be the same as when the pans are empty ; in fact, a difference is almost always observed. 1 The sensitiveness is then found with 50 grams in each pan, and finally with 100 grams in each pan. Thus, in an actual case, a balance which was sensitive with the pans empty to the extent of 1-3 divisions per eg., was affected to the extent of 1-6 divisions per eg. with 20 g. in each pan, 1*4 divisions 1 It will be shown in 1T 22 that the effect of a load on the sensitive- ness of a balance cannot be anticipated ; hence the student who records faithfully what he sees, not what he expects to see, will here as elsewhere in Physical Measurement, be likely to obtain the most accurate results. (See 30.) / / * 3 -4 j ' -6 -7 '8 \ So 60 t+0 20 \ \ x \ jXj \ ^ "Zs FIG. 16. H 22.] SENSITIVENESS OF A BALANCE. 33 per eg. with 50 g. in each pan, and 1-2 divisions per eg. with 100 g. in each pan. These results are re- corded, as before, by crosses in the proper places (see Fig. 16), and a curve is drawn by a bent ruler through these crosses. This curve enables us to find approximately the sensitiveness of the balance un- der any ordinary load by the method explained in 59. When we know the sensitiveness (s) of a balance to 1 eg., a single observation of the pointer is suf- ficient to determine exactly the weight indicated. If w is the lighter weight (in the pan toward which the pointer inclines) and x the number of scale- divisions between the resting point of the index and the middle of the scale, the load (I) indicated is found by substituting s for x -\- y and .01 for a in the formula of ^[ 20 ; or ^[ 22. Conditions on which the Sensitiveness of a Balance Depends. In order that a balance may move perceptibly under the influence of a very small weight added to either pan, the central knife-edge (6, Fig. 15) on which the beam turns must not only be sharp (^[ 19), but must pass nearly through the centre of gravity. If the centre of gravity is above this knife-edge, the balance will be "top heavy." This difficulty must be remedied by attaching a bit of sealing-wax to the pointer below the knife-edge i, or by lowering the centre of gravity in any other 34 THE BALANCE. [Exp. 6. manner. 1 If on the other hand the centre of gravity is too low, the balance will be too steady, and it will not respond sufficiently to a small change in the load. In this case it is necessary to fasten a small weight to the balance beam, somewhere above the knife- edge 5, or otherwise to raise its centre of gravity. When the balance-pans are loaded, new considera- tions come in. Since in all positions of the beam the loads hang vertically beneath their respective knife- edges, the result is the same as if they were concen- trated at those knife-edges. Let us suppose that the instrument has been adjusted so as to be sufficiently sensitive when the pans are empty. In order that it may remain equally sensitive when loaded, the three knife-edges must be in the same straight line, as m J., Fig. 17. If the two outer knife-edges wnich - --- /TF7\ FIG. 17. bear the loads (see a", c" in (7) are distinctly above the central knife-edge (6"), the combined effect of the loads will be towards unstable equilibrium ; or if the outer knife-edges (see a', c' in B), are below the central knife-edge (&'), the combined effect of the loads will be to steady the balance, and hence to diminish its sensitiveness. There are therefore three types to which a balance beam may belong, repre- 1 A movable screw or counterpoise is provider! in some balances for the purpose of raising or lowering the centre of gravity. 723.] RATIO OF THE BALANCE ARMS. 35 sented by the three diagrams, A, B, and 0. In the first, the load does not affect the sensitiveness, except in so far as friction may be concerned ; in the second, it lessens it ; in the third, it may increase the sensi- tiveness until the balance actually becomes u top heavy." A common balance may belong successively to all three of the types, (7, A, and B. Let us suppose that with the pans empty the extremities of the beam are bent upward, as in C. With a medium load, the beam may be straightened, as in A, and with a still greater load the ends may be bent downward, as in B. Such a balance would be more sensitive with a small load in each pan than when the pans were empty ; because a small load, being insufficient to straighten the beam, would raise its centre of gravity 1 as in C ; but when already heavily loaded, so that the beam is bent downward as in B, the further addition of weight would lessen its sensitiveness. The curious shape of the curve found in the last section (Fig. 16), is thus accounted for. ^[ 23. Determination of the Ratio of the Arms of a Balance. The balance is now readjusted if necessary as in ^[ 19, so that the pointer swings accurately about the central division of the scale when the pans are empty, and the 100 gram weight is balanced against its equivalent as before, only that small weights are added to one side or to the other to 1 A balance, though stable with a heavy or with a medium load, as well as when the pans are empty, may actually become "top heavy," with a small load in each pan. In such a case, the centre of gravity should be permanently lowered. 36 THE BALANCE. [Exp. 6. bring the pointer as nearly as possible to the central division, and the exact weight estimated as in ^[ 21, considering as the load, Z, that weight which is ap- parently the larger. The loads in the two pans are now interchanged, readjusted by the use of the small weights, and compared exactly as before. The pans being once more emptied, the pointer should swing about the central division, otherwise the balance must be readjusted and the process described in this sec- tion must be repeated until the equilibrium of the balance remains undisturbed. The object of testing the balance, as above, with equal weights in the opposite scale-pans, is to discover any inequality which may exist in the length of the balance arms (a b and b c, Fig. 17). Such an inequal- ity might seriously affect the accuracy of results, and we have no right to neglect it even in ordinary weigh- ings without some test similar to the one described. It is true that by the method of double weighing (see 44), errors due to the inequality of the balance arms may be eliminated ; but double weighings are sometimes impracticable, as in the case of a body of variable weight, or in a very long series of determina- tions. In such cases the inequality of the balance arms should be found by a careful and extended series of observations. For the purposes of this course of experiments, a single determination will suffice. The ratio of the balance arms is calculated therefrom as explained in the next section. ^[ 24. Calculation of the Ratio of the Balance Arms. If the arms of a balance are unequal, it is impor- 124.] RATIO OF THE BALANCE ARMS. 37 tant to know from which arm the unknown weight is suspended. To avoid the necessity of mentioning in each case the pan containing the load in question, it is customary to place the unknown weight at the left hand whenever a single weighing is to be made. In this way the known weight, consisting generally of several small pieces, is conveniently adjusted by the right hand. To find the proportion which the weight on the left arm always bears to the weight on the right arm, we need only a single comparison between two known weights. As these weights are inversely as their respective arms (see 113), the proportion in ques- tion is equal to the ratio of the right arm to the left arm. Thus if (in an extreme case) 101 grams in the left-hand pan balance 100 grams in the right-hand pan, the right arm must be y|}^ or 1.01 times as long as the left arm. All weights in the left-hand pan are therefore 1% greater than those which balance them in the right-hand pan ; hence to find the value of an unknown weight in the left-hand pan we multiply that of the known weight in the right-hand pan by 1.01. The ratio of the balance arms is in general that number by which the known weight must be mul- tiplied in order to find the unknown weight "which balances it. We usually require, as we have seen, the ratio of the right arm to the left arm. This is found by dividing a known weight in the left-hand pan by a known weight in the right-hand pan which balances it. The object of interchanging the two weights in 38 THE BALANCE. [Exp. 7 ^f 23, each nominally equal to 100 grams, is to avoid mistakes arising from a difference between the two weights in question. If no such difference exists, the interchange will not affect the result. Otherwise to find the ratio of the balance arms, we take the average of the two weights in the left-hand pan, and divide it by the average of the two weights in the right-hand pan. In taking these averages we accept the nominal values of the weights in question, any errors in which are practically eliminated by the method of interchange ( 44) here adopted. EXPERIMENT VII. CORRECTION OP WEIGHTS. ^ 25. Process of Testing a Set of Weights. The brass 1 gram weight is first balanced against all the smaller weights, which should together be equal to 1 gram ; then each 2 gram weight against the 1 gram plus the smaller weights; then the 5 gram weight against the two 2 gram weights plus the 1 gram ; then in. the same way the 10, 20, 50, and 100 gram weights, each against its equivalent. Whenever there are two ways of making an equivalent, that selection is made by which the fewest weights may be employed. (See 36, 2d ed.) The 100 gram weight is finally balanced against a standard. 1 In 1 The standard should be of the same material as the set of weights employed, that is, of brass; but if any other material is used, a cor rection must be made for the unequal buoyancy of the atmosphere upon the loads in the two pans. See 07 and Table 21. U 26.] ESTIMATION OF TENTHS. 39 each case, where two weights are balanced, the differ- ence between them is estimated by the method of vibration (^[ 20), and recorded as will be explained below. To avoid corrections named in the last ex- periment, the method of double weighing is used in every case. ^f 26. Estimation of Tenths in "Weighing. In a long series of weighings, as in testing a set of weights, it is hardly thought to be advisable (see, however, 33) to record each turning-point of the index as in ^[ 20. The student who wishes to make any extended use of the balance should learn to estimate correctly the point of the scale about which the index is swinging, and hence the number of divisions from the middle of the scale * to the point where the index tends to rest; to carry this number in the head while finding by inspection of figure 16 (see ^[ 21 and 59) the sensitiveness of the balance under the load in ques- tion, 2 and to divide mentally the number thus carried in the head by that representing the sensitiveness of the balance, or the effect of 1 eg. (See general rules for interpolation, 41.) He will thus find the frac- tion of a centigram necessary to make the index swing about the middle-point of the scale, and will 1 Instead of adjusting the balance as in U" 19, so that the index may swing about the middle-point of the scale, the advanced student may often prefer to observe accurately the point about which the index actually oscillates when the pans are empty, and to measure all distances from this point. 2 It is sometimes quicker to add one centigram to the lighter pan, and thus to re-determine the sensitiveness. In many cases the sen- sitiveness may be recalled from memory with a sufficient degree of exactness. 40 THE BALANCE. [Exr. 7. record the number of milligrams nearest to that frac- tion with the proper algebraic sign. Thus if with a weight marked 10 g l in the left- hand pan and with 10 g 2 in the right-hand pan, the index swings about a point corresponding to 10'3 of the scale, that is, 0-3 divisions to the right of the middle-point, and if the sensitiveness of the balance with a load of 10 grams is about 1-5 divisions per centigram (see Fig. 16, ^[ 21), the weight 10 ^ is clearly heavier than 10 g 2 by 0-3 -^ 1-5 = | eg. or 2 mgr. We record such an observation as follows : 10 g i = 10 g. 2 -f 2 mgr. In the same way we enter the result of placing 10 g L in the right-hand pan and 10 g 2 in the left- hand pan ; and if there is any difference, we find the average excess of 10 g : over 10 # 2 , or the reverse. ^[ 27. Calculation of the Corrections for a Set of Weights. Any one familiar with algebra can find the relations existing between the different weights of a set from a series of equations obtained as in the last section. The following suggestions may how- ever be useful. Call the value of the 1 gram weight G : find the total value of the smaller weights (100 eg.} in terms of this. For instance, let 100 eg. = G + 1 mgr. Then find the value of the 2 gram weights, 2 g l and 2 g a in terms of G. If for example, 2^, = 100 eg. -f G 1 mgr., we find, substituting for 100 eg. its value, G + 1 mgr., 2^ = + 1 mgr. + G 1 mgr. = 2 G; t27.] CORRECTION OF WEIGHTS. 41 and if still further, it has been observed that 20, = 2&-f 2w0r., we find similarly 2fc=2G-f2*itfr. Again, if by observation we have 5#=2G+2G+2 mgr. + G + 1 mgr. = 5 In the same way we find the values of all the weights in terms of G, until we come finally to the standard. Knowing the standard in terms of G, we find G in terms of the standard. The corrected value of G should be expressed in grams and carried out to five places of decimals. Substituting this value in all the equations, we obtain finally the correction in mgr. for each weight belonging to the set from 1 gram upwards. This method of framing and reducing equations is not peculiar to a set of weights. The student may substitute for it, if he prefers, the correction of a set of standard electrical resistances, which he will learn how to compare in Experiment 87. The same method may be applied to any other standards capable of being arranged like a set of weights, so that each one may be compared with an equivalent made up of the others below it. The general principle by which such a standard set is corrected is one of the best illustrations of the method of multiplication ( 39) upon which nearly all measurements are founded. 42 THE BALANCE. [Exp. 8. EXPERIMENT VIII. WEIGHING WITH A BALANCE. ^[ 28. Determination of Weight in Air by a Balance. The apparent weight of a body in air may be found approximately, as has been explained in Ex- periment 1, by placing it in one pan of a balance the left being understood unless otherwise stated (see ^[ 24) and finding by trial (^[ 2) the requisite number of weights to counterpoise it. The accurate determination of weight in air differs from this rough method chiefly in the delicacy of the instrument em- ployed, and in the consequent care of manipulation (see ^[ 19). In this, as in all other accurate deter- minations with the balance, unless otherwise stated, it is assumed that the method of weighing by oscil- lations is employed (^[ 20). The object recommended for this experiment is a glass ball, the weight of which will be needed later on in the course. To prevent it from rolling out of the pan, it may be set in the middle of a small ring of known weight, which we will suppose to be coun- terpoised with one of equal weight in the opposite pan. It is necessary in this experiment either to know the ratio of the balance arms (see ^[ 23), or to employ the method of double weighing ( 44) as in Experi- ment 7. The density of air must also be determined by an observation of the barodeik (^[18), or by an observation of the atmospheric pressure, temperature, T 29. THE HYDROSTATIC BALANCE. 43 and humidity (*f[^[ 13-15). We must also know the material, and hence approximately the densities of both the object weighed and the weights with which it is counterpoised. These densities may be found with a sufficient degree of accuracy by referring to Tables 8-11. The correction of apparent weights to vacuo is then made as explained in 68. EXPERIMENT IX. THE HYDROSTATIC BALANCE, I. *J[ 29. Determination of the Density of Solids by the Hydrostatic Balance. An arch is placed over a balance pan as in Fig. 18, so as not to inter- fere with its free vibration ; and on the middle of the arch is set a beaker. The glass ball weighed in the last ex- periment is now bound in a net- work of fine wire and suspended by a single strand from the hook of the balance, so as to clear the bottom of the beaker. The latter, being moved if necessary so that its sides may not touch the ball, is filled with a quantity of distilled water sufficient to cover, 1 in all positions of the balance, both the ball and its network of wire. All bubbles of air 1 A small loop of wire, projecting above the surface, may com- pletely ruin a determination. FIG. 18. 44 THE HYDROSTATIC BALANCE. [Exp. 9. clinging to the ball, or wire, must now be removed with a camel's-hair brush. The suspending wire, being likely to attract grease or other foreign matter which repels water, is cleaned if necessary, so that it may be kept wet for a distance of about one centi- metre above the level of the water, by the continual oscillation of the balance. The capillary phenomena already noticed in ^[ 5 are thus reduced to a small and nearly constant amount. 1 By these adaptations the instrument which we employ has been completely transformed into a " hy- drostatic balance," by which the weight of the ball and wire in water may now be found, as in the last experiment, by counterpoising it with weights in air (see Fig. 15, ^[ 19). The method of weighing by oscillations is not, however, recommended in the case of a hydrostatic balance ; but rather a direct obser- vation of the pointer in its position of equilibrium, which, owing to fluid friction, is quickly reached. Apart from friction, the sensitiveness of a hydro- static balance is always somewhat less than that of the same balance when used for measuring weights in air, 2 and must therefore be re-determined by adding a centigram to the smaller of the two loads when nearly balanced and observing the result (see ^[ 21). In this, as in all experiments with the hydrostatic 1 The use of spirits of wine to diminish still further the capillary action (Trowbridge, " New Physics," page 17), is not recommended to beginners, on account of the danger of its mixing with the water and thus affecting its density. 8 The variable amount of water displaced by the suspending wire tends to increase the stability of the balance. f 29.] THE HYDROSTATIC BALANCE. 45 balance, the temperature of the liquid should be observed both before and immediately after finding the weight of a solid in it. The weight of the wire in water must be found separately in the same manner and under the same conditions as before. 1 The ball is removed from the network of wire so as to leave the latter undisturbed in so far as possible, and water is added to the beaker in order that the same amount of wire may be sub- merged in each case. It may even be necessary, if a coarse wire is used, to adjust the level of the water exactly to a given mark, and if the network is bulky, to raise or lower the. temperature of the water to the same point as before. The apparent weight of the ball in water is found by subtraction, and reduced to vacua by the principle of 67. The difference between the apparent weights in air and in water gives the apparent weight of wa- ter displaced ( 66), and hence the volume displaced (see Table 22). The difference between the weight of the ball in vacua (^[ 28) and its weight in water (reduced to vacua as explained above) gives, by a strict interpretation of the Principle of Archimedes ( 64), the weight in vacua of water displaced, and hence also its volume (by Table 23). We have thus two methods of calculating volume, of which the first is more generally useful, as it does not require any previous reduction of weights to vacua; but the 1 Precautions similar to those which follow are necessary when- ever a method of difference is employed. For further illustration 46 THE HYDROSTATIC BALANCE. [Exr. 10. second is more rigorous, because, depending upon weights in vacuo, the results will not be affected by variations of apparent weight due to changes in at- mospheric density. The latter should therefore be employed when any considerable time elapses between the determinations of weight in air and in water. The density (or average density) of the ball is finally calculated (see ^f 1) by dividing its weight in vacuo by its volume. (See ^[ 4, also 68.) EXPERIMENT X. THE HYDROSTATIC BALANCE, II. ^[ 30. Determination of the Density of Liquids by the Hydrostatic Balance. The experiment consists essentially of a repetition of Experiment 9, substitu- ting, however, for distilled water some other liquid of greater or less buoyancy. Various modifications of this experiment may be necessary according to the nature of the liquid used ; for instance in the case of strong acids, platinum wire must be substituted for iron, which would be speedily dissolved, and even platinum cannot be used in aqua regia. To avoid fumes in the balance case, the suspending wire is sometimes carried down through a series of small holes to a beaker below. To avoid evaporation, in the case of volatile liquids, the beaker should alwa} T s be covered with cork or cardboard perforated for the suspending wire. The same precaution should be taken when moisture is H31.] THE HYDROSTATIC BALANCE. 47 likely to be absorbed. In some liquids scarcely any bubbles are formed ; in others, such as glycerine, it may take hours to remove them, though their forma- tion may be prevented if the glycerine is poured in a continuous stream down the sides of the beaker. In most liquids the effects of temperature are greater than in the case of water (see Table 11), hence the thermometer must be read with the greatest care. It is well to warm or cool the liquid (and hence also the ball) to the temperature of the water in Experi- ment 9, to avoid all corrections for temperature. ^[31. Calculation of the Density of Liquids by the Hydrostatic Method. We find in the same way as in Experiment 9, the apparent weight of the ball in the liquid, allowing for the wire as before ; and from this we subtract the weight of air displaced by the brass weights (see 67), to find the true weight of the ball in the liquid. The difference between its true weight in the liquid and that in vacua, already found (^[ 28), is equal to the weight in vaciw of the liquid displaced. This follows from the Principle of Archimedes ( 64). The volume of liquid displaced is of course equal to the volume of the ball, which will not differ per- ceptibly from the value previously determined (see end of ^[ 29) if the temperatures of the two experi- ments are nearly the same. If this is not the case, it is necessary to allow for an expansion or contrac- tion of the glass, at the rate of about one part in 40,000 for every degree Centigrade. (See Table 8 b and 83.) 48 THE HYDROSTATIC BALANCE. [Exp. 10. The weight in vacua of the liquid displaced is finally divided by its volume to find its density. The weight in vacua may be checked by calculating the apparent weight of the liquid displaced, as in Ex- periment 9, then reducing at once to weight in vacua by applying the necessary factor from Table 21, as explained in 68, using the density already calcu- lated. This latter method is slightly inaccurate, as has been stated before (^[ 29), on account of its dis- regarding variations of atmospheric density during the course of experiments. In determining the density of water by the hydro- static balance, the weight displaced may be found as in Experiment 9 or 10 ; but the volume displaced cannot be calculated in the manner explained above, because the tables which we employ themselves de- pend upon the density of water. It is necessary to calculate the volume of the solid immersed from actual measurements of its dimensions 1 (see ^[ 1). By this method, essentially, with the aid of instru- ments of precision, accurate determinations of the density of water have been made (see Table 25). The student will have an opportunity in Experiment 19, to confirm these determinations within the limit of accuracy of the instruments which he employs. 1 The volume, , of the glass ball may be calculated from its diameter, d, by the formula, v = -5236 d 3 . In place of the glass ball we may use, for purposes of illustration, the rectangular block whose volume has already been determined in Experiment 1. If it floats in water, a lead sinker may be attached to it. The sinker must remain in place after the block is removed, in order that its weight may be allowed for. A spring balance maj' be used to find roughly the weight of water displaced. See Exercises 7-10 in the Descriptive list of Experiments in Physics published by Harvard University. - 32.) CAPACITY OF VESSELS. 49 EXPERIMENT XI. CAPACITY OF VESSELS. ^[ 32. Determination of the Capacity of a Specific Gravity Bottle. Any bottle with a solid stopper of ground-glass may be used for finding the specific gravity of liquids ; but when solids are to be intro- duced, one with a wide mouth will be needed. The capacity of the bottle is determined in the following manner. The bottle is first washed in perfectly pure water, then dried with a cloth inside and out, and afterwards still more thoroughly dried with a hot air- blast. 1 The weight of the bottle is found within a centigram, then the bottle is alternately dried and weighed until by the agreement of two successive weighings, the drying is known to be complete. The last weight found, if confirmed by the method of double weighing as in ^[ 28, is the apparent weight of the bottle in air. It is understood that the stop- per is always weighed with the bottle. In this case, it should be placed in the scale-pan beside the bottle, so that the density of the air may be the same inside and out. The bottle, which will be warmed by the hot air-blast, must be allowed time to cool to the temperature of the room before the weighing is completed, since otherwise currents of hot air might seriously affect the result (see ^[ 19). 1 When a hot air-blast cannot be had, the bottle may be dried by rinsing it out several times with a small quantity of alcohol, and exposing it for a few minutes to a draught of air. 4 50 SPECIFIC GRAVITY BOTTLE. [Exp.il. The bottle is then filled with distilled water at an observed temperature, not far from that of the room ; then closed in such a manner (see Fig. 19) as to allow all bubbles of air to escape. 1 The outside of the bottle is then carefully dried with a cloth or blotting-paper. The weight is again found with the same degree of ac- curacy as before, and immediately afterward the temperature of the water and the density of the air The difference between the two apparent weights of the bottle con- taining air and water, respectively, is equal to the apparent weight in air of the water which it contains ( 66) ; this weight of water multiplied by the space occupied (at the higher of the two observed temperatures, see ^[ 33) by a quantity of water weighing apparently 1 gram (in air of ttte observed density, see Table 22), gives the total space occupied by the water, or in other words the capacity of the bottle at the observed temperature. ^[ 33. Effects of Varying Temperature on a Specific Gravity Bottle. It is hardly necessary, in the experi- ments which follow, to allow for the expansion of the glass bottle due to changes of temperature which 1 If the shape of the stopper makes this impossible, it must be altered by grinding or by filling up any hollows in it with paraffine or other material not acted upon by ordinary liquids. In this case the weight in air must be re-determined. 133.] EFFECTS OF TEMPERATURE. 51 it is likely to undergo. 1 In a laboratory, maintained as it should be at a nearly constant temperature, these changes will be slight. Unless, however, spe- cial precautions are taken to keep the water in the bottle at a constant temperature, serious errors are likely to arise. These errors will be still greater in the case of certain other liquids which we shall em- ploy. The expansion of alcohol, for instance, will be found to be several hundred times as great as that of glass (see Table 11). Let us first suppose that the liquid which fills a closed bottle is gradually cooling, and hence in the process of contraction. A bubble will soon be formed. This need not, however, give rise to apprehension if the initial temperature (at which the bottle was filled) has been correctly observed; for the weight of the liquid will not be changed by its contraction, and the bubble weighs practically nothing. We may there- fore determine the weight of a liquid which fills a bottle a,t an observed temperature, after it has fallen below that temperature. Now, let us suppose that the liquid is growing warmer; and hence, expanding, that it forces its way out by the stopper, yet clings to the bottle. Unless the liquid is volatile or hygroscopic, 2 its weight 1 The capacity of a vessel increases by the same amount as the volume of a solid of the same material which would exactly fill the vessel. In the case of glass, this increase is at the rate of about 1 part in 40,000 per degree Centigrade. 2 Hygroscopic liquids, such as sulphuric acid or chloride of cal- cium, should be slightly warmed before the experiment, so that they may be weighed while cooling. 52 SPECIFIC GRAVITY BOTTLE. [Exp. 11. will be unchanged, and hence may be determined at leisure. If, however, the liquid evaporates immedi- ately (as many liquids do) on contact with the air, there will be a continual loss of weight. In such cases, we must find the temperature as nearly as pos- sible at the time of weighing, when it will be seen that the quantity of liquid weighed exactly fills the bottle. In practice, both the initial and final temperatures are usually observed ; the former just before the in- sertion of the stopper, the latter immediately after completing the weighing. We notice that with a non-volatile liquid, the initial temperature is always required; and the same statement applies to a volatile liquid which is cooling ; but with a volatile liquid in general it is the maximum temperature which we wish to determine. In no case do we take the mean of the two temperatures before and after the ex- periment. The liquids which we employ should be warmed or cooled if necessary, so that they may be nearly at the same temperature as the room ; since otherwise the rapid changes of temperature which must ensue ( 89) would make an accurate observation of the thermometer impossible. Errors in weighing might also be introduced, owing to currents of hot or cold air (^[ 19). In the case of certain liquids (as ether) which are apt to become cold through evaporation, 1 1 Care must be taken in general to prevent evaporation; and especially in the case of impure liquids, tlie strength of which would be affected by the escape of the more volatile ingredients. f 34. J DISPLACEMENT. 53 there is danger that moisture may be condensed on the sides of the containing vessel (see ^[ 17). Par- ticular care must be taken in the case of water, when below the temperature of the room ; lest through the humidity of the air or from other causes it should fail to evaporate as fast as it is driven out of the bottle. Any moisture collected around the stopper should be removed with blotting-paper before making a final adjustment of the weights. EXPERIMENT XII. DISPLACEMENT I. ^[ 34. Determination of Displacement by the Specific Gravity Bottle. The experiment consists essentially of a repetition of Experiment 11, with a bottle al- ready partly full of sand, or any other substance insoluble in water. The capacity of the bottle for water is evidently less than before by an amount exactly equal to the space which the sand takes up ; hence the latter can be found by subtracting the new capacity from the old. This method of deter- mining volume is especially convenient in the case of powders, which cannot easily be suspended from a hydrostatic balance. Certain modifications of the methods used in Experiment 11 are introduced when finely divided substances are employed. Even with sand consid- erable difficulty may be found in removing the bubbles of air which cling to it under water. By 54 SPECIFIC GKAVITY BOTTLE. [Exp. 12 continual shaking with water in a well-stoppered bottle, this air may generally be freed from the sand. 1 To obtain dry sand, it should be heated before the experiment to a temperature above 100. The same process may be used to dry various powders not easily melted or decomposed by heat; but others require special precautions belonging to the province of Chemistry rather than Physics. It may be observed that the apparent weight of the solid used in this experiment is incidentally determined ; for we have only to subtract from the apparent weight of the bottle with it that of the bottle without it as found in the last experiment. The density of the solid may therefore be calculated as in Experiment 9. ^[ 35. Illustration of the Principle of Archimedes, To understand what is meant by the water displaced by a solid, the bottle may be filled with water as in Experiment 11, then the solid may be introduced ; water will be literally displaced, and if the whole quantity thus driven out of the bottle could be col- lected and weighed, we should have a direct measure- ment of the water displaced lay the solid. In practice we prefer to find this by difference. If we call s the apparent weight of the sand, b that of the bottle, w that of the water which fills it, and d that of the water displaced by the sand, the weights observed are (1) b and (2) b -\- w in Experiment 11, 1 An air-pump greatly facilitates the process, but unless special precautions are taken the water is apt to bubble over into the re- ceiver and to find its way into the valves of th pump. f 35.] DISPLACEMENT. 55 (3) b + s and (4) b -f s + w> d in Experiment 12. The apparent weight of water which fills the bottle is the difference between the first and second obser- vations, or (2) (1), but when the sand is already in the bottle the quantity of water required is the difference between the last two observations, or (4) (3) ; hence the quantity displaced is [(2) (1)J -[(4) (3)]. Now the weight of the sand in air is evidently the difference between the first and third observations, or (3) (1) ; its apparent weight in water is the dif- ference between the second and fourth, 1 or (4) (2) ; its loss of weight in water is therefore [(3) (1)] [(4) (2)]. This is seen by comparison to be identical with the expression above for the weight of water displaced. The student who finds difficulty in realizing how the apparent weight or loss of weight of a solid in water can be found by the specific gravity bottle may repeat these measurements with a hydrostatic balance, using a cup to hold the sand in place of the network of wire employed in Experiment 9 to hold the glass ball ; or he may find the weight and loss of weight in water of the steel balls used in Experiment 4 by means of the specific gravity bottle. The Prin- ciple of Archimedes ( 64) states that loss of weight 1 In both observations we have the same weight of the bottle, and the same hj'drostatic pressure of the water upon the bottom or sides of the bottle ( 63) ; the only difference is the downward pressure of the sand, which is present in (4) and absent in (2). This pressure exerted under water is what we call the weight of the sand in water. 56 SPECIFIC GRAVITY BOTTLE. [Exp. 13. in water (which we think of as determined by hydrostatic methods) is equal to the weight of water displaced (which we think of as determined by a specific gravity bottle). The agreement of the results obtained by hydrostatic methods with those from the specific gravity bottle may serve therefore either as an illustration of this principle or as a mutual confirmation of these results. EXPERIMENT XIII. DISPLACEMENT II. If 36. Determination of the Volume and Density of Solids Soluble in Water. When owing to the solubility in water of the substance employed, the method ex- plained in the last experiment cannot be applied, it remains only to find some other fluid of known den- sity in which that substance is insoluble. The vari- ous products of the distillation of petroleum are especially suited to this purpose, since they dissolve few (if any) ordinary substances which are soluble in water. We may occasionally, with great care, use a saturated aqueous solution of the substance whose density is to be determined, or a liquid which has been allowed to act chemically upon an "excess" of that substance, since in either of these cases the liquid will have no further action on the solid. Gases may also be employed ; but on account of the diffi- culty of measuring their weight correctly even by the most delicate balances, it is customary to estimate T37.] SPECIFIC VOLUMES. 57 the quantity present by a direct or indirect measure- ment of its volume. 1 Owing, however, to the ten- dency of certain substances to absorb large quantities of gas, all such methods may lead to erroneous and even absurd results. For sake of simplicity we will choose the liquid whose density has been determined in Experiment 10, and for the solid some substance insoluble in that liquid ; and in order that the density of the liquid may be the same as before, it should be warmed or cooled if necessary to the temperature observed in Ex- periment 10. With such a solid and liquid, Experi- ment 12 is to be essentially repeated. ^[ 37. Calculation of Volume and Density by the Use of Specific Volumes. We have already seen how the weight of water displaced by a solid may be found either by the hydrostatic balance (Experiment 9) or by the specific gravity bottle (Experiment 12). By the same methods we may obtain the weight of any other fluid displaced by a solid. We have al- ready applied this principle in Experiment 10 for determining the density, of a liquid. Knowing the weight in grams and the number of cubic centimetres displaced, we found by division the weight of 1 cu. cm. It would have been equally simple to inter- change the divisor and dividend, and thus to find the space in cu. cm. occupied by 1 gram. This is some- times called the specific volume of a liquid. The mutual relations existing between the weight 1 For a description of the " Volumenometer," see Trowbridge's New Physics, Experiment 31. 58 SPECIFIC GRAVITY BOTTLE. [Exp. 14. w, the volume v, the density c?, and the specific vol- ume s, of any substance are given by the equations . s = -, v w s, etc. d The specific volume is therefore technically the " re- ciprocal " of the density. To find it we divide unity by the density already determined in Experiment 10, or by that which we may find from Experiment 14. We have already used specific volumes in Table 23 (see 1" 29), and we know that the weight in vacua of the liquid displaced, multiplied by its specific vol- ume, 1 gives the actual volume displaced, which is of course equal to that of the solid causing the displace- ment. The volume of the solid enables us to reduce its apparent weight to vacuo ( 67), and hence to cal- culate its density ( 68). EXPERIMENT XIV. DENSITY OP LIQUIDS. IF 38. Determination of the Density of a Liquid by the Specific Gravity Bottle. We have already found the weight of a bottle containing water and air, and we have calculated its capacity ; it remains only to find its weight when filled with any other fluid, in order 1 The student should bear in mind that the specific volume here employed is the space occupied by a quantity of liquid weighing 1 gram in vacuo, not that which weighs apparently 1 gram in air. True specific volumes must be multiplied by true weights in vacuo to find actual volumes. Apparent specific volumes (see Table 22) are in- tended to give the same result with apparent weights in air. 139.] THE DENSIMETER. 59 that the density of that fluid may be determined. For the purpose of comparison we will choose the liquid already used in Experiments 10 and 13, and warm or cool it, as nearly as may be convenient, to the temperature of those experiments. The actual tem- perature should be observed for reasons explained in If 38, both before and immediately after weighing. The barodeik should also be read, in order to make sure that no great change has taken place in the course of our experiments with the specific gravity bottle, since otherwise its apparent weight in air must be re-determined. The apparent weight of a quantity of alcohol suffi- cient to fill the bottle is found by subtracting that of the bottle with air from that of the bottle filled with alcohol, and is reduced to vacua as explained in 67. The density is then calculated by dividing the weight in vacua by the capacity of the bottle, from IT 32. The strength of the alcohol is finally found by ref- erence to Table 27, using a process of double inter- polation (see 58). The strength of the alcohol may also be calculated from the data of Experiment 14 ; and even if the temperatures in Experiments 10 and 14 differ considerably, the two results should agree in respect to strength. EXPERIMENT XV. THE DENSIMETER. ^[ 39. Hydrometers and Densimeters. There are various kinds of hydrometers employed in the arts. 60 DENSITY OF LIQUIDS. [Exp. 15. Nicholson's has been already described, and is the type of a "hydrometer of constant immersion ;" that is, one which in use is always made to sink in a liquid to a given mark. A common glass hydrometer is, on the other hand, an example of "variable immer- sion." The distance it sinks in a fluid depends upon the density of the fluid, and is read by a scale at- tached to the stem of the instrument. The scales used in the arts are generally arbitrary. The principal ones are those invented by Baume', Beck, Cartier, and Twaddell, which are compared in Table 40 with a scale of density. The instruments most convenient for scientific purposes carry a scale which indicates at once the density of the liquid, and hence bear the name of densimeters. The sensitiveness of a densimeter evidently depends upon the smallness of the graduated stem, compared with the whole displacement of the instru- ment ; but if we make the stem too small, a single hydrometer of the ordi- nary length can cover only a very lim- ited range of densities. A set of three instruments is often used, one for liq- uids lighter than water, one for liquids heavier than water, and one for liquids of intermediate density. There are also sets of twelve or more hydrometers, covering together the whole range of densities from sulphuric acid (1.8) to ether (0.7). With these great accuracy and rapidity may be 140.] THE DENSIMETER. 61 attained, even without applying any of the ordinary corrections ; 1 but if rapidity be the chief object, a single instrument with a " specific gravity scale" will be found most convenient. Such a one is often called by dealers a " Universal hydrometer " (see Fig. 20). The errors of such instruments are not so great as one might expect, considering that the scales are printed in quantities from originals none too care- fully made, fitted to tubes of by no means uniform bore, regardless within certain limits of their size, and fastened to these tubes at a point too high or too low, as the case may be. Still, even if the read- ing in water is found to be nearly correct, considerable errors may be discovered in other parts of the scale. As these errors depend largely upon the calibre of the tube, the process of correcting them may be properly called calibration ( 36). ^[ 40. Calibration and Use of a Densimeter. The reading of the instrument is taken while floating successively in at least three standard liquids of known density, such as water, alcohol, and glycerine (see Tables 25-27), then in a number of other liquids whose density is to be determined. As with a Nich- olson's hydrometer, the under surface of the liquid is (when possible) used as a sight (see Fig. 6, ^f 6) ; and the same precautions are taken to avoid friction against the sides of the jar, and the effects of capil- 1 It should be remembered that changes of atmospheric density influence only that portion of a hydrometer which is above the liquid, and hence will not generally affect even the fourth place of decimals. The effect of a narrow range of temperature in changing the volume of a glass hydrometer is equally unimportant. 62 DENSITY OF LIQUIDS. [Exp. 15. lary action due to the stem's becoming dry near the surface of the liquid. Both the densimeter and the thermometer (which is invariably read in every ob- servation) must be washed after immersion in each liquid, either under the faucet or in three changes of water; they should also be carefully dried before immersion iu a new liquid ; otherwise more or less dilution or mixture is sure to take place. The cor- rections of the densimeter are then calculated and applied as explained in the next section. ^[ 41. Treatment of Corrections by the Graphical Method. Correction and error are by definition ( 24) equal and opposite. ]f the observed value of a quantity is greater than its real value, we say that the error is positive, the correction negative. Thus, by subtracting the observed from the tabulated densities of water, alcohol, and glycerine at a given temperature, we find the several corrections for the instrument by which these densities were observed. The correction of an instrument will gen- erally vary according to the reading in question ; hence, to find the correction for every reading, it is necessary to con- struct either a table of corrections or a curve. Thus, in Fig. 21 the three points indicated by crosses represent (see 59) cor- rections of a particular densimeter corresponding to c 02 01 CC 01 1 BALANCING COLUMNS. three densities : namely, for alcohol, density 0.80, correction .004 ; for water, density 1.00, correc- tion .002 ; for glycerine, density 1.25, correction -{-.004. The curve drawn by a bent ruler through the crosses enables us to find approximately the cor- rection of this instrument for all intermediate densi- ties by the general rules of the graphical method ( 59). Thus for an ammoniacal solution of the density 0.9 or thereabouts, the correction would be not far from .003. Corresponding corrections should be applied to each of the liquids whose density has been determined by means of the densimeter. EXPERIMENT XVI. BALANCING COLUMNS. If 42. Determination of Density by Methods of Balancing Columns. The ordinary method of balancing col- umns is illustrated in Figure 22. Some mercury, for instance, is poured into a U-tube, then into the longer arm some water. Suppose the mer- cury is thus forced up to a level, 5, in the shorter arm, and down to a level, , is to be determined, w the length of the column of water which balances it in vacuo, and d the density of this water at the observed temperature (see Table 25), we have, solving the inverse proportion mentioned above, . IT 44.] DENSITY OF GASES. 67 EXPERIMENT XVII. DENSITY OP AIR. ^[ 44. Determination of the Density of Air. A stout flask provided with a stop-cock (Fig. 24) is made thoroughly dry (see ^[ 32), and weighed with the stop-cock open. The flask is then connected with an air-pump, and as much air as possible is exhausted. The stop-cock is now closed ; and the flask, having been disconnected from the air-pump, is re-weighed. It should be left on the balance long enough to prove that there is no perceptible gain of weight from leakage of air into it, then quickly opened under water as in Fig. 25. The stop-cock is closed by some mechan- ical contrivance while the flask is still completely submerged ; then the flask is dried outside and weighed with the water which has entered. The temperature of the water is now observed. Finally the flask is rilled completely with water and re-weighed. When all these obser- vations have been recorded, an ob- servation of the barodeik (see ^[ 18) is made for purposes of comparison. Having found the proportion of air exhausted, we calculate its density, as explained below. FIG. 24. FIG. 25. 68 DENSITY OF GASES. [Exp. 17. ^[ 45. Theory of the Partial Vacuum. When a flask from which the air has been partially exhausted is opened under water as in Figure 25, the water is forced inwards until the residual air is sufficiently compressed to resist the atmospheric pressure from outside. If the temperature is constant, as will be essentially the case when the flask is surrounded by water, the pressure depends chiefly on the density (see 78) ; hence the residual air is compressed until its density is the same as that of the outside air. The space which it then occupies, compared with the whole capacity of the flask, will then represent the proportion of air remaining in it ; and the amount of water which enters compared with the total amount necessary to fill the flask will represent the proportion of air exhausted. The flask must not be plunged too deep below the surface of the water, for if it is the air within it may be perceptibly compressed ; but it is well to submerge it to a depth of 10 or 20 cm., to offset the expansion of the air caused by its taking up vapor from the water with which it comes in contact (see Table 13). The less air there is, the less will be its expansion. To obtain accurate results, we must therefore exhaust nearly all the air, or else substitute for water some less volatile fluid. It may be observed that the water which enters the flask replaces, bulk for bulk, that portion of the air which has been exhausted. The weight of this air is the difference between the weights of the flask before and after exhaustion ; the weight of the equivalent IT 46.] DENSITY OF GASES. 69 bulk of water is the difference between the last two weighings, before and after the admission of water. We notice that in this experiment, unlike those which precede it, the water enters the flask without dis- placing any air whatever; hence no allowance is made for the weight of air displaced. Both the weight of air exhausted and that of the water which takes its place are affected by the buoyancy of the atmosphere upon the brass weights ( 65), and in the same proportion ; hence their quotient is unaf- fected, and represents the true specific gravity of the air referred to the water. This should agree closely l with the atmospheric density indicated by the barodeik. EXPERIMENT XVIII. DENSITY OP GASES. ^[ 46. Determination of the Density of a Gas. A light flask, as large as the balance pans will admit, is made perfectly dry (see ^[ 32), and weighed with its stopper beside it. To determine the density of the air within the flask, an observation of the barodeik is made (see ^[ 18). Then the flask is filled with coal-gas conducted through a rubber tube reaching as far as possible into the flask. To prevent the escape of the coal-gas, which is lighter than air, the 1 The true specific gravity of any substance referred to water at any temperature must strictly be multiplied by the density of water at that temperature (see 69), to find the density of the substance in question. In the present case, the multiplication will hardly affect the last significant figure of the result. 70 DENSITY OF GASES. [Exp. 18. flask is held iu an inverted position throughout the process ; after which the tube is drawn slowly out of the flask without checking the flow of gas (see 6, Fig. 26), and the stopper (a) is immediately inserted. The weight of the flask is again determined. More gas is then passed into the flask as before until it reaches a constant weight. The temperature of the gas in the flask is then found by a ther- mometer inserted through a bored stopper ; and the pressure is deter- mined by an observation of the ba- rometer. Finally the flask is filled with water and weighed for the purpose of finding its capacity. The last weighing and the observation of tempera- ture which should accompany it may be comparatively rough; but the weighings with air and with gas should be made with the utmost precision, since the difference between them, upon which the result depends, is so slight that even a small error would affect this result in a very considerable proportion (see 36). If ordinary prescription scales are used, tne result should depend upon the mean of at least five double weighings in each case. When great accuracy is desired, a counterpoise should be used consisting of a second flask, hermetically sealed, equal to the first in volume and nearly equal in weight. Small weights added to the counterpoise should bring about an exact adjustment. By using such a counter- poise, changes in atmospheric density are eliminated, 11 47.] STANDARD OF LENGTH. 71 since the air will buoy up the contents of both pans alike. The capacity of the flask is then calculated as in ^[ 32, and the density of coal-gas at the observed temperature and pressure is found by the formula of 70, using the density of the air indicated by the barodeik. The result is then reduced to and 76 em. pressure by the formula of 81. EXPERIMENT XIX. MEASUREMENT OF LENGTH. Tf 47. Selection of a Standard of Length. A care- ful comparison of the various scales which we have hitherto employed for the measurement of length will generally show cases of disagreement. These may sometimes be explained as the result of expan- sion by heat (see Table 8 i) ; for, though a scale should be correct at 0, unless otherwise stated, there is no agreement to this effect among manufacturers. 1 In other cases errors are discovered which may be traced to the machine by which the scales are di- vided. It will not do to assume that the most carefully finished scales are the most accurate. Those printed in large quantities on wood compare very 1 English measures are generally adjusted (if at all) to a tempera- ture of about 62 Fahrenheit. Certain French manufacturers main- tain that all standards are supposed to be correct at 4 Centigrade. In the case of brass metre scales, discrepancies of nearly half a millimetre may sometimes be traced to the temperatures at which they have been adjusted. 72 MEASUREMENT OF LENGTH. [Exp. 19. favorably with common varieties of " vernier gauge " (see Fig. 27). The latter, in particular, need to be tested as will be explained below. For this pur- pose, " end standards " are made by various manufac- turers with a considerable degree of precision. In place of these, however, the student will find it more instructive to use one depending, as follows, upon his own measurements. The volume, v, of a glass ball has already been determined (^[ 29) ; from this the diameter, d, may be calculated by geometry, using the formula J d = 1.2407 #v. In calculating the diameter of a sphere from the cube root of its volume, great accuracy may be obtained (see 36). Thus if the volume is really 40.00 cu. cm., and owing to an error of 1 eg. in weigh- ing, the observed value is 40.01 cu. cm., the calculated diameter will be 4.2435 cm, instead of 4.2432 cm. The difference (.0003 cwi.) between the calculated and the true value would be imperceptible. If the ball which we .employ is not perfectly spherical, an average diameter will be given by the formula. We shall see in ^[ 50, I. how slight ir- regularities can be allowed for. We may therefore obtain from our experiments in hydrostatics a stan- dard, in the form of a sphere, by which it is possible to correct the reading of a vernier gauge, or any other kind of caliper. 1 This is derived from the ordinary formula T48.] TESTING CALIPERS. 73 1" 48. Testing Calipers. A caliper is an instrument intended especially to determine by contact the diameter of bodies, generally the outside diameter. It is provided with two points called "teeth" or " jaws," one of which at least is movable. In one class of calipers the jaws are hinged together, their motion being magnified in some cases by a long index ; in another class there is a sliding motion, as in the vernier gauge used in Experiment 1 (see Fig. 27) ; in a third class the motion is produced by a screw, as in the micrometer gauge (Fig. 28). The instrumental errors ( 31) likely to arise dif- fer, of course, according to the special construction of the gauge in ques- tion ; but there are certain classes of de- fects common to all calipers, and hence it is well, before begin- ning any series of measurements, to make a regular ex- amination of each in- strument, covering the , ,. . . , FIG. 28. following points: (a) DISTORTION. The shank of a vernier gauge (ad, Fig. 27) should appear per- fectly straight to the eye, when " sighted " in the ordinary manner, and perfectly free from twist. A micrometer screw (cd, Fig. 28) should similarly ap- pear straight, so that the tooth c may be .accurately centred in all positions. MEASUREMENT OF LENGTH. [Exp. 19. (5) CONTACT. The jaws of a gauge must be able to touch each other at some point (as pp r Fig. 29) convenient for measurement. The shape of these jaws may be modi- fied, if necessary, by the use of a file, or by the application of solder, FIG. 29. in order that this condition may be fulfilled. The location of the point of contact is found by examining the streak of light between the jaws. (c) PERPENDICULARITY. The surfaces of the teeth or jaws at the point of contact should be at right angles with the shank of the gauge. In the case of a microm- eter, any obliquity immediately appears when the screw is ro- tated. To detect it in a sliding gauge it is necessary to reverse one of the jaws (as 5 in Fig 30), and to see whether the two inner surfaces remain parallel. (d) GRADUATION. The uniformity of the thread of a micrometer screw is sufficiently established if it turns in the nut, when well oiled, with equal facility throughout its entire length. The graduation of a vernier gauge is most easily tested by the vernier it- self ; for if the latter always subtends exactly the same number of divisions on the main scale, these may be assumed to be sensibly uniform. (e) LOOSENESS. A gauge should slide freely from one position to another ; but any looseness in the moving parts must be prevented. For this pur- FJG. 30. T49.] PRECAUTIONS IN THE USE OF CALIPERS. 75 pose a set screw (a, Fig. 27) is usually attached to a vernier scale. In the absence of any equivalent ar- rangement, a nut may often be tightened successfully by pinching it slightly in a vice. If the defects here mentioned cannot be over- come, the caliper should be discarded for the purposes of the exact measurements which follow. IT 49. Precautions in the Use of Calipers. (a) WARMTH. In ordinary measurements with a vernier gauge, the warmth of the hand will hardly cause a perceptible expansion ; but with micrometers, considerable care must be taken to avoid errors from this source. The usual method is to hold the instru- ment with a cloth, but it is still more effective to mount it in a vice, and thus to leave both hands free for making the necessary adjustments. (5) CLAMPING. When a caliper has been " set " on a given object, it is customary to clamp it before making a reading, lest in the mean time dislocation should take place. There is danger, however, that in the very act of clamping any instrument, its " set- ting" may be disturbed. Vernier gauges, unless specially provided with springs to keep the moving parts in place, are troublesome in this respect. The difficulty is lessened by keeping a moderate pressure on the clamp while the setting is taking place. In all instruments, the accuracy of a setting should be tested after clamping. ( 31 ) which determine the zero reading of the gauge (see Fig. 29). These are generally the most prominent points of the inner surfaces ; hence the rule, place the object to be measured where it fits with the greatest difficulty. (e) OBLIQUITY. The line pp' (Fig. 31) is neces- sarily parallel to the shank of the gauge ; hence also the diameter of any object which coincides with it. If, however, through any mistake in the above adjust- ment, the diameter to be measured is perceptibly inclined with respect to the line pp\ a considerable IT 50.] CORRECTION OF CALIPERS. 77 error is likely to be introduced into the result. It may be shown by trigonometry that if the inclina- tion is less than 1, the error will be less than one six-thousandth part of the quantity measured ; and hence practically insensible. Since the eye can de- tect under favorable circumstances an obliquity even less than 1, the following rule will be found suffi- ciently accurate : make the diameter to be measured sensibly parallel to the shank of the gauge. (/) POSITION. An object may be fitted between the teeth of a caliper in various ways, and care must be taken that the diameter thus measured is the one sought. In the case of a rectangular block, for in- stance, a minimum diameter is usually required, and care must be taken not to place it corner wise ; in the case of a sphere, however, a maximum measurement is wanted, and to secure this, especially when the teeth are rounded (as in Fig. 32), many trials must be made and with the greatest care. (g) PARALLAX. Errors of parallax ( 25) may be avoided when two scales are mutually inclined, by holding the eye or the gauge in such a position that the lines appear parallel, as in A, Fig. 33, not in- FIG. 32. ,. , T-J clmed as m B. ^[ 50. Correction of Calipers. i i , I j iifli ml' In M| i in 1 1 It is important to determine |<"'T""l /""W'l the reading of a gauge or caliper F IG . 33. when the jaws are in contact (see Fig. 29). This is called the "zero reading," because it corresponds to a distance zero between the 78 MEASUREMENT OF LENGTH. [Exp. 19. points p and p' where contact takes place. A gauge need not be condemned simply because the " zero reading " is not exactly zero. The fulfilment of this condition is in fact exceedingly rare. It is only neces- sary that the zero reading shall be accurately deter- mined, in order to avoid (by subtraction) all errors from this source ( 32). I. VERNIER GAUGE. The general method of reading a vernier gauge has been explained in ^[ 3. We have seen in 37 how the tenths of the millimetre divisions on the main scale are read by means of a " vernier." In case, however, the indication of the vernier lies between two numbers, it becomes necessary in all exact measurements to estimate fractions of tenths. We have already found a rough way of representing such fractions (see ^T 6). A more exact method is described in 37. To obtain success in applying this method to a vernier reading to tenths of a millimetre, the rulings of the scale should be fine, and a hand lens (such as is represented in Fig. 34) should be used to magnify the ver- nier and main scale divi- sions so that the difference between them may be plainly visible to the eye. The student will find it difficult, at first, to select the diagram in 37 most re- sembling the case of coincidence in question ; J but with 1 One of the chief difficulties in conducting this experiment lies in the tendency of students to hold a gauge more or less obliquely, so that all cases of coincidence may appear to be exact, or (what is nearly as hopeless) precisely alike. To an accurate observer, no two settings present in general exactly the same appearance. If 50, L] THE VERNIER GAUGE. 79 a little practice most of his errors should be confined to a range of one or two hundredths of a millimetre. If the zero of the vernier comes opposite a point below the zero of the main scale, the reading is negative. For convenience, however, the negative sign is applied (as in logarithms) only to the whole number indicated on the main scale, the fraction remaining positive. Thus if the zero on the vernier passes the zero on the main scale by .02 mm. when the jaws are brought into contact, the reading of the vernier should be .98 ; and in this case, the zero read- ing is 1.98, according to the general rule given in ^[ 3. When the zero reading has thus been found within one or two huudredths of a millimetre, a body of FIG. 36. known diameter is set between the jaws of the gauge. The glass ball, for instance, used in Experiments 8 and 9 is to be placed (see Fig. 31), so as to reach between the points p and p' by which the zero reading was determined (see Fig. 29). Looking at the jaws endwise, we should see the ball symmetri- cally situated, as in Fig. 35. If the ball is not perfectly round, we shall need at least 10 measurements of its diameter; and these 80 MEASUREMENT OF LENGTH. [Exp. 19. measurements should obviously be distributed as uni- formly as possible over the surface of the sphere. The student will do well to mark in ink ten points upon the ball as in B, Fig. 36, which are to be brought successively under the point p (Fig. 35), in one jaw of the gauge. After each measurement, the corre- sponding mark should be erased, to prevent confusion. As to the manner of spacing the ten points in ques- tion, the student is advised to begin with a 20-sided paper weight (A, Fig. 36), to place a number in the middle of each of the ten faces visible from a given point of view, then to copy these marks on the glass ball J5, so that they may appear to be spaced in the same manner in both cases. The geometrician will observe that there is one way and only one way of distributing ten diameters uniformly over the sur- face of a sphere, and that this way has been here practically adopted. In each of the ten measurements, a reading is made to hundredths of millimetres ; then the zero reading is re-determined. From the mean of the ten measurements above, the mean zero reading is sub- tracted. We thus find the average diameter of the ball according to the gauge. Dividing this observed diameter by that obtained by the hydrostatic method (which we will suppose to be the true diameter see ^[ 47), we obtain an important factor, namely, the average space in millimetres occupied byeach milli- metre division in a certain part of the gauge. If the gauge is uniformly graduated (see ^[ 48, d), it is ob- viously possible to correct all measurements made 1J50, II.] THE MICROMETER GAUGE. 81 with the gauge at the same temperature by means of the factor thus found. In practice, however, it may be assumed that a gauge has been selected in which these corrections are too small to be considered. II. MICROMETER GAUGE. In place of the glass ball of Experiments 8 and 9, the student may use the steel balls of Experiments 3 and 4, provided that the displacement of these balls has been confirmed by the specific gravity bottle, as suggested in *fi 35. The joint volume of these balls is then found by the use of Table 22 (see ^[ 29), then the average volume, from which (the balls being uniform in size) the av- erage diameter is calculated by the formula of ^[ 47. The diameters of these balls may now be measured by means of a micrometer gauge (see Fig. 28). The tests to be applied to a micrometer and the precau- tions to be followed are essentially the same as with any other kind of caliper (see ^[ 48 and ^[ 49). The zero reading is found as in the case of a vernier gauge by bringing the teeth into contact. Then the teeth are separated by turning the head of the screw (Fig. 28) until the ball whose diameter is to be measured fits symmetrically between these teeth as in Figure 37. FlG " 3L The whole number of revolutions of the screw should correspond with the number of main scale divisions on the nut d, uncovered by the barrel e. The hundredths of a turn maybe read by the gradua- tion on the edge of the barrel, using as an index a mark running along the nut. Care must be taken 6 82 MEASUREMENT OF LENGTH. [Exr. 19. to avoid a mistake of a whole turn in reading the gauge ; if, for instance, nine whole divisions (nearly) are uncovered by the barrel, and the index points to 98 hundredths, the reading is 8.98 (not 9.98). It is safer with many micrometers to confirm the whole number of revolutions by actually counting them. In reading the micrometer the divisions correspond- ing to hundredths of a revolution should be divided into tenths by the eye ( 26). A micrometer with a millimetre thread thus indicates the thousandth part of a millimetre. In the case of a negative zero reading, as with the vernier gauge, the minus sign should be applied only to the whole number of turns. The diameter of each of the steel balls is deter- mined in this way to thousandths of a turn of the screw ; and from the average reading we subtract the average zero reading, observed before and after the above with an equal degree of precision. We find in this way the average number of turns and thousandths of a turn actually made by the screw. Dividing the average diameter of the balls (from the hydrostatic method) by the corresponding number of turns of the screw, we have finally the distance through which the micrometer screw advances in each revolution. This is called the "pitch of the screw." We shall assume that a micrometer has been found, reading to millimetres and thousandths so accurately that in the case of objects of small diameter, no correction need be applied. I 51.] ZERO READING OF A SPHEROMETER. 83 EXPERIMENT XX. TESTING A SPHEROMETER. IT 51. Determination of the Zero Reading of a Spher- ometer. A spherometer (Fig. 38) is essentially a mi- crometer (see IT 50, II.) supported by three legs (d, /, #). The verti- cal screw (ce) has a head (5) di- vided into a hundred parts, the tenths of which may be estimated by the eye ( 26). The thou- sandths of a revolution may thus be read by means of an index (a). This index carries a vertical scale (a/), on which the head of the FlG 38> micrometer (6) registers the whole number of revo- lutions made by the screw. Both on the scale (/") and on the micrometer, the indications should in- crease as the screw is raised. It is well to renumber the main scale if necessary, so that negative readings may be avoided. The zero reading of a spherometer is its reading when the point of the central screw is in the plane of the three feet. To find it, the instrument is set on a piece of plate glass (Fig. 39) of sensibly uniform thick- ness, selected by the aid of a micrometer gauge, and the screw of the spherometer is raised or lowered until all four points seem to touch the glass at the same time (see Fig. 40). If the central screw is driven too far forward, the instrument will not stand firmly 84 THE SPHEROMETER. [Exp. 20. upon the glass, but will have a tendency to rock. This will be noticed especially if one of the feet be held down by the finger, while the other two feet are subjected to an alternating pressure. In fact, the con- ditions upon which rocking depends are so delicate that a change of a thousandth of a millimetre may cause it to appear or to disappear. When the instru- ment has been adjusted so that rocking is barely per- ceptible, the reading is estimated in millimetres to three places of decimals, in the same manner as in the case of a micrometer gauge. FIG 39. FIG. 40. On account of possible irregularities in the glass, at least five readings should be taken in different parts of one surface ; and as plate glass is apt to warp slightly in the process of manufacture, five more readings should be taken on the other surface. The mean of the values thus found on a piece of glass of uniform thickness gives the zero reading of the spherometer, and should be determined after as well as before any series of measurements such as will be described in the next section, in order to avoid errors due to change of temperature and to the wearing away of the points upon which the instrument rests. T 53.] PITCH OF SCREW. 85 Tf 52. Determination of the Pitch of the Screw. A spherometer with a screw of known pitch can be used in place of a micrometer to measure the diameter of small objects. These are placed upon the plate glass already used to determine the zero reading, and the screw is adjusted so as to touch them from above (see Fig. 41). If the point of the screw is very sharp, FIG. 41. and the surface of the object in question convex, great care is needed in finding the maximum diameter. To determine the pitch of the screw, we select an object of known diameter by means of a vernier or micrometer gauge ; we may determine, for instance, the diameter of a steel bicycle ball. This is then fitted as above (Fig. 41) beneath the point of the screw, and the reading of the spherometer accurately determined. Subtracting the zero reading, we have the number of turns made by the screw in traversing the diameter of the ball. Dividing this diameter by this number of turns, we have (as in 1" 50 II.) the pitch of the screw. Assuming that the screw has a uniform pitch, it is evident that the distance traversed by the point of the screw will always be given by the product of the number of turns and the pitch of the screw. ^[ 53. Determination of the Span of a Spherometer. The span of a spherometer, or the average distance of its three feet (d, f, and g, Fig. 38) from the central screw (e) in its zero position (Fig. 40) is an impor- tant element in all calculations relating to curvature 86 THE SPHEROMETER. [Exp. 20. (see next experiment). It may be determined roughly by a series of measurements with an ordi- nary vernier gauge. If difficulty is found in measur- ing directly the distances in question from centre to centre, an impression of the feet and central screw may be taken on paper, and the distances thus indi- rectly determined. 1 For this purpose the student will doubtless prefer to use a glass scale, if one can be obtained, graduated in millimetres and tenths. In such a scale the rulings should be placed next the paper, and examined with a mag- nifying glass. If the feet are blunt (as a and b in Fig. 42), the point of contact will be uncertain. FIG. 42. j n suc \ l a case the feet should be sharp- ened, and the zero reading re-determined. 1 54. Testing a Spherometer. We have seen that a spherometer may be fitted to a plane surface (IT 51) ; in the same way it may be adjusted to a curved sur- face. To bring this about, the central screw must be driven forward, if the surface is concave, or turned backward if the surface is convex. The distance through which it must be moved obviously depends upon the curvature of the surface in question. The spherometer can therefore be used to determine the curvature of surfaces. There are, however, various sources of error in the use of a spherometer, and to 1 Some authorities prefer not to measure directly the distances (ed, ff, eg. Fig. 38) of the three feet from the central screw, but to calcu- late the span by multiplying the average of the three distances (df, fg, gd) between the feet by the square root of one third, or 0.57735. H 54.] TESTING A SPHEKOMETER. 87 detect these, the instrument is first of all adjusted to a surface of known curvature, as for instance that of the sphere used in Experiments 8 and 9, (see Fig. 43), or if that is not large enough, to some other sphere of known diameter. The central screw is set as in f 51, so that rocking is barely percep- tible, and the reading of the instrument is determined with the same degree of precision as before. At least ten set- FlG - 4a tings should be made on different portions of the spherical surface. In reducing the results we find first the average reading of the spherometer, then subtracting the zero reading we find the number of turns which the screw has made, and hence the dis- tance in millimetres through which the point of the screw has retreated from its zero position, since the pitch of the screw has been already determined in IT 52. If this distance is d, and the diameter of the sphere D, the square (s 2 ) of the span of the spher- ometer may be calculated by the formula (see 1" 56, II.), s a _ Dd (P. In this formula, all measurements should be ex- pressed in millimetres. The result should confirm that obtained by squaring the span actually observed in If 53. Slight discrepancies may sometimes be traced to obliquity or excentricity of the central screw, or to irregularities in the shape of the three feet. 88 THE SPHEROMETER. [Exp. 21. EXPERIMENT XXI. CURVATURE OF SURFACES. If 55. Determination of the Radius of Curvature of a Spherical Surface. It is frequently required in optics to know the curvature of the surfaces of a lens ; for this curvature, together with the nature of the glass of which a lens is made determines its power of bringing light to a "focus" ( 103-104) ; and conversely, if the curvature and focussing power are known, we may find what sort of glass the lens is composed of. This subject will be fully treated of in Experiments 41 and 42. It is necessary at present only to point out that as the surfaces of lenses are generally ground to resemble portions cut out of a sphere, their curvature may be determined in the same way as that of any other spherical surface. The spherometer is set upon the lens as in Figure 44, and adjusted so that rocking is barely perceptible as in IT 51 and 1" 54. Ten settings are thus made on each side of the lens, the curvatures of which, even if both are convex, are by no means necessarily the same. Be- tween successive measurements the position of the spherometer should be varied somewhat, so as to determine as well as possible the average curvature of each surface. The results are then averaged for each surface ; the mean zero reading subtracted from each, and the IT 56.] CURVATURE OF SURFACES. 89 distance (c?) between the point of the screw and the plane of its three feet thus determined. From this, the diameter, D, of the sphere of which the surface in question forms a part is calculated by the formula (see IF 56, I.), D = a + 2 -r- a. where s 2 is the square of the span already calculated in the last article. The " radius of curvature " is found by halving the diameter. *[T 56. Theory of the spherometer. The formulae of the last two articles depend upon the following con- siderations : Let a, Figure 45, be the point of the central screw of a spherometer, and b one of the three feet lying in the plane 5c, and let ad be a diameter of the sphere abd intersecting the plane be at c; then if the screw is properly adjusted, acb and bed will be right triangles. Now abd is also a right triangle, being measured by half the semi- circular arc ad; hence the angles cba and bdc are equal, both being complementary to cbd ; the right triangles abc and bdc are therefore similar and we have ac : be : : ~bc : cd, whence cd = be* -f- ac, and ad = ac -j- cd = ac -f- Id* -r- ac. I. We are thus able to calculate the diameter of a sphere (ac?) if we know the span of the spherometer FIG. 45. 90 EXPANSION OF SOLIDS. [Exp. 22. , and the distance, ac, between the point of the screw, a, and the plane of the three feet, be. We can also calculate the square of the span be, by the formula, easily derived from the above, be* = ~^d X ac oc 2 . II. EXPERIMENT XXII. EXPANSION OF SOLIDS. ^[ 57. Determination of the Coefficient of Linear Ex- pansion. By measuring the length of a rod at two different temperatures, the amount of linear expan- /r-^ ' e T ^ <^ ' ^ A IS FIG. 46. sion due to heat may obviously be determined. To make the expansion measurable, a long rod must be employed ; and even then delicate instruments are needed to measure the expansion accurately. A mi- crometer gauge, especially constructed for this pur- pose, is represented in Fig. 46. It consists of a rectangular wooden frame, bcon, capable of admitting a metallic rod, gi, 1 metre long, between the fixed point fg and the point of the micrometer screw, i.j. If 57.] LINEAR EXPANSION. 91 The rod is surrounded with a tube, also 1 metre long, held in place by the supports, k and m. The tube is closed at both ends with corks, thinner near the middle than at the edges, and serving to keep the rod in position. A setting of the micrometer is first made with the rod in position, and the reading determined (see Tf 50, II.) ; the temperature of the rod is then found by means of a thermometer, A, passing through a cork, e, in the side of the tube. To determine the pitch of the micrometer, 1 it is turned backward (as in ^[ 52) until an object of known diameter fits be- tween it and the end of the rod. A new reading is then made, and the pitch of the screw is calculated as in the case of an ordinary micrometer gauge a 50, no. The screw of the micrometer is now withdrawn, to allow room for the expansion of the rod, and steam from a generator (a) is passed through the tube from the inlet (c?) to the outlet (Z). As soon as a steady current of steam appears at the outlet, a new setting of the micrometer is made. Subtracting from the last reading of the microme- ter the original reading, we find the number of turns made by the screw. From this, knowing the pitch of the screw (^[ 52), we find the expansion of the rod in mm. Subtracting the original temperature (let us say 20) from the final temperature (100, nearty, see, however, Table 14) we find the rise of temper- 1 By using the same micrometer as in If 52, a determination of pitch will be rendered unnecessary. 92 EXPANSION OF SOLIDS. [Exp. 22. ature which has caused this expansion. To find the expansion of 1 mm., we divide the total expansion by the length of the rod in mm. (1,000 mm.) ; and we divide the quotient by the rise of temperature in degrees (80 in this instance) to find the expan- sion in mm. of 1 mm. 1 for 1. The result is called the coefficient of linear expansion of the material of which the rod is composed ( 83). ^[ 58. Errors in the Determination of Linear Ex- pansion. In determining the temperature of a metallic rod by a thermometer beside it, a consider- able error is likely to arise unless the temperature of the surrounding air is constant, and the observation prolonged. Air is, as we shall see (Experiment 31), a comparatively poor conductor of heat. To attain greater accuracy in this experiment, the tube may be filled with water, as it is found that an equilibrium of temperature is reached much more quickly with water than with air (see ^[ 65, (6) ). A still more accurate method is to replace the tube by a trough packed with melting ice or snow. The mixture should be stirred vigorously for a few minutes, so that the rod may acquire a nearly uniform tempera- ture, not far from 0. If this method is followed an observation of the thermometer will be unnecessary. For rough purposes, the temperature of the steam which fills the tube in the second part of the experi- ment may be assumed to be 100 ; but this tempera- 1 The student should note that the expansion of 1 mm. in mm. is numerically the same as that of 1 cm. in cm. The result does not therefore need to be reduced to the C. G. S. System. 1[ 58.] LINEAR EXPANSION. 93 ture really depends more or less upon the barometric pressure. The thermometer cannot be depended upon to give this temperature correctly, particularly if the bulb only is surrounded by steam. When ac- curacy is desired, an observation of the barometer must be made (see IT 13). The true temperature of the steam may then be found by Table 14, as will be explained in Experiment 25. It is obviously impossible for the whole rod, gi (Fig. 46), to be in contact with the steam or ice surrounding it ; for even when the corks are hol- lowed out, as shown in the figure, so as to leave nearly the whole surface of the rod uncovered, there must still be a small portion at each end which the steam or ice can never reach. The expansion of the rod will not therefore be as great as it should be. On the other hand, the points fg and ij\ being heated by contact with the rod, will expand some- what, and thus make the expansion of the rod ap- pear to be greater than it really is. To diminish the conduction of heat, the teeth may be protected by the use of insulating material, or by simply pointing them. In all cases contact should be maintained only as long as may be necessary to make a reading of the micrometer. There is always more or less un- certainty as to temperature when a hot and a cold body are in contact. To. eliminate errors arising from this source, it would suffice to construct a new ap- paratus, which should be as short as possible, but otherwise similar to the first, and to calculate the results from the difference of expansion in the two 94 EXPANSION OF LIQUIDS. [Exp. 23. cases, according to the general method suggested in 32. There is, however, no way to allow for the expan- sion of the sides of the gauge, caused by the warmth of the steam jacket. We meet here, in fact, one of the fundamental difficulties in the accurate measure- ment of expansion, namely, changes in the length of the instruments by which expansion is measured. To avoid errors from this source, a glass tube is sometimes substituted for the metallic tube repre- sented in Fig. 46, so that the expansion of the rod may be observed from a distance. In the most ac- curate determinations, the gauge or standard used for comparison is insulated from all sources of heat, and even, in some cases, maintained artificially at a uni- form temperature. The expansion of a gauge constructed, like that shown in Fig. 46, principally of wood (see Table 8, 6), and with sufficient space for the circulation of air, will be found in practice to be very slight ; but, in the absence of special precautions, the student should not expect his results to contain more than three significant figures ( 55). EXPERIMENT XXIII. EXPANSION OF LIQUIDS, I. ^[ 59. Determination of the Coefficient of Expansion of a Liquid by the Method of Balancing Columns. A convenient form of apparatus for this experiment IT 59.] BALANCING COLUMNS. 95 (see Fig. 47) consists of two vertical metallic tubes, ch and^)', about one metre long, with horizontal elbows (cd, ef, hi, and ify at each end. The lower elbows are connected together with a rubber tube (z), while each of the upper elbows is joined to one end of a differential gauge (aJ) by one of the rubber couplings (d and e). Each of the tubes ch and/)' is sur- rounded with a larger tube, or ''jacket" which can be filled either with melting ice, with water, or with steam. The spouts g and k are to be used either as inlets or as out- 3 lets, as the experiment may require. The liquid whose expansion is to be investigated is first freed from any air which may be held in solu- tion, by boiling it, then poured steadily through a funnel into the tube a until, after completing the circuit (adchijfeb), it issues in a continuous stream from b. The whole apparatus is now inclined first to the right and then to the left, so that any bubbles of air which may be lodged in the horizontal tubes may have an opportunity to escape. A little liquid is next poured out, until the column stands at the level b. This level should be the same, at first, on both sides of the gauge. FIG. 47. 96 EXPANSION OF LIQUIDS. [Exp. 23. Steam is then admitted to the jacket eg through the spont g ; and the jacket fk is filled with water from a faucet by a tube connected to the spout k. The temperature of the water is observed after it reaches the top of the tube, /. The height of the liquid in each side of the gauge (a and 6) is measured as soon as it becomes stationary, by means of a milli- metre scale, as in Experiment 16. (See ^f 42.) The vertical length of the tube (ch} is finally measured be- tween the elbows (cd and hi), from centre to centre, as close as possible to the jacket. This measurement should (strictly) be made while the tube is still heated by steam. When the apparatus has become sufficiently cool, the water is emptied out of the jacket fk, which is, in its turn, filled with steam, while the jacket eg is cooled by water from the faucet. The tem- perature of the water and the reading of the gauge are observed as before ; in this case, however, the vertical distance fj is measured. The object of interchanging the jackets is (see 44) to elimi- nate errors due to capillarity, or in fact any cause which might tend constantly to raise or lower the level of the liquid on one particular side of the gauge. Instead of admitting steam to one of the jackets, melting ice may be employed, or water at various temperatures, which must, of course, be observed. The other jacket is always maintained at a tempera- ture not far from that of the room, by the water with which it is filled. If 60.] BALANCING COLUMNS. 97 ^[60. Precautions in determining Expansion by the Method of Balancing Columns. It is evident that the temperatures employed in this experiment must not be higher than the boiling-point nor lower than the freezing-point of the liquid in question, and that this liquid must not be such as to act chemically on the tubes which contain it. Even a very slight action may generate a quantity of gas sufficient to impair the accuracy of the results. The air dissolved in the liquid must be completely boiled out before the experiment, since otherwise bubbles are apt to form when heat is applied. The tubes should be large enough to allow the escape of any air which may be carried into them while they are being filled ; but small bubbles can sometimes be dislodged only by jarring the whole apparatus. The tubes should be completely surrounded with the steam, water, or melting ice by which their tem- perature is to be regulated. There should be a free vent through one of the spouts (g or k~) for the water formed by the melting of the ice, otherwise the tem- perature of the mixture may rise above 0. If steam is admitted through one of these spouts, the jacket should be partly covered, leaving only a small opening through which the steam should escape in a slow but continuous stream. If the jackets contain water, the latter should be stirred vigorously to secure a uni- formity of temperature. It is well also, in this case, to find the reading of a thermometer at different levels. This will require either a self-registering thermometer, or one with a very long stem. If the 7 98 EXPANSION OF LIQUIDS. [Exp. 23. temperature is not uniform, the average temperature must be calculated. 1 The jackets (eg and /&) should be made vertical by a plumb line, as nearly as the eye can judge, and also both branches of the gauge (ai). The tubes cd, ef, and hj should be perfectly horizontal, in those portions at least which are affected by the flow of heat to or from the jackets. The gauge (6) should be maintained at a uniform temperature (the same always as that of one of the jackets) by surrounding it, if necessary, with water. The tubes of which this gauge is constructed should be of the same uniform calibre, and both perfectly clean, otherwise the effects of capillary action may not be perfectly eliminated. It is well to make sure, both before and after the ex- periment, that the liquid stands at the same level on both sides of the gauge when the temperature in the two jackets is the same. To obtain the most accurate readings of such a gauge, a double sight should be employed, as in the case of a standard barometer. The setting is always made so that the plane of the sights may be tangent to the meniscus, or curved surface of the liquid (see ^[13 and ^ 42). The sights may be provided with a vernier reading to tenths of a millimetre. ^[ 61. Theory of Balancing Columns at Unequal Temperatures. The difference in hydrostatic pressure between the two liquid columns, eA andfj, is balanced 1 The average temperature will be indicated sit once by an air thermometer of sufficient length, which the student himself may be interested to construct. See Experiment 26. f 61.] COEFFICIENTS OF EXPANSION. 99 by the pressure of a column of liquid reaching from a to 5, or more strictly, by the difference between the hydrostatic pressure of such a column and that of an equally long column of air. The latter, being exceed- ingly light, may be left out of the account. To sim- plify calculations, we will suppose all the tubes to have a cross-section of 1 sq. cm. Then if d is the difference in cm. between the two levels (a and b) in the gauge, when it is maintained at the same tem- perature () as the jacket fj ; and if I is the length of the column ch at a higher temperature, t a ; then I cu. cm. of the liquid at the temperature t. 2 plus d CM. cm. at the temperature t v balance I cu. cm. at the temperature t r It follows that I cu. cm. at t must balance (I d*) cu. cm. at tf. Now two columns of liquid of the same cross-section cannot balance one another unless they have the same total weight ; hence the same quantity of liquid which occupies (I cf) cu. cm. at t must expand by the amount d cu. cm. when heated to t. 2 , since it then occupies I cu. cm. If an expansion of d cu. cm. is caused by a rise of ( 2 ^) degrees, 1 would cause an expan- sion in the average ( 2 ^) times less than d cu. cm.; and since the expansion of 1 cu. cm. would be (Z c?) times less than that of (I cT) cu. cm., the expansion (ef of 1 cu. cm. for 1 would be This expression becomes somewhat modified when the gauge is at the higher temperature, t. 2 . We have, 100 EXPANSION OF LIQUIDS. [Exp. 23. then, (Z + d) cu. cm., all at the temperature 2 , balancing I cu. cm. at the temperature ^. The expansion is as before, d cu. cm. ; but the quantity expanding is no longer {I d), but I cu. cm. The expansion e" per cu. cm. per degree is therefore e"=^J II. We have assumed so far that the tubes have a cross-section of 1 sq. cm. ; but the principles of hy- drostatic pressure are independent of cross-section (see 63) ; hence the solutions found in one case may be applied to all. The method of balancing columns is the only one which enables us to measure the expansion of a liquid without taking into account changes in the capacity of the vessel in which the liquid is contained. The object of this method is to determine an aver- age coefficient of expansion between two tempera- tures rather than the true coefficient of expansion ( 83) at any particular temperature. The results may differ considerably from those contained in Table 11, which refers in nearly all cases to the expansion of liquids from to 1 Centigrade. We consider, moreover, the expansion of a quantity of liquid measuring 1 cu. cm. at the lower of the two temperatures observed instead of at 0. The result given by the formulae of this section should, therefore, be designated as the relative coefficient of expansion from t to t, that is, from the lower to the higher temperature. II 62.] COEFFICIENTS OF EXPANSION. 101 EXPERIMENT XXIV. EXPANSION OF LIQUIDS, II. ^| 62. Determination of the Coefficient of Expansion of a Liquid by means of a Specific Gravity Bottle. The experiment consists essentially of a repetition of Experiment 14, with a given liquid at two or more different temperatures. These temperatures should be separated from one another as widely as possible, in order that the densities observed may differ by an amount large enough to be accurately measured. The temperatures themselves must be determined with the greatest care, particularly if they are far above or far below the temperature of the room ; for in this case rapid changes will take place and must be guarded against. A convenient way of heating a liquid in a specific gravity bottle to a uniform temperature, is to sur- round the bottle up to the neck with $iot water. To prevent evaporation, the bottle should be closed temporarily by a cork, with a hole made in it sufficiently large to admit freely the stem of a ther- mometer, to which a brass fan is attached (see Fig. 50, ^[ 65). By this means the liquid is continually stirred until a maximum temperature is reached. As soon as the reading of the thermometer has been observed, the stopper is inserted, with due care not to enclose bubbles of air (see ^[ 32, Fig. 19). The bottle is then carefully dried, and weighed at leisure (see ^[ 33), after cooling to the temperature of the room. 102 EXPANSION OF LIQUIDS. [Exp. 24. The student is advised not to attempt determina- tions of density below the temperature of the room, on account of the obvious difficulty of preventing the loss, especially in the case of a volatile liquid, of the portion which is forced out of a specific gravity bottle by .its gradual rise of temperature. He should, however, make at least two determina- tions of density above the temperature of the room, with the liquid already employed in Experiment 14 ; and he should repeat rapidly the determination made in that experiment at the temperature of the room, to make sure that the result has not been seriously affected by atmospheric changes, or by variations of the density of the liquid due to evaporation or other causes. Coefficients of expansion are then calculated and reduced as explained in the next section. ^[ 63. Calculation of Coefficients of Expansion. Let j, 2 , t s , etc., be the temperatures at which the densities d n c? 2 , rf g , etc., respectively, have been deter- mined and calculated, essentially as in ^[ 38. The results are first represented by points plotted on co- ordinate paper (see Fig. 48), and connected by a curve drawn with a bent ruler, essentially as in 59. The necessary forces should be applied to the ruler as near the ends as possible, in order that the curve may be continued downward as far as 0. The density of the liquid (d ) at is now inferred by means of this curve (see 59). The specific volumes, v , v t , v 2 , v 3 , etc., correspond- ing to the densities d , d l9 c? 2 , d s , etc., are now found by the formulae derived from ^f 37, IT 63.] COEFFICIENTS OF EXPANSION. 103 v a = 1 -i- d u ; v i = -i- l ; v 2 = -j- c 2 ; v 8 == -5- J 3 , etc. Evidently a certain quantity of liquid expands by the amount (v 9 vj cu. cm. when heated from the temperature ^ to the temperature t z ; that is, ( 2 ^) degrees. The expansion per degree is therefore (v z Vl ) -f- ( 2 ^). Since the quantity of liquid &-Q011- 0012 0013-OOtV 00(5 FJG. 48. FIG. thus expanding occupies v cu.cm. at 0, the ex- pansion (e) of a quantity occupying 1 cu. cm. at that temperature would be one v ih as large, or The coefficient e which determines the expansion of a quantity of liquid occupying the unit of volume at the standard temperature (0) is a true as distin- guished from a relative coefficient of expansion (see ^ 61) ; it expresses, however, the average expansion between the two temperatures t^ and 2 . We find in the same way the average coefficient of expansion from 2 to 3 by substituting, in the formula above, 2 , 3 , t> 2 , and v 3 , for ,, 2 , v^ and v 2 , respectively. Each result may be represented on co-ordinate paper by a cross, at the right of a point half-way between the two temperatures in question, and under the correspond- 104 EXPANSION OF LIQUIDS. [Ex p. 25. ing coefficient of expansion (see Fig. 49). A line drawn through these points represents approximately the coefficient of expansion at any given tempera- ture. It is clear, however, that with only two de- terminations of the coefficient of expansion, we can- not tell even whether this line should be straight or curved. EXPERIMENT XXV. THE MERCURIAL THERMOMETER. ^[ 64. Preservation of a Mercurial Thermometer. It would seem hardly necessary to point out that a mercurial thermometer is an exceedingly fragile in- strument ; but in the processes of manipulation about to be described, it is frequently required that a ther- mometer should be subjected to forces very near the limit of its strength, and which, even in skilled hands, may break it. The student is therefore ad- vised to experiment with thin tubes or strips of win- dow-glass, before attempting the calibration of a thermometer ; and to examine the almost micro- scopic thickness of the glass constituting the bulb, before subjecting it to any considerable pressure. In respect to its resistance to a blow endwise, the bulb of a thermometer may perhaps be compared to the point of a lead-pencil when moderately sharp. In attempting to move the mercury in the thermome- ter by centrifugal force, the student should limit himself to such velocities as he might give to a palm-leaf fan. More thermometers are broken by t 65.] THE MERCURIAL THERMOMETER. 105 suddenly arresting than by suddenly creating the necessary velocity. If a glass thermometer be tem- porarily mounted on a wooden support, like an or- dinary house thermometer, it may be much more roughly treated with the same safety. The full heat of a flame should never be applied immediately to any glass instrument, since fracture will almost inevitably result. By giving to a flame a waving motion,, heat may be applied as slowly as may be desired. As soon as the glass acquires a dull- red heat the danger of fracture is past. There will, however, be no occasion for so high a temperature in the case of a thermometer. The student is particu- larly cautioned against plunging a cold thermometer into hot mercury, 1 or a hot thermometer into any cold liquid whatsoever. In applying heat to the bulb of a thermometer, care must be taken not to drive out more mercury than there is room for in the expansion chamber at the top of the instrument. The temperature of the mercury should not be raised above its boiling- point 2 (350 C.) in any part of the thermometer ; for the pressure of the vapor, being transmitted to the bulb, will be likely to cause an explosion. ^[ 65. Precautions in the Use of a Mercurial Ther- mometer. (1) TEMPER. In addition to the dan- 1 The thermometer should be placed in the mercury while cold, and gradually heated with the mercury. On account of its rapid conduction of heat, mercury is more likely to cause fracture than other liquids. 2 Special thermometers are now constructed so as to read safely a.s high as the boiling point of sulphur (440 C.). 106 EXPANSION OF LIQUIDS. [Exp. 25. ger of fracture, the accuracy of a thermometer may be greatly impaired by any wide change of tem- perature, especially if the change be sudden. After a thermometer is freshly made, there is found to be a gradual contraction of the bulb, which continues perceptibly for months and even for years. This accounts for the fact that nearly all old thermome- ters stand somewhat too high, although they are not supposed to be graduated until 'the contraction of the bulb has ceased. The value of a thermometer evidently depends partly on its age or "temper." This value may be completely destroyed by a sudden change of temperature. (2) CHANGE OF FIXED POINTS. In fact, when a thermometer is simply heated to the temperature of steam, then cooled as gradually as possible, the readings are almost always affected to the extent of one or two tenths of a degree. In the course of a month the thermometer may return to its former reading, but the change is gradual. It is therefore customary to test a thermometer in ice, for in- stance (see ^j 69, II.) after testing it in steam (see ^[ 69, I.), or in fact after subjecting it to any consid- erable change of temperature. (3) CONTINUITY OF THE MERCURIAL COLUMN. Errors in reading a thermometer frequently arise from a break in the mercurial column, which can be guarded against only by inspection. A slight jar- ring is usually sufficient to make the column reunite ; but when a small bubble of air interrupts the col- umn, or when in the expansion chamber a globule 165.] THE MERCURIAL THERMOMETER. 107 becomes separated from the rest of the mercury, special precautions are necessary (see ^[^[ 65, 67). (4) TEMPERATURE OF THE STEM. To make an accurate determination of temperature with a mercu- rial thermometer, it is necessary that the mercury, in the stem as well as in the bulb, should be raised to the temperature in question. In a thermometer reading to 10 C., for instance, if the bulb only is heated, the errors, even if the thermometer is correctly graduated, will be as follows: at 50, 0.5 ; at 100, 2.0 ; at 200, 7.6 ; at 300, 17 ; etc. As the temperature rises, more mercury flows into the stem, and it becomes still more important to heat this mercury to the given temperature (see ^[ 84). (5) UNIFORMITY OF TEMPERATURE. In nearly all determinations of the temperature of liquids, it is necessary to make use of some stirring apparatus, to secure a uniformity of tem- perature. A small fan of thin sheet brass is customarily attached to the stem of the thermometer, just above the bulb. . The stirring is accomplished by twisting the stem of the thermometer. Special de- vices are necessary when finely divided substances are employed, though the stem of the thermometer itself may (with due care) occasionally be used, especially in mixtures, as of powdered ice and water, where the resistance will be exceedingly small. (6) TIME REQUIRED. The length of time required to attain an equilibrium of temperature depends largely upon the conductivity of the surrounding medium, and 108 EXPANSION OF LIQUIDS. [Exp. 25. upon the degree of accuracy which is aimed at. Let us suppose that a thermometer is taken out of a mixture of ice and water, and placed in air at 32 ; if at the end of one minute it rises 16, that is, half-way towards its final temperature, we may expect it to accomplish in another minute half of what is left, or 8, according to the general law explained in 89. The temperatures attained would thus be as follows : in 1 m., 16; in 2 m., 24; in 3 m., 28; in 4 m., 30; in 5 m., 31; in 6 m., 31 J, etc. At the end of 10 minutes the reading would differ from 32 by only 3^ of a degree, a quantity hardly perceptible to the eye on an ordinary thermometer. Now, if the ther- mometer had been placed in water at 32 instead of in air, the temperature would have reached 16 in a few seconds; and at the end of a minute it would have indicated 32 within a very small fraction of a degree. Again, a mixture of hot lead and cold water may take several minutes before the temperature is practically equalized. One almost always knows, at least roughly, what the final temperature will be. A useful rule is to observe how long it takes the temperature to reach a point half-way between its original and its final value; then to allow from ten to twenty times as long a time before making a determination of the temperature, according to the degree of accuracy required. (7) OTHER PRECAUTIONS. The necessity of shielding a thermometer from radiation has been already alluded to (^f 15). Delicate thermometers 1 66 J THE MERCURIAL THERMOMETER. 109 may be perceptibly affected by mechanical, hydro- static, or even barometric pressure on the bulb, and by mercurial pressure from within. Such thermome- ters should be tested both in a vertical and in a horizontal position. Other special precautions will be mentioned as the necessity for them arises. ^[ 66. Selection of a Mercurial Thermometer. For the purpose of calibration, it is best to select a glass thermometer, graduated on its own stem (be, Fig. 51), in degrees at least 1 mm. long, from to 100 centigrade, with a few divisions above 100 and below 0. The bulb (a&) should have a volume 1 of nearly 1 cu. cm. ; and the expansion chamber (c) at the top of the thermometer should have about ^ of this FIG. 51. capacity. The bulb (a6) should for convenience be elongated as in the figure, so as to pass freely through a hole in a cork fitted to the stem of the thermometer. The expansion chamber should be pear-shaped (see c, Fig. 51), since otherwise particles of mercury are likely to lodge there. The shape and size of the tube must be such that mercury may be made to flow, with a little jarring, from one end to the other ; and the quality of the mercury such that there is no tendency for the column to break up into small fragments. 1 The volume of a thermometer bulb may be estimated by the quantity of water displaced in a small measuring glass (IT 85). A small bulb usually implies a stem of small calibre, which may give rise to difficulty in calibration. HO EXPANSION OF LIQUIDS. [Exp. 25. v ^[ 67. Manipulation of a Thread of Mercury. It is frequently required in the calibration of a ther- mometer to separate from the rest of the mercury in the stem of the thermometer a thread or column of a given length, and to place it in a given part of the stem. When a thread has been broken off, it may be easily moved (by sufficiently inclining or swinging the thermometer) under the influence of its own weight or inertia. For slight motions, jarring is often efficient. The place where the thread breaks off is generally determined by a microscopic bubble of air. To find the location of this bubble, the ther- mometer is inverted. If a thread of mercury sepa- rates at once from the rest, the position of the bubble is evident ; if the mercury runs in an unbroken col- umn into the expansion chamber, a small quantity of air will probably be found in the bulb ; and if the mercury flows easily back again, there is probably a little air in the expansion chamber. The (nearly) empty space in the bulb caused by the flow of mercury into the expansion chamber has in any case the appearance of a bubble, which may be made to rise into the neck (5, Fig. 51) by sud- denly turning the thermometer into an upright posi- tion. If it really contains air, it may be worked up into the stem by jarring the thermometer, especially before all the mercury has had time to flow back from the expansion chamber. If the experiment has been successful, a thread of mercury may now be broken off by inverting the thermometer, and tapping it gently on the table. IF 67.] THE MERCURIAL THERMOMETER. Ill In the absence of air in the bulb or in the stem, it remains only to make use of air in the expansion chamber. As much mercury as possible is first made to flow into the expansion chamber, and detached from the rest by jarring the thermometer while in a horizontal position. Then the rest of the mercury is returned to the bulb. If there is any air in the expansion chamber, a part of it will now flow into the bulb ; and when the globule of mercury is once more returned to the bulb by centrifugal force (see Tf 64), a thread of mercury can probably be separated. The presence of a bubble of air 1 in the neck of the bulb (6) greatly facilitates the adjustment of the length of the thread of mercury which will break off when the thermometer is inverted. If the bulb is slowly heated or cooled by a certain number of degrees, the mercury will usually flow by the bubble without dislodging it, thus lengthening or shortening the thread by that same number of degrees. The surest way, however, of shortening a thread of mer- cury by a few degrees is to hold the thermometer upright and jar it slightly (see ^[ 64), so that the bubble may rise farther and farther into the stem. If at the same time the bulb is gradually cooled, one may be perfectly sure of shortening the thread to any extent. There is no certain method of increasing the length of a thread of mercury, except by trans- ferring it to the expansion chamber, and adding to 1 Few, if any, thermometers will be found to be entirely free from air. 112 EXPANSION OF LIQUIDS. [Exp. 25. the globule thus formed more or less mercury from the stem. The globule is then detached and forced backward into the stem, as has been previously de- scribed. To prevent it from all. returning to the bulb, the latter should be warmed somewhat. The thread will now, probably, be much too long ; but may, as we have seen, be shortened at pleasure. Certain difficulties which are occasionally met in these manipulations may be avoided by the cautious application of heat (*f[ 64). It is sometimes impos- sible to force mercury from the expansion chamber into the stem either through its weight or through its inertia, especially when through accident the ex- pansion chamber has been allowed to become com- pletely full. Heat should then be applied to the top of the expansion chamber until the mercury is driven out by the pressure of its own vapor. When a thread of mercury can be broken off in no other way, heat may be applied to the stem of the thermometer at the point where a separation is desired. When the mercury refuses to leave the bulb, the flow may be started by slightly warming it ; in fact, any desired quantity of mercury may be forced into the expan- sion chamber in this way (see, however, *fl" 65, (1)). When the calibration of a thermometer has been finished, as will be explained in the next section, it is well to remove the bubble of air from the mer- cury. This is done either by cooling the bulb in a freezing mixture (as, for instance, ice and salt) until no mercury remains in the stem ; or if this is impossible, by heating the bulb until the air is driven : 68.] THE MERCURIAL THERMOMETER. 113 ito the expansion chamber. In either case a slight jarring should free the bubble from the mercury. If the bubble is too small to respond to this treat- ment, it will hardly affect the accuracy of results, unless it actually causes a break in the mercurial column (see ^[ 65, (3) ). ^[68. Calibration of a Mercurial Thermometer. A thread of mercury, about 50 in length, 1 is placed so as to reach first from upwards, then from 100 downwards. The reading of the end near 50 is taken to a tenth of a degree in both cases, as will be explained below. This enables us to detect any dif- ference in calibre between the upper and lower parts of the thermometer. Next, a thread about 25 long is made to reach first from 0, then from 50 upwards, then also from 50 and from 100 downwards, with exact readings of the end near 25 or 75, as the case may be. These will enable us to compare the different quarters of the tube from to 100. It is not necessary, for most purposes, to carry the pro- cess of calibration any further. To avoid parallax ( 25) the eye may be held so that the divisions of the scale seem to coincide with their own reflections in the thread of* mercury. One end of the thread is always placed so as to coincide ex- actly with a given division line of the scale (0, 50, or 100), so that any error in the estimation of tenths of degrees will be confined to the reading of the other end. To reduce this error to a minimum, 1 A thread from 49 to 51 will answer. In cases presenting special difficulty, a greater latitude may be allowed. 114 EXPANSION OF LIQUIDS. [Exp. 25. the student is advised to study or to construct for himself diagrams like the following (Fig. 52), show- ing the appearance of a mercurial column when dividing the space between two lines into a given number of tenths, and to identify the reading in each case with the diagram which it most resembles. Before calculating a table of corrections (see *f[ 70) from the results of calibration, it is necessary to de- termine two " fixed points " on the scale of the ther- mometer, as will be explained in the next section. FIG. 52. If 69. Determination of the Fixed Points of a Thei> mometer. 1 I. The mercurial thermometer is placed in a steam generator (Fig. 53) so that the bulb and nearly the whole of the stem may be surrounded with steam. Only the divisions above 99 project above the cork (a) by which the thermometer is held in place. When the greatest accuracy is desired, the sides of the generator are made double, as in Fig. 54. By this means the inner coating, being surrounded on both sides with steam, will have a temperature of 100 nearly, and there will be no radiation of heat between it and the thermometer, since radiation de- pends upon a difference of temperature ( 89). It is 1 The student who is interested in the changes produced in a ther- mometer by the application of heat will do well to observe the freezing-point before as well as after the boiling-point. 169.] THE MERCURIAL THERMOMETER. 115 important also to construct a shield of some sort so that the boiling water in the bottom of the apparatus may not be spattered upon the bulb of the thermom- eter. Such a shield is moreover useful in prevent- ing the thermometer from dipping into the water. It must be borne in mind that the temperature of boiling water is very uncertain, being sometimes FIG. 53. FIG. 54. several degrees above the true boiling temperature, even when the water is perfectly pure, owing to the adhesion of the liquid to the sides of the vessel con- taining it. On the other hand, the temperature at which steam condenses depends only upon the pres- sure to which it is subjected. 116 EXPANSION OF LIQUIDS. [Exr. 25. It is possible, with an apparatus like that shown in Fig. 53, particularly if the spout (6) be small, to generate steam so rapidly that the pressure may be perceptibly greater within the generator than it is outside. Care must be taken to check the supply of heat until the feeblest possible current of steam issues continuously from the spout. The atmospheric pressure is then to be observed by means of a barom- eter ( [4] Fig. 53), and the reading of the thermom- eter determined within a tenth of a degree (see ^[ 68, Fig. 52). If the barometer happens to stand at 76 cm., this reading is called the "boiling-point" of the thermometer, otherwise a correction must be applied, as will be explained in the next section. II. The thermometer is now allowed to cool as slowly as possible to the temperature of the room, so as not to destroy its " temper" (^[ 65, (1) ), then surrounded in a beaker with a mixt- ure of water and finely-powdered ice (Fig. 55), well stirred and covering the scale within one or two divisions of the zero mark. The melting-point of ice is not n ff O XT perceptibly affected by barometric or ordi- nary mechanical pressure. The ice must be pure and clean. The bulb of the thermometer must not be jammed by the ice (^j 65, (7) ). The reading is to be accurately observed (^[ 68). This reading is called the " freezing-point " of the thermometer. The boiling and freezing points are called the two "fixed points" of a thermometer, and from them, with the results of calibration, a complete table of t 70.J THE MERCURIAL THERMOMETER. 117 corrections should be calculated, as will be explained in the next section. ^[ 70. Calculation of a Table of Corrections for a Thermometer. The correction of a thermometer at is found at once by reversing the sign of the read- ing in melting ice (see ^ 69, II., also ^ 41). If, for instance, the reading in melting ice is -fO.9, the correction at is 0.9. The correction at 100 is found by subtracting (algebraically) the actual read- ing in steam from the true temperature of steam cor- responding to the barometric pressure observed. (See Table 14.) Thus if the thermometer reads 99. when the barometer stands at 72 cm., since the true temperature of steam at this pressure is 98.5, the thermometer stands too high by 0.5, and the cor- rection is 0.5. It is obvious that under the nor- mal pressure (76 cm.) the thermometer would indicate 100. 5 instead of 100.0^ hence the standard boiling- point is 100.5 on this thermometer. We find the standard boiling-point in general by adding (numeri- cally) to 100.0 the correction (at 100) if the ther- mometer is found to stand too high, or subtracting the same if the thermometer stands too low. Let us now suppose that in the calibration of the thermometer a given thread of mercury reached from to 49. 5 ; if the bottom of this thread had been placed at the observed freezing-point (-}-0 .9) instead of at the mark 0, it would evidently have reached farther up the tube. Since the length of the thread can hardly vary by a perceptible amount when it is moved less than one degree, even in a tube with 118 EXPANSION OF LIQUIDS. [Exp. 25. considerable variations of calibre, we may assume that the thread would reach a point just nine tenths of a degree higher than before ; in other words, it would reach from 0.9 to 50. 4. In the same way, if the thread is found to reach from 100 to 50.7, we infer that it would have reached from the standard boiling-point (found by observation to be at 100.5) to a point five tenths of a degree above 50. 7, or 51. 2. Between 50.4, and 51. 2 we find a half-way point 1 on the thermometer, namely 50. 8. If the thread of mercury had been four tenths of a degree longer it would have reached to this half-way point, either from the freezing-point or from the boiling- point. We infer that the volume of the tube in- cluded between the boiling and freezing points is exactly halved at 50. 8. Now, by definition, the tem- perature at which the mercury reaches this point is 50.0, according to a perfect mercurial thermometer; hence the correction for the thermometer at 50 is -0.8. In the same way we find the correction of the thermometer at 25, then at 75, by considering how far the shorter thread (25 long) would have reached if one end had been placed at -}-0 .9 instead of 0, at 50.8 instead of 50, or at 100.5 instead of 100. We thus find two points near 25, and half-way be- tween them a third point, showing where the ther- mometer would stand at a temperature of 25, 1 This point is sometimes called the " middle point " of a ther- mometer ; but some authorities mean by the " middle point " one half-way between the divisions numbered and 100 respectively. U 71.] THE AIR THERMOMETER. 119 according to a perfect mercurial thermometer ; we find also the indication of the thermometer for a temper- ature of 75 ; and hence also the corrections at 25 and 75. The corrections at 5, 10, 15, etc., up to 100 are finally calculated by interpolation. Thus if the cor- rection at 25 is found to be 0.8, and at 75, 0.7, we should find the following table : TABLE OF CORRECTIONS. 0.9 25 0.8 50 0.8 75 0.7 50 o.9 30 0.8 55 0.8 80 0.7 IQO 0.9 35 0.8 60 0.8 85 150 _oo.8 40 0.8 65 0.7 90 20 0.8 45 0.8 70 0.7 95 250 o.8 60 0.8 75 0.7 100 0.5 EXPERIMENT XXVI. THE AIR THERMOMETER, I. ^[71. Calibration of an Air Thermometer. A simple form of air thermometer consists of a glass tube (ac, Fig. 56) about 40 cm. long, and 2 mm. in diameter, closed at one end (a). The tube has an a . / .- THE AIR THERMOMETER. 127 The coefficient of expansion of all permanent gases is in the neighborhood of .00367. EXPERIMENT XXVII. THE AIR THERMOMETER, II. *[[ 75. Construction of an Absolute Air-Pressure Thermometer. A form of air thermometer depend- _ ent almost entirely upon pressure is represented in Fig. 60. It consists of a U-tube (aic), with a large bulb (c) blown at the end of the shorter arm, and a somewhat smaller bulb (a) at the end of the longer arm. The apparatus is sealed at the at- mospheric pressure with enough mer- cury to fill the smaller bulb more than half-full. It is evident that at the absolute zero of temperature (see 75), in the absence of any pressure in either bulb, the mercury must stand at the same level in both arms of the U. To lo- cate the absolute zero accordingly, mercury is poured back and forth from one bulb to the other until no difference in the level is observed when the thermometer is returned FIG. 61. FIG. 60. 128 PRESSURE OF GASES. [Exp. 27. to a vertical position. The zero of a millimetre scale is now adjusted to this level (see Fig. 61). By pour- ing mercury into the bulb a (Fig. 60), and suddenly restoring the thermometer to an upright position, the mercury in the tube will be found to stand above its level in the cistern, owing to the compression of air in c and its rarefaction in a. This process is re- peated with more or less mercury in a until the column reaches a point b on the scale corresponding to the absolute temperature (see ^f 72). The ther- mometer should now indicate any temperature cor- rectly on the absolute scale, and has the advantage over that employed in Experiment 26 of being un- affected by atmospheric pressure. In practice, the bulb c is made so much larger than the tube (6) that no account need be taken of the variation of the mercury level in c. The height of the mercurial column is measured accordingly by a fixed scale. The expansion of the air in the bulb c is also disregarded, together with the compression of the air in a. All these causes tend to diminish the sensitiveness of the thermometer. The air thermometer represented in Fig. 60 depends upon the principle ( 76) that the pressure of a gas which is prevented from expanding increases in pro- portion to the absolute temperature. When both bulbs (a and c) contain gas, the pressure in each increases, and hence also the difference in pressure between them increases with the absolute temper- ature. It follows that the height of the mercurial column which can be maintained by the difference IT 76.] THE AIR THERMOMETER. 129 of pressure in question itself varies as the absolute temperature. ^[ 76. Determination of Temperature by the Pressure of Confined Air. 1 A tube (, we find t = !QQ~ z = ~ 100 t e h * h i *i 100 h It is believed that in the case of a perfect gas the coefficient which determines the increase of pressure per degree should be the same as the coefficient of expansion (Experiment 26). In practice, differences are observed even with the most permanent gases ; but these differences are small in comparison with the errors of observation which the student is likely to make. It is interesting to compare the temperature, , in- dicated by an air-pressure thermometer with that indicated by a mercurial thermometer, and to test the accuracy of the work by calculating the tempera- ture (z), at which air would be wholly devoid of pressure, as well as the coefficient e, relating to change of pressure. If the results agree with the values given in ^[ 74, within one or two per cent, the student will be justified in applying a correction to the mercurial thermometer. 132 PRESSURE OF VAPORS. (Exp. 28. EXPERIMENT XXVIII. PRESSURE OP VAPORS, I. ^f 77. Application of the Law of Boyle and Mariotte in the Air Manometer. One of the most important applications of the Law of Boyle and Mariotte ( 79) is in the construction of a pressure-gauge, or manometer. A simple form is represented in Fig. 62. It consists of a U-tube, closed at one end \\ (l an( * ^ e( * w * tn mercur y U P to a certain i JJ y level, corresponding to No. 1 on the gauge. The open end of the U-tube is connected * fej w ith the interior of a vessel, the pressure IG ' ' in which is to be determined. If the mer- cury stands as before at No. 1, we know that the vessel must be at the ordinary atmospheric pressure. If, however, the air in the closed arm is compressed to half its original volume, we know that the pressure must amount to 2 atmospheres ; if the air is reduced to one-third its original volume, the pressure is 3 atmospheres, etc. If, on the other hand, the air expands, the pressure must be less than 1 atmo- sphere. The pressure in atmospheres may therefore be indicated directly on a scale properly spaced. No. 2 is, for instance, half-way between the closed end of the tube and No. 1 ; No. 3 is one-third way ; No. 4 one-quarter way, etc. Such a gauge is useful in experiments where it is necessary to know roughly the pressure in a closed vessel, as, for instance, a IF 78.] THE AIR MANOMETER. 133 steam boiler. When accuracy is desired, it is neces- sary to increase the length of the tube, to calibrate it (see ^[ 71), and to allow for the hydrostatic pressure of the liquid in the bend. The tube already calibrated (^[ 71), for the purpose of measuring the expansion of air, may serve as a manometer. The manometer may be surrounded (if necessary) with water, to prevent the temperature from varying perceptibly in the course of the ex- periment. ^[78. Testing an Air Manometer. The tube () of the vapor 1 must be ^ 82. Evaporation and Boiling. The student will notice the regular increase of the quantity of aqueous vapor in the air as the temperature is increased, until finally, as the water approaches its boiling-point, scarcely any air remains in the flask. It is interest- ing to push the experiment still further, and to expel all the air by actually boiling the water. Boiling may be distinguished from evaporation by the presence of bubbles of pure steam. Unlike the bubbles of air set free from the water by the application of heat, the bubbles of steam may at first completely condense with a crackling sound before reaching the surface of the liquid. When, however, the whole liquid is raised to the boiling-point, the bubbles expand as they escape from the liquid, and if the supply of heat 1 We neglect in this formula the pressure 4.6mm. of aqueous vapor at 0. IT 82.] EVAPORATION AND BOILING. 139 is sufficient, furnish a steady current of steam which issues from the neck of the flask. The stopper is in- serted before boiling has ceased, but, to avoid explo- sion, not until the source of heat has been removed. When the vapor is condensed by pouring cold water on the bottom of the flask (Fig. 68), ebul- lition will take place even after the water within the flask is no longer warm to the touch. If the experiment has been suc- cessful, a peculiar metallic sound will be heard on shaking the water in the flask. This sound is called the water-hammer, and indicates an almost total absence of air. If the flask is opened under water, it should be completely filled. If opened in air, the space not already occupied by water will be filled with air. The student may be interested to make a rough determination of atmospheric density by weighing the flask before and after the admission of air (see ^[ 44). The capacity of the flask for air is found from the quantity of water which must be added to that already present in order to fill the flask (see ^[ 45). The principal objection to a deter- mination of density by this method lies in the fact that an unknown quantity of aqueous vapor may be taken up by the air which enters the flask. Its ad- vantage consists in the nearly perfect vacuum which is produced by the condensation of aqueous vapor. For further illustrations of evaporation and boiling, see Exercise 22 of the " Elementary Physical Experi- ments," published by Harvard University. 140 BOILING AND MELTING POINTS. [Exp. 30. EXPERIMENT XXX. BOILING AND MELTING POINTS. ^[ 83. Determination of Boiling and Melting Points. The heater already used to determine the boiling- point of water on a mercurial thermometer may also be employed to find the boiling-points of other liquids. The chief objection to this apparatus is the change of composition which results from boiling away an impure liquid, owing to the fact that the more vola- tile ingredients are the first to escape. It becomes necessary to condense the vapor before it escapes from the spout, and to make the liquid thus formed re- turn to the boiler. There are, moreover, two practical objections to the use of such an apparatus, the difficulty of obtaining a sufficient quantity of liquid to fill the boiler, and the danger of fire. These objections are met by boiling the liquid in a long test-tube, as in Fig. FIG. 69. gg^ pj ie va p 0r condenses on the sides but does not escape, and the danger of fire is avoided by the use of hot water instead of a flame as a source of heat. Alcohol, for instance, will boil freely if the test- tube is plunged in water at or near the temperature of 100, since the boiling-point of alcohol is between 78 and 80. As the water cools it may be used successively to find the boiling-points of chloroform t 84.] BOILING AND MELTING POINTS. 141 (58-61), bisulphide of carbon (47-48), and ether (35-37). It is well to have the water about 20 warmer than the boiling-point of the liquid which is to be determined. The same apparatus, or one with a shorter tube, may be used to determine melting-points. A piece of a paraffine candle rnay be melted in the test-tube by hot water; then, as it begins to harden, the temperature is observed. Again, by the use of hot water, the paraf- fine is gradually heated, and the temperature noted at which it begins to melt. Owing to impurity of the paraffine, certain constituents usually congeal more easily than others. It has, therefore, no definite melt- ing point. A certain variety of commercial paraffine melts, for instance, between 53 and 57. The results are to be corrected as explained below. ^[ 84. Precautions and Corrections in Determining Boiling and Melting Points. To prevent radiation to or from the bulb of the thermometer, and to avoid all danger of spattering (see ^[ 69, I.), a shield may be constructed out of thin sheet brass, small enough to fit into the test-tube. The bulb must not dip into the liquid, but must be surrounded with vapor. The level of the vapor will be distinctly visible through the sides of the tube. It should reach a point a little beyond the end of the mercurial column in the stem of the thermometer, but must in no case reach the open end of the test-tube. A slight escape of the vapor, due to evaporation, cannot be avoided ; but a continuous current must be instantly arrested by removing the source of heat. 142 BOILING AND MELTING POINTS. [Exr. 30. In finding melting-points, the bulb and stem of the thermometer should be surrounded with liquid up to a point just below the end of the mercurial column. If the stem be dipped any farther into the liquid, it may become impossible to read the thermometer. The student is advised not to attempt the deter- mination of boiling-points above 100 C., 1 on account of the danger of accidents. It may, however, be in- structive to explain how a temperature above 100 can be determined with a thermometer reading only to 100. A thread of mercury not over 100 in length is first broken off and stored in the expansion chamber (c, Fig. 51, ^[ 66). The thermometer is then tested in steam (^[ 69, I.). Its reading will be somewhat above ; let us say 15. Then all the readings of this thermometer will be about 85 too low. It is possible, therefore, to determine temperatures up to 185. We should, however, remember that a column measuring 85 at a temperature of 100 will measure more or less than that amount, according to the tem- perature in question. Let the length of the thread of mercury, in degrees, be I, and let the temperature at which this thread is actually observed be t (100 in the instance above) ; then if , is the tempera- ture to be determined, the correction in degrees is .00018? (t O- This follows from the value of the coefficient of expansion of mercury ; for if a thread 1 Chloroform should be substituted for turpentine (which boils at about 160) in the second Experiment in Physical Measurement in the list published by Harvard University. IT 84.] BOILING AND MELTING POINTS. 143 1 long when heated 1 centigrade expands by the amount 0. 00018, then a thread 1 long when heated (t Zj) would expand I X (tt J times as much. Thus the correction in determining the boiling-point of turpentine (160) with a thread 85 long, broken off and measured at the temperature 100 instead of 160, would be .00018 x 85 X (160 100), or a little over 0.9. Instead, therefore, of adding 85 to the reading of the thermometer (let us say 74) we should add, strictly, 85.9, that is, the actual length of the thread of mercury at the temperature observed. In- stances have already been given (^[ 65, (4)) of errors resulting from heating only the bulb of a thermometer to a given temperature. The corrections in such cases are calculated by the rule given above. That is, we multiply the length of the thread exposed to the air by the difference in temperature between the air and the bulb of the thermometer, to find the cor- rection which should be applied. In all determinations of temperature, the readings of the thermometer are made to tenths of a degree (^[ 68), and corrected by the table already calcu- lated (T 70). The boiling-points of all liquids are affected more or less by atmospheric pressure. A reading of the barometer should always accompany such determinations. 144 CALORIMETRY. [Exp. 31. EXPERIMENT XXXI. METHOD OF COOLING. ^[ 85. Determination of Rates of Cooling. A Cal- orimeter (Fig. 70) is usually constructed of two (or more) metallic cups, one inside of the other. A vertical section of the calorimeter is shown in Fig. 71, and a horizontal section in Fig. 72. The inner cup, generally made of thin brass, has its outer sur- face brightly polished to lessen radiation; and for the same reason the outer cup should be polished inside. To prevent the conduction of heat from one FIG. 72. FIG. 70. FIG. 71. cup to the other, the cups are separated by pieces of cork, which should be sharpened to a point, and held in place by wires. A large flat cork serves to cover both cups, and thus in a great measure to prevent loss of heat ; for if the top of the calorimeter were open, a considerable quantity of heat would be carried away by currents of air. In some cases Ti 85.] METHOD OF COOLING. 145 a small stopper is also used, to close the inner cup water-tight. We prefer for most purposes a calorimeter de- pending (like that shown above) upon air spaces for its insulation, to one in which these spaces are filled with wool, or other non-conducting material; 1 for though air transmits more heat than wool, it absorbs much less. The heat absorbed by insulating materials is a continual source of error in calorimetry, because there is no simple way of allowing for it. On the other hand, the heat transmitted through the sides of a calorimeter can, as we shall see, be easily determined. (1) The inner cup is to be filled with hot water, be- tween 90 and 100, and the temperature of the water is to be found by a thermometer passing through a hole in the cork cover (Fig. 71). The stirrer at- tached to the stem of the thermometer is used to keep the water in continual agitation ; and a stopper is employed to prevent any of it from being spilled over the edges of the cup. Observations of tem- perature are made at intervals of one minute, 2 and should be continued until the thermometer indicates 30 or 40 degrees. The temperature of the room is then observed ; and the quantity ol water which has 1 When no allowance is to be made for loss of heat by the calori- meter, the use of felt is to be recommended. See Experiment 10 in the Descriptive List of Chemical Experiments published by Har- vard University. 2 A clock especially constructed to strike a bell once a minute will be found serviceable in the determination of rates of cooling. Simultaneous observations of time and temperature may thus be made (see 28). 10 146 CALORIMETRY. [Exp. 31. been used is determined by weighing the calorimeter with and without it. (2) The experiment is now to be repeated with a much smaller quantity of water, just enough, let us say, to cover the bulb of the thermometer and the stirrer. The calorimeter is to be inclined in every possible direction between the observations of tem- perature, so as to bring the hot water in contact with every part of the inner cup. (3) The experiment is again repeated with the same quantity of water as in (2), but without inclin- ing the calorimeter. The stirrer is to be used as in (1), but simply to secure a uniform temperature in the water. (4) Finally, the calorimeter is to be filled with glycerine or turpentine, warmed by hot water (see Tf 83). The depth of the liquid, and the method of agitation should be the same as in (1). The tem- peratures and weights are to be observed as before. The results of (1), 10 20 3o m ^o j-p 60 yo . (2), (3), and (4) are to be reduced as will be explained in ^[^[ 86-89. Tf 86. Effect of the Temperature and Ther- mal Capacity of a body on its Rate of Cooling. -T 1 g. I O. (1) The results of ^[ 85 (1) are to be represented by a curve (aJ, Fig. 73), drawn on co-ordinate paper as in 59. The s \ \ ff \ ^ ft Xjjjg \ ^ "^ X '-., "-. Ter l>er. & rr o f W. * vo om; |3ir: 1 86 (1).] NEWTON'S LAW OF COOLING. 147 scale at the top of the paper corresponds to the num- ber of minutes which have elapsed since the first observation was taken; the scale at the left of the paper represents the observed temperature of the water in degrees. The temperature of the room (22|) is shown by the dotted line, which the curve (6) should approach as a limit, that is, without ever reaching it. It is advantageous for many purposes that the scale of degrees at the left of the paper should repre- sent, not the temperatures actually observed, but the differences between those temperatures and that of the room ; l since the rate of cooling depends upon the differences in question (see 89). If this method is adopted, the first observation should be one about 50 above the temperature of the room. In any case the student should satisfy himself that Newton's Law of Cooling ( 89) is approximately fulfilled. 2 Thus the calorimeter may cool (see 5, Fig. 73) between the 5th and the 10th minute from 75 to 70, that is, 5 in 5 minutes ; while between the 50th and the 60th minute it may cool only from 44 to 40o, or 4 in 10 minutes. In the first case, when the average temperature (72^) is 50 above that of the room (22) we have a rate of cooling equal to 1 per minute ; in the second case, with an average temperature (42) nearly 20 above that 1 This method of plotting the curves must be adopted if the temperature of the room varies considerably in the course of the experiment (IF 85). 2 Departures of 20% have been observed in a range of 60. See Everett's Units and Physical Constants, Art. 143. 148 CALORIMETRY. [Exp. 31. of the room, the rate of cooling is | per minute. Obviously, 50:20 :: 1 : f . In the same way, with 20 grams of water in the calorimeter, the rate of cooling should be found to vary in proportion to the excess of temperature above that of the room. The rate of cooling is, however, very different in different cases, as it depends upon the quantity of water which the calorimeter contains. Let us next consider the relation between this quan- tity of water and the rate of cooling. (2) The fundamental principle underlying all de- terminations by the method of cooling is that the number of units of heat ( 16) lost by a calorimeter per unit of time is proportional to the difference in temperature between the inner and outer cups. It does not, therefore, depend upon the contents of the calorimeter except in so far as the nature or quantity of these contents may modify the temperature of the inner cup. Let us first suppose that in both experiments, ^[ 85 (1) and (2), the water is agitated sufficiently to bring it in contact with every portion of the inner cup, so that a perfectly uniform temperature is the result ; then if the outer cup is unchanged in temperature the flow of heat from one cup to the other corresponding to a given reading of the ther- mometer must be in both cases the same. How, then, do we account for the marked differences which we observe in the rates of cooling? T 86 (2).] RATES OF COOLING. 149 The supply of heat in a calorimeter may be com- pared to the quantity of water in a leaky pail. Given the rate of the stream flowing out of the pail, the time it takes for the water-level to fall one inch is evidently proportional to the horizontal section of the pail. In the same way, with a given flow of heat from a calorimeter, the time required for the temper- ature to fall 1 must be proportional to what we call the thermal capacity ( 85) of the calorimeter and its contents. It is obvious from Figure 73 that with 80 grams of water the cup must cool more slowly than with 20 grams. In the first case it takes, for instance (see aft, Fig. 73), 60 minutes to cool from 80 to 40 ; if in the second case only 20 minutes are required to cover the same range of temperature, the natural inference is that the thermal capacity in the first case is to that in the second case as 60 is to 20, or as 3 is to 1. The thermal capacity in question is in no case simply proportional to the quantity of water which the calorimeter contains ; for the inner cup, the ther- mometer, and the stirrer all possess a certain capacity for heat. We may estimate this capacity roughly by the method of cooling. Let us call it c. Then in the first case the total thermal capacity is 80 -\- c and in the second case it is 20 -j- c; hence we have 80 -f c : 20 -f c :: 3:1, a proportion which can be satisfied only if c = 10. We infer, therefore, that the calorimeter, thermom- 150 CALORIMETRY. [Exp. 31. eter, and stirrer are together equivalent, in thermal capacity, to about 10 grams of water. We may assume provisionally that this inference is correct ; but for accurate calculations, we prefer a determination of thermal capacity made as will be described in Experiment 32. ^[ 87. Calculations concerning Loss of Heat by Cool- ing. We have found in the last section (^[ 86, 1), that when a certain calorimeter contains 80 grams of water at an average temperature 50 above that of the room, the rate of cooling is 1 per minute. We have also found (^[ 86, 2) that the calorimeter itself is equivalent in thermal capacity to about 10 grams of water ; hence the total thermal capacity is 80 -f- 10, or 90 units. The heat lost under these conditions is therefore 90 X 1, or 90 units per minute. Let us now suppose that the average temperature is only 1 above that of the room, instead of 50; then by Newton's Law ( 89) the rate of cooling will be -ffe of 1 per minute ; hence the loss of heat will be 90 X -gV, or 1'8 units per minute. It follows from the fundamental principle of the method of cooling (^] 86, 2) that the loss of heat at a given temperature is the same, no matter what substance or substances the calorimeter may contain, provided that every part of the inner cup is brought in contact with the mixture. The rate of flow cor- responding to difference in temperature of one degree between the inner and outer cups is accordingly an important factor in calculations (see ^[ 93, 3) relating to loss of heat by cooling. 187.] RATES OF COOLING. 151 Unless the calorimeter is filled, as in ^[ 85 (1), or its contents sufficiently agitated, as in (2), the inner cup will not be uniformly heated throughout. When a glass vessel is used (as in Exp. 38), only those por- tions nearest the liquid may be perceptibly warmed or cooled by it ; and even with metallic vessels, es- pecially when thin, differences of temperature can frequently be recognized by the touch. The result is a considerable diminution in the rate of cooling. To estimate the effect in question, we may utilize the results of ^[ 85 (3). From these results the curve ac (Fig. 73) is to be plotted in the same manner as ab (If 86, 1). If in both curves (as in Fig. 73) the first observation utilized is about 80, we shall find a point of inter- section, a, nearly opposite 80 and minutes. We may notice that ab takes 60 minutes to fall from 80 to 40, while with ac only 30 minutes are re- quired ; hence the rate of cooling represented by ac is twice as great as in the case of a, so that when reduced to 1 difference in temperature, it will be 5^ of 1 per minute. Now let the weight of water be 20 grams ; then since the calorimeter is equivalent to 10 grams, 1 we have a total thermal capacity of 30 units. The loss of heat is therefore, not 1-8, as before, but 30 X 5% or 1-2 units per minute. These figures are sufficient to show the importance, in the method of cooling, of comparing two quantities 1 We should remember, strictly, that if only a portion of the inner cup is heated, the thermal capacity will be somewhat less than 10 units. 152 CALORIMETRY. [Exp. 31. under exactly the same conditions. Let us suppose that we were to calculate the thermal capacity of the calorimeter from the results of IF 85 (1) and (3), in which the conditions are not the same. Since the rate of cooling is twice as great in (3) as in (1), we might infer that the thermal capacity of the calorimeter with 80 grams was twice that with 20 grams. This would make the thermal capacity of the calorimeter alone 40 units instead of 10 (see ^[ 86, 2). ^[ 88. Construction of a. Series of Temperature Curves. From an extended series of results 1 it would be possible to construct a series of curves similar to those shown in Fig. to\nu*es 74. It is not, however, necessary that each of these curves should be the result of observa- tion. From two of them, the rest may be obtained with more or less accuracy by differ- Fm 7 ent processes of inter- polation. Let acegi and abdfh be the two curves already obtained (see Fig. 74), corresponding respectively to 80 grams and to 20 grams of water, and let it be required to draw a curve corresponding to 50 grams of water. Then since 50 is midway between 80 and 1 The teacher may, for the sake of illustration, have a series of curves constructed from the results of a large class of students using different quantities of water. / 9 7.0 30 40 5~0 fo 70 V *1 K d ^ I 3^ 7 ^ '>^ A. X ^ /-" o*>~~- fo Tern f,pr atfi YP f ffl p rc om nV T88.J CURVES OF COOLING. 153 20, the curve in question may be placed (roughly) midway between the other two ; and in the same way other curves may be drawn so as to divide the distance equally into still smaller parts. This method of interpolation is, however, obviously inaccurate, and especially so between such wide limits. A more accurate method depends upon the prin- ciple (see ^[ 86, 2) that the time of cooling is (other things being equal) proportional to the thermal ca- pacity of the calorimeter and its contents. Since 80 grams require, for instance, 10 minutes to cool from 80 to 70, and 20 grams take only five min- utes (see Fig. 74), we may infer that 50 grams would require 7^ minutes; or in other words, that the distance be would be bisected by the 50-gram curve. In the same way the other horizontal dis- tances, de,fg, hi, etc., would be bisected. To obtain the intermediate curves, accordingly, the horizontal distances, be, de, fg, etc., are each to be divided into a given number of equal parts. The curves may then be drawn through the points of division. It is easy to show that this method of interpolation, though more accurate than the first, may still lead to considerable errors, when we consider differences in the flow of heat from the calorimeter. With 80 grams of water, 1 above the temperature of the room, we have calculated that the loss of heat amounts to 1-8 units per minute (see ^[ 87) ; with 20 grams we have found similarly 1 -2 units per min- ute. Let us assume that with 50 grams the loss is midway between these two numbers, or 1-5 units 154 CALORIMETRY. [Exp.31. per minute. Then since the total thermal capacity is 60 units, the temperature must fall at the rate of 1-5 -T- 60 or^g-of 1 per minute. The time required to fall 1 at this rate would be 40 minutes ; in the case of 80 grams it would be 50 minutes (see ^[ 87) ; in the case of 20 grams it would be 25 minutes. The times required for 80, 50, and 20 grams to fall through a given range of temperature would be, accordingly, proportional to the numbers 50, 40, and 25, respectively. Since 40 is by no means midway between 50 and 25, the 50-gram curve must be con- sidered as only approximately bisecting the horizontal distance between the other two. It is evident that if the system of curves shown in Fig. 74 were to be relied upon for exact calcula- tions, it would be necessary to confirm the position of the 50-gram curve, at least, by direct observations. As a matter of fact we shall refer to Fig. 74 only for the purpose of making small corrections for cool- ing ; so that we may disregard any errors in these curves which are likely to arise from an interpola- tion depending upon a division of horizontal distances into equal parts. ^| 89. Calculation of Specific Heat by the Method of Cooling. I. A set of curves is to be constructed essentially as in ^[ 88, using, however, in connection with the curve acegi (Fig. 74) representing the re- sults of ^[ 85 (1), a curve abdfh, derived from the results of ^[ 85 (2), and not (as in Fig. 74) from the results of ^[ 85 (3). The intermediate curves will then represent rates of cooling corresponding to 189.] CALCULATIONS FROM COOLING. 155 different quantities of water when brought in contact with every part of the inner cup. The results of ^[ 85 (4) are next to be plotted on tracing-paper, with a horizontal line (as in Fig. 73 to) represent the temperature of the room. This line is then super- posed (by moving the tracing-paper) over a similar line in the new series of curves ; and at the same time the curve on the paper is made to pass through the common point of intersection of the series in question (see a, Fig. 74). A curve thus obtained with, let us say, 75 grams of turpentine, may be made to coincide, not with the 70-gram curve, nor with the 80-gram curve (see Fig. 74), but with one rather which would correspond to 30 or 40 grams of water. Under the conditions of the experiment, the heat lost by the calorimeter must be the same whether it contain turpentine or water (see ^[ 86, 2) ; hence equal rates of cooling imply equal thermal capacities (ibid.}. Since the calori- meter has the same total thermal capacity with the turpentine as with the water, the 75 grams of turpen- tine must be equivalent to 30 or 40 grams of water ; and 1 gram of turpentine must be equivalent to a quantity of water between f and $% of a gram ; or let us say 0.4 -f- grams. In other words, the specific' heat ( 16) of turpentine must be 0.4+- In the same way the specific heat of any other liquid might be calculated. It is evident that the curves of ^[ 88, if thus treated, would not have given an accurate result. 20 grams of water might be found, for instance, 156 CALORIMETRY. [Exp. 31. under the conditions of ^[85 (3), to cool as slowly as the 75 grams of turpentine in ^[ 85 (4) ; but this would be due, not simply to the fact that water has a greater thermal capacity than turpentine, weight for weight, but also to the fact that a much smaller amount of surface is heated by the water. Obviously the 20 grams of water cannot be equivalent in thermal capacity to the 75 grams of turpentine, be- cause their rates of cooling, though equal, have been compared under dissimilar conditions. II. Another method of calculating specific heat depends upon a comparison of the rates of cooling of two liquids when equal volumes are employed. Let us suppose that the time occupied by 75 grams of turpentine in cooling from 80 to 60 in ^ 85 (4) is really the same as that of 20 grams of water in ^[ 85 (3), that is, 10 minutes (see ac, Fig. 73), while that required in ^f 85 (1) for 80 grams of water (see a5, Fig. 73) is 20 minutes ; then since the conditions are nearly the same in (1) and (4), the total thermal capacities in question must be to each other as 10 is to 20 (^[ 86, 2). If the calorimeter is equivalent (see ^[ 86, 2) to 10 grams of water, we have with 80 grams of water a total thermal capacity of 90 units; hence with the turpentine the total thermal capacity must be -|$ of 90 units, or 45 units. Sub- tracting from the 45 units the 10 units due to the calorimeter, we find a remainder of 35 units, which must be the thermal capacity of 75 grams of turpen- tine. Hence the specific heat of turpentine is 35 -f- 75, or 0.4 +. IT 90(1).] THERMAL CAPACITY. 157 The method of cooling has been applied to the determination of the specific heats of solids in the form of powder, as well as to liquids; but it is gen- erally thought to be less reliable than the methods of mixture about to be described (Exps. 33 and 34). EXPERIMENT XXXII. THERMAL CAPACITY. ^[ 90. Determination of the Thermal Capacity of a Calorimeter. (1) We have already seen that the thermal capacity of a calorimeter may be calculated roughly from data obtained by the method of cooling (see ^[ 86, 2) ; but that a very slight change in the conditions of the experiment may make the result worthless. For this reason the method of cooling is hardly to be counted as a practical method for finding the thermal capacity of a calorimeter. The experimental determination of thermal capacity may be made by either of the following methods : I. The whole calorimeter is to be weighed, includ- ing (see ^[ 85, Fig. 71) the inner and outer cups, the cork supports and cover, and the thermometer and stirrer. The temperature of the inner cup is now found by observing the thermometer, after it has remained within this cup for some time (see Tf 65, 6). Then water at an observed temperature, between 30 and 40, is poured rapidly (^[ 92, 4) into the cup until it is nearly full (f 92, 8). The 158 CALORIMETRY. [Exp. 32. cork is immediately inserted (^[ 92, 6) and the time noted (Tf 92, 9). The water is then stirred (Fig. 50, ^[ 65) by twisting the stem of the thermometer, until two successive observations of the thermometer a minute apart (see ^[ 92, 10) agree as closely as in ^[ 85 (1), at the same temperature (see ^| 92, 8). The resulting temperature is then observed, and the time again noted (^j" 92, 9). The whole apparatus is then re-weighed to find how much water is in the calorimeter (see also ^f 92, 5). There are two practical objections to the method just described : first, that the change in temperature of the water is almost too small to be measured accu- rately with an ordinary thermometer ; and second, that the quantity of heat absorbed by the calorimeter may be small in comparison with that lost by cooling (^[ 93), which can only be roughly allowed for. The change of temperature of the water may be increased by using a smaller quantity of it ; but this is objectionable, as will be seen by comparing the results of ^[ 85, (2) and (3), unless the water can be well shaken in the calorimeter, or unless the object of the experiment be a determination of thermal capacity of the calorimeter when partly full. A ther- mometer graduated to tenths of degrees will be found useful in this and other experiments where it is ne- cessary to measure small changes of temperature. II. Another method of finding the thermal capa- city of a calorimeter consists in heating the inner cup instead of the water. This may be done by fill- ing the cup with hot lead (or better, copper) shot, f90(2)] THERMAL CAPACITY. 159 the temperature of which is to be determined by two or three observations of a thermometer at intervals of a minute (see ^[ 92, 10). The shot must be well shaken between these observations, to secure a uni- formity of temperature (see ^[ 92, 8) ; it is then poured out, and immediately replaced by water at an observed temperature near that of the room. The resulting temperature is then determined, and the weight of water used is found as before. The change in temperature of the water may be made practically five or ten times as great in II. as in I., and the correction for its cooling will be com- paratively slight. The principal source of error in this experiment is the rapid cooling of the inner cup while empty (see ^f 92, 4). (2) The results of an experimental determination of thermal capacity should in all cases be confirmed by a calculation based upon observations of the weights and specific heats of the substances employed in the construction of the calorimeter. The inner cup is to be weighed, also the stirrer (Fig. 50, ^f 65) ; and the amount of water displaced by the thermometer is to be found by the aid of a small measuring- glass (Fig. 75). The glass should be filled with water so that the thermometer may be immersed to the same depth as when it is used to determine the temperature of liquids in the calorimeter. The level of the water is then carefully observed with and without the thermometer. It will be assumed that the thermometer is constructed of glass and mercury; 160 CALORIMETRY. [Exp. 32. the calorimeter and stirrer of brass; otherwise the materials in question must be noted. From these data the thermal capacity of the calorimeter may be calculated (see ^ 91, III.)- IT 91. Calculation of Thermal Capacity. We have already considered a method by which thermal ca- pacity may be roughly computed through a compar- ison of rates of cooling (If 86, 2). This section relates to the calculation of thermal capacity from the observations made in IT 90. If, as in the first method (If 90, I.), ^ is the orig- inal temperature within the calorimeter, w the weight of water used, t 2 its temperature just before it is poured into the calorimeter, and t the resulting tem- perature, then, since w grams of water cool (t. 2 ) degrees by coming in contact with the calorimeter, they must give up to it w X (t z ) gram-degrees, or units of heat ( 16). This raises the temperature of the calorimeter (t Q degrees; hence to raise it 1 would require a quantity of heat, c, given by the formula tpxft-Q j t t This is, by definition ( 85), the thermal capacity of the calorimeter. To find the temperatures t and 2 , at the time when the water is introduced into the calorimeter, allowances for cooling should be made (see 1T^). The second method (1" 90, II.) differs from the first in that fy grams of water are warmed (t >) de- grees, and hence must receive w X (t 2 ) units of IT 91.] THERMAL CAPACITY. 161 heat from the calorimeter, the temperature of which is thereby reduced (^ t) degrees ; hence to reduce it 1 would require a quantity of heat, c, given by the formula x(t-Q. (,-<) This formula is evidently reducible to the same form as I. In the last method (^1 90, 2) if w 1 is the weight of the inner cup, w z that of the stirrer, and w s the weight (or volume) of the water displaced by the thermome- ter ; if furthermore s t and s 2 are the specific heats, respectively, of the materials of which the inner cup and the stirrer are made, 1 and s a the thermal capacity of a quantity of mercury and glass equal in volume to a gram of water ; 2 then the thermal capacity of the inner cup is u\ s { ; that of the stirrer, w 2 s 2 ; that of the thermometer, w 8 s 3 ; hence the total thermal capacity of the calorimeter (c) is given by the formula, c = w l s 1 + w 2 s 2 + w s s 3 . III. If, for example, the inner cup contains 100 g. of brass, of the specific heat .094, its thermal capacity is 1 The inner cup and t stirrer are usually made of brass (an alloy of copper and zinc), the specific heat of which may be taken as .094. 2 It will be noted that though the specific heat of mercury (.033) differs greatly from that of glass (0.19), the thermal capacity of equal volumes is very nearly the same. Since 1 cu. cm. of mercury weighs 13.6 grams, it will require 13.6 X .033, or 0.45 units of heat, to raise it 1. In the same way, since 1 cu. cm. of ordinary glass weighs not far from 2.5 grams, it would require about 2.5 X 0.19, or 0.47 units of heat to raise it 1. In calculating the thermal capacity of a thermometer, there will be, accordingly, no appreciable error in assum- ing for s 3 a mean value, 0.46. 162 CALORIMETRY. [Exp. 32. 100 X .094, or 9.4 units ; if the stirrer is made of thin brass weighing 2 grams, its thermal capacity is similarly 0.2 units ; and if the thermometer displaces 0.9 grams of water, its thermal capacity is (see 2d footnote, page 161) 0.9 X 0.46, or about 0.4 units. The total thermal capacity of a calorimeter thus con- structed would be 9.4 + 0.2 + 0.4 = 10.0 units. The first method is apt to give too high results, since the cooling of the water, due to evaporation and other causes, is attributed to contact with the calorimeter. The second method usually gives too low results, on account of the rapidity with which heat escapes from the calorimeter while empty. If, however, the outer cup becomes heated indirectly by the shot, a portion of this heat may be radiated back to the inner cup when filled with water. It is possible, therefore, that the results may be too great. The last method generally gives too small a re- sult, because we neglect the heat absorbed by the materials surrounding the inner cup. If, however, only a portion of the inner cup is to be heated, we may easily over-estimate its thermal capacity. In the latter case, we prefer an experimental deter- mination of thermal capacity ; but when the inner cup is made of very thin metal (as is desirable for accurate work), the thermal capacity may be so slight that it cannot be exactly determined by ex- periment. In such cases, we usually depend upon a calculation based, as in the last method, upon the weights and specific heats of the materials composing the calorimeter. 1T 92 (2).] PRECAUTIONS. 163 ^[ 92. Precautions Peculiai to Calorimetry. In nearly all experiments in calorimetry two bodies, of known weights and temperatures, are brought together so that by the flow of heat from one to the other (see Experiments 33 and 34) or by the action of one on the other (see Experiments 35-38) a third tempera- ture results. There are, accordingly, many precau- tions common to these various experiments. (1) CHEMICAL ACTION. It is evident that the substances employed should exert no chemical action on the sides of the calorimeter. With strong acids, a glass vessel should generally be employed. Instead of a brass stirrer, one of platinum may be used. In the case of mercury, iron will do even better. A coating of asphaltum is often sufficient to prevent metals from being attacked by acids. When two substances are placed together in a calorimeter, neither should act chemically upon the other unless the object of the experiment be to meas- ure the heat developed by the reaction. The chemical relations between two substances thus employed must frequently be investigated by a separate experiment. (2) COMPARISON OF THERMOMETERS. The gen- eral precautions necessary to the accurate observation of temperature have been already considered (^[ 65), and must be observed. In addition to these precau- tions, certain others are required when simultaneous observations of temperature are to be made. In such cases it may be necessary to employ as many ther- mometers as there are temperatures to be determined ; and these thermometers have to be compared with 164 CALORIMETRY. [Exp. 32. one already tested by a process of calibration (^[ 68). To do this, the several thermometers are to be placed in boiling water, in ice-water, and in water of at least three intermediate temperatures. A large quantity of water should be used (see (3)), and it must be well stirred in each case. The indications of each ther- mometer are to be read in turn ; then again read in the inverse order. There should be regular intervals (let us say 30 seconds each) between the observa- tions. The two readings of each thermometer are to be averaged, and the^ averages compared. Know- ing (from Experiment 25) the corrections for one of the thermometers, we may easily calculate the cor- rections for the others. For example, if three ther- mometers, A, B, and (7, gave the following readings : A, 76.0 ; B, 75.7 ; <7, 75.l ; C, 74.7 ; B, 74.5 ; A 74.0; the average for A would be 75.0 ; for B, 75.l ; for O, 74. 9. These averages evidently correspond to the same point of time. We should therefore sub- tract 0.l from the correction of A at 75 to find that of B ; and we should add 0.l to find that of C. The object of making observations in the order given above is to eliminate errors due to cooling. (3) CONSTANT TEMPERATURE. The difficulty of making accurate observations of temperature at a given point of time increases with the rate of cool- ing. The use of large masses of water (see (2)) is one of the most general methods of avoiding rapid changes of temperature. In certain experiments in T 92 (4).] PRECAUTIONS. 165 calorimetry, special devices are frequently employed. When, for instance, one of the temperatures to be observed is in the neighborhood of 100, a steam- heater may be employed (see Fig. 77, also Fig. 79, T[ 94). Again, a body may be maintained at by surrounding it with melting ice ; or it may be kept indefinitely, without special precautions, at the tem- perature of the room, provided that the latter be constant. By the use of devices for maintaining a con- stant temperature, thermometric observations become greatly simplified. One or more temperatures may be known by definition, as in the case of ice, or steam at a certain pressure ( 4). In the absence of cooling, a series of observations for each temperature will not be required, and the temperatures of several bodies at a given point of time may be found from successive observations with the same thermometer. The least constant temperature should be observed nearest the time in question. (4) EXPOSURE TO THE AIR. When a body is transferred from a heater or from a refrigerator to a calorimeter, there is always more or less heat gained or lost from exposure to the air. The time of exposure should evidently be made as short as possible. In pouring liquids, a glass funnel may be employed ; but the funnel must be warmed to the same temperature as the liquid, otherwise it would take from it more heat than the air. Water may be guided conveniently from a beaker to a calorimeter by a wet glass rod, ale, bent as in Figure 76. To 166 CALORIMETRY. [Exp. 32 . prevent the water from following the side of the beaker, the lip should be greased at the point b. The wet stem of a thermometer may also be used as a conductor, and with this advantage, that, since the thermal capacity is easily found (IT 90, 2) the heat required to raise it to a given temperature may be 76 calculated. We may notice, how- ever, that if the thermometer is immediately afterward placed in the calorimeter, it will give up most if not all of the heat which it has absorbed, and that the remainder may be neglected. Hot shot may be poured directly from a heater suit- ably shaped (see Fig. 79, ^[ 94) into a calorimeter ; but it is safer to use a paper funnel, to prevent the possibility of losing a portion of the shot. Most of the shot should enter the calorimeter without touch- ing the funnel ; and the remainder should be in con- tact with it only for an instant. In this case the heat absorbed by the paper may be neglected. A hot body may also be suspended by a thread, and thus trans- ferred from one place to another. It is obvious that the calorimeter should be brought as near the heater or refrigerator as is possible with- out danger that its temperature may be affected by radiation, conduction, or convection from the heater ( 89). A common pine board makes an excellent shield. In Regnault's apparatus l (Fig. 77) the 1 For a fuller description of Regnault's apparatus, see Cooke's Chemical Physics, page 470. T92(4).] PRECAUTIONS. 167 calorimeter (at the left of the figure) can be brought directly under the large steam heater (at the right of the figure). The steam heater rests upon a sup- port, serving to shield the calorimeter from radiation. The support is made hollow, so that it may be kept cool by a current of water. The inner chamber of the heater contains hot air. The temperature within it is observed by means of a thermometer passing through a cork by which the top of the chamber is FIG. 77. closed. The bottom of the chamber is closed by a non-conducting slide. By drawing the slide a body suspended by a thread in the hot-air chamber may be lowered directly into the calorimeter. The cal- orimeter is then immediately removed to a sufficient distance from the heater, so that the resulting tem- perature may be accurately determined. By devices similar to those alluded to, the gain 168 CALOKIMETRY. [Exp. 32. or loss of heat by exposure to the air may be almost indefinitely reduced, but never completely avoided. The student is advised riot to attempt any correction for this heat; because a greater error might easily result from applying such a correction than from neglecting it altogether. At the same time, it is well to estimate roughly the quantity of heat gained or lost, with a view to determining what figures of the final result are likely to be affected. For this purpose two experiments may be made. In one, a body is transferred in the ordinary manner from the heater or from the refrigerator to the calori- meter. In the second experiment, it is passed back and forth let us say 5 times each way, and finally placed in the calorimeter. The body is thus to be exposed to the air in one case about 11 times as long as in the other case, and under similar condi- tions ; so that from the difference in the results we may infer the effect of an ordinary exposure (see IT 93, 4). (5) Loss OF MATERIAL. In rapidly pouring a liquid into a calorimeter, or in rapidly lowering a hot solid into a liquid already contained in a calori- meter, there is danger that a portion of the liquid or solid may be lost. It is accordingly desirable to weigh, both before and after each addition to the contents of the calorimeter, not only the calorimeter itself, but also the vessel in which the substance in question was originally contained. The student will do well also to make sure that the space between the inner and outer cups is empty, both before and after Tf 92 (7).] PRECAUTIONS. 169 the experiment ; for if any of the substance finds its way into this space, its loss will not be apparent from the weighings. (6) EVAPORATION. A considerable portion of the heat lost by a liquid when poured into a calori- meter may be caused by evaporation. When once the liquid has been transferred to the calorimeter, all further loss of heat by evaporation should be prevented by immediately corking the inner vessel. It will be assumed that the inner vessel is never uncorked, except when necessary for the purposes of manipulation. Of two liquids, the denser is usually the less volatile, and hence should be heated in pref- erence to the other. For the same reason, a solid should be heated in preference to a liquid. A com- bustible liquid should, as we have seen (Exp. 30), never be heated directly by a flame, but indirectly by hot water. (7) TEMPERATURE OF THE ROOM. The loss of heat which takes place from the gradual cooling of a calorimeter and its contents depends, as we have seen in Experiment 31, upon the difference of tem- perature between the inner cup and its surroundings. To diminish the loss of heat in question, it has been proposed that the outer cup should be placed in water at the same temperature as the inner cup. More ac- curate results might be expected from calorimetry if some means were perfected by which we could ad- just the temperature of surroundings to the needs of an experiment. In practice, however, the experi- ment must be adapted to the temperature of the air in 170 CALORIMETRY [Exp. 32. which it is to be performed. When considerable time is required to obtain an equilibrium of -temperature (see (8) ), it is important that the average temperature within the calorimeter should agree as closely as possible with that of the room. The weights and temperatures of the substances employed in calor- imetry, are, therefore, frequently chosen so as to give a final temperature between 20 and 25. It is much easier to prevent than to allow for losses of heat by cooling ; and it may be stated as a general rule in calorimetry that we must avoid in so far as possible all differences of temperature between bodies under observation and the objects by which they are surrounded. (8) EQUILIBEIUM OF TEMPERATURE. It has al- ready been pointed out that to obtain a uniform temperature throughout the inner cup of a calori- meter, the cup should be completely filled. If this is not done, special precautions must be taken to bring its contents into contact with every portion of its surface (see ^[ 85, 2). The necessity of stirring these contents has also been alluded to (^f 65, 5). When a mixture (like lead shot and water) is of such a nature that an ordinary stirrer cannot be used, the inner cup must be closed water-tight, so that the contents may be shaken. The thermometer should in this case fit tightly into the stopper which closes the inner cup, and should reach into the body of the mixture. Solids, if any be used, should be finely divided, so that there may be no risk of break- ing the thermometer. 1 92 (9).] PRECAUTIONS. 171 We prefer, moreover, finely divided solids, on account of the comparative rapidity with which an equilibrium of temperature may be reached, or a process of fusion, solution, or chemical combination completed. When a solid sinks in a fluid (as is generally the case), it is well if it can be warmer than the fluid, on account of the manner in which convection currents are formed ; and for the same reason we prefer that the denser of two liquids should have the higher temperature. It is always desirable that the denser of two substances should be poured into the other, so that, as it passes through, as much heat as possible may be communicated from one to the other. The various processes in calorimetry should in general be completed in the shortest pos- sible time, especially when they cannot be conducted at the temperature of the room, since otherwise large losses of heat are apt to occur. Throughout the processes in question, stirring must be interrupted from time to time, in order that rough observations of temperature may be made. When two successive observations agree, or when they' differ by an amount which may be attributed to the regular cooling of the calorimeter (see Exp. 31), the equilibrium of temperature should be complete. The student will do well, however, to make sure that the temperatures at the top and bottom of the calori- meter are the same, before proceeding to make exact observations of the thermometer. (9) TIMING OBSERVATIONS. When observations of temperature are taken regularly at intervals of one 172 CALORIMETRY. [Exp. 32. or two minutes throughout an experiment, we may infer the time when a given process begins and when it ends ; but to avoid errors due to the possible omis- sion of one or more observations, it is well to note the beginning and end of each process in hours, min- utes, and seconds. In any case, the time should be thus noted, (1st) when all the bodies have been transferred to the calorimeter, and (2d) when, after an equilibrium of temperature has been reached, the resulting temperature is first observed. (10) SERIES OP TEMPERATURES. It is well in all cases to make several observations of the final tem- perature within a calorimeter, in order that the result may not depend upon one alone (see 51). The series should be made at intervals of one minute, so that, as in ^f 93 (2), the rate of cooling may be found and allowed for. If the calorimeter contains water only, we may utilize the temperature curves already plotted (see ^[ 93, 1) ; or if we have deter- mined, as in ^[ 87, the flow of heat from the calori- meter, we may make an allowance for the heat lost as in ^[ 93 (3). In the absence of any previous determination under the same conditions as in the actual experiment, a series of observations of the temperature of the calorimeter will be required. In the same way, if the temperature of a body is changing perceptibly before it is placed in a calori- meter, it must be determined by a series of observa- tions. The intervals in all such series would natur- ally be one minute each ; but when the temperatures of two or more bodies are to be found, the observa- 193(1).] CORRECTIONS. 173 tions must be taken in turn. When special precau- tions concerning equilibrium of temperature (see (8)) have to be observed, the student is advised not to attempt observations at intervals of less than one minute. The temperatures of the several bodies con- cerned are to be reduced in all cases, as in ^[ 93 (1), to the time when they are first enclosed in the calori- meter. After this time, losses of heat are to be cal- culated as above, from the known rate of cooling of the calorimeter. Tf 93. Corrections for Cooling. (1) GRAPHICAL METHOD. When a calorimeter contains water only, as in the determination Mi miles, of thermal capacity above (^[ 90, I.) or in parts of various experi- ments which follow, the temperature at one point of time may be inferred from an obser- vation taken at another point of time by using one of the curves in Fig. 74, TT 88. Let ab (Fig. 78) be the curve corresponding to the quantity of water which the calorimeter con- tains, and let c be the observed temperature. We first find a point d on the curve at the right of c, then a point e above d. Then we measure off a dis- tance efon the scale of minutes corresponding to the length of time during which the calorimeter has been cooling. Then we find a point g on the curve below /, and finally the temperature h, at the left of g. 10 2 M soe 40 fo 60 10 \ : \ s ; \ *fl N < \ ^ i R r ft& r?. f.M e yo orn ZH* FIG. 78. 174 CALORIMETRY. [Exp. 32. This temperature corresponds in the figure to a time / earlier than e ; but by laying off the distance ef to the right of e, we could find, if we chose, the temper- ature at a later point of time. A more exact method would be to start with a point c (in Fig. 78), corresponding to a temperature as far above that of the room (22|, Fig. 78) as the actual temperature observed was above the observed temperature of the room. The number of degrees included between c and 6 gives approximately, in any case, the fall of temperature which takes place in an interval of time corresponding to the number of minutes between e and /. (2) ANALYTICAL METHOD. When several tem- peratures have been recorded at regular intervals, we may infer the temperature at a point of time before the beginning or after the end of the series as follows : The observations are first written down in a column, as in the example below ; then the tem- perature of the room is subtracted from each, and the results entered in a second column ; then a third column is formed from the differences between each pair of consecutive numbers in the second column ; then each number in the third column is divided by the one just below it in the second column, to find what per cent must be added to that number in order to obtain the one above it ; these per cents are arranged in a fourth column and averaged ; then each number in the third column is divided by the number in the second column just above it, to find what per cent must be subtracted from that number to obtain f 93 (2).] CORRECTIONS. 175 the number just below it ; the per cents to be sub- tracted are then arranged in a fifth column and averaged. We may now extend the second column upwards by adding to the first number in it the aver- age per cent from the fourth column, and we may extend it downward by subtracting from the last number the average per cent found in the fifth column. When the second column has been thus extended, the corresponding numbers in the first column may be found by adding in the temperature of the room. The temperature at a time which would come between the observations in the series thus extended may evidently be found by simple interpolation. For example, when the temperature of the room is 26, the observations below would be reduced as follows : Temperatures Temperatures Fall of Per Cent Per Cent to Observed. less 26. Temperature. to be Added, be Subtracted. 40 .0 38.0 2o.O 5.3 5.0 ;:i 4i 60.5 340.5 . , .o 590.0 330.0 % 56.0 30 .0 Average 4.9 4.7 To extend the second column upwards we add to the first number in it 4.9 per cent of itself. Since 4.9 of 40.0 is 2.0, the number above 40.0 should be 40o.O + 2.0, or 42.0 ; and since 4.9 per cent of 42.0 is 2.l, the next number should be 44.l, etc. To extend the second column downwards, we sub- 176 CALORIMETRY. [Exp. 32. tract from the last number (30. 0) in it not 4.9 per cent but 4.7 per cent of 30. ; that is 1.4 ; this gives 28.6 ; and subtracting from this 4.7 per cent of itself, or 1.3, we find 27.3 for the number following, etc. Adding 26 to the new numbers in the second column, we infer, finally, that the temperatures pre- ceding 66.0 in the first column should be 68.0 and 70.l, while those following 56.0 should be 54.6 and 53.3, etc. Let us suppose that the temperatures were ob- served at intervals of one minute; then to represent the temperature for instance 1.5 minutes before the first recorded observation, we should take a number half-way between 68.0 and 70.l, or 69.0 nearly. If, however, the intervals between observations were two minutes each, then 1.5 minutes would be three fourths of one interval, and we should add to 66 three fourths of the difference (2) between it and the next temperature above it in the series to find the temperature (67.5) in question. The discovery of various methods by which the calculations described above may be shortened, es- pecially by the use of logarithms, may be left to the ingenuity of the student. The method here described is important, as an illustration of the fact that when a body is steadily cooling its temperature falls, not a given amount in each minute, but a certain per cent (approximately) of the number of degrees which lie between it and the temperature of the room (see IF 86, 1). The accuracy with which a series of observations 1 93 (4).] CORRECTIONS. 177 may be extended by analytical methods evidently grows less as the number of new terms increases. It may be said in general that the new terms should not be more numerous than those obtained by actual observation. (3) HEAT LOST BY COOLING. We must distin- guish between the rate of cooling of a calorimeter and the number of units of heat lost by it. The latter may be found without knowing the nature of the mixture which the calorimeter contains, provided that the inner cup is completely filled by the mixture, or filled to a known depth ; for we have only to refer to the results already found with water at the same depth in Experiment 31. If, for example, a calorimeter, nearly filled with a mixture of lead shot and water, has been cooling for ten minutes at an average temperature about 20 above that of the room, we reason that since at a temperature 1 above that of the room it was found (Tf 87) to lose 1.8 units of heat per minute, at a temperature 20 above that of the room it would lose 20 times 1.8, or 36 units per minute; that is, 360 units in ten minutes. If, therefore, the first accurate observation of temperature was taken ten minutes after the introduction of the mixture, we should add 360 units to the amount of heat apparently given out by the hot body, or if more convenient we may sub- tract 360 units from the quantity of heat apparently absorbed by the cool body (see ^[ 98). (4) METHOD OP MULTIPLICATION. When two ex- periments are made, in one of which a body is exposed, 12 178 CALORIMETRY. [Exp. 33. let us say, 11 times as long or 11 times as often to the air as in the other experiment, in which we give it the ordinary exposure, the difference between the results obtained in the two cases should correspond to the effect of 11 less 1, or 10 ordinary exposures. Hence, if this difference be divided by 10, we may estimate roughly the correction to be applied to the result obtained with the ordinary exposure. If, for example, the thermal capacity of a calori- meter is found to be 10.1 units when warm water is poured into it directly, and 11.1 units if the water is first poured back and forth five times each way, then the effect of cooling due to 10 transfers is 11.1-10.1, or 1 unit in the result ; and the effect of a single transfer is about 0.1 unit. The true thermal capacity is, therefore, about 10.1-0.1, or 10.0 units. If the cooling due to transferring a substance from one place to another is thought to affect the figure in the tenths' place, as in the example, it is evident that the hundredths will not be significant (see 55). EXPERIMENT XXXIII. SPECIFIC HEAT OF SOLIDS. Tf 94. Determination of the Specific Heat of a Solid by the Method of Mixture. I. A quantity of lead shot sufficient to half fill the calorimeter (Fig. 70, ^[ 85) is first weighed, then put into a steam heater (Fig. 79), and covered by a cork. A thermometer, passing through the cork into the midst of the shot, IT 94, L] METHOD OF MIXTURE. 179 FIG. 79. is allowed to remain there until it ceases to rise. Meanwhile the temperature within the calorimeter is determined by a second thermometer (^[ 92, 2). The calorimeter is then weighed, and a vessel containing a mixture of ice and water is also weighed. This vessel should be provided with a strainer, so that water may be poured from it without danger of particles of ice fol- lowing the stream. The ice and water should be thoroughly stirred just before the experiment, to secure a uniform temperature of 0. The time should now be noted (^[ 92, 9). The thermometer and corks are then removed from the heater, and the shot is poured as rapidly as pos- sible (^[ 92, 4) into the calorimeter. Immediately ice-cold water is added, the quantity being nearly sufficient to fill the calorimeter. A thermometer is then pushed cautiously into the middle of the shot through a small stopper, closing the inner cup water- tight (^ 92, 8). The large cork cover (Fig. 71, ^[ 85) may then be added, and the time again re- corded. The mixture must now be carefully shaken. The temperature indicated by the thermometer is to be noted at intervals of 1 minute, until it begins to fall steadily (^[ 92, 8 and 10). Then the calorimeter is re-weighed with its contents ; and the vessel origi- nally containing the water is also weighed (^[ 92, 5). 180 CALORIMETRY. [Exp. 33. II. Instead of finding the temperature of the shot in the heater, as in I., we may determine it by a series of observations in the calorimeter, before the ice-water is added (^[ 92, 10). It is necessary in this case to cork the inner cup, and to shake the shot between the observations of temperature (^[ 92, 8), in order that there may be a uniform temperature not only in the shot, but also in the inner cup, the thermal capacity of which must be considered. The ice-water is finally added, and the temperature of the mixture determined as before. III. Instead of pouring the hot shot first into the calorimeter, we may begin by introducing ice-water. In this case the proper quantity of water must be determined beforehand. It will probably be found that the water should fill the calorimeter about half- full. In other respects this method is the same as I. IV. Instead of assuming that the temperature of the water is the same as that within the vessel origi- nally containing it (that is, 0), we may find its temperature after it has been transferred to the cal- orimeter. In this method, however, as in the second method, the thermal capacity of the cup must be con- sidered. To avoid the necessity of making a separate series of observations (^[ 92, 10) between which the water in the calorimeter must be shaken up (^[ 92, 8), it is customary to use water at the temperature of the room. In this case, the mixture will be above the temperature of the room ; hence its rate of cool- ing must be allowed for (^[ 93). V. Other methods of determining specific heat IT 94.] SPECIFIC HEAT OF SOLIDS. 181 may easily be devised, depending upon the use of hot water and cold shot. We have in fact already made use of such a method in finding the thermal capacity of a calorimeter (^f 90, I.). On account, however, of the practical difficulties arising from evaporation (^[ 92, 6), the high temperature of the ^mixture (*[f 92, 7), and the small change of tempera- ture produced, these methods are generally avoided. The principal use which can be made of them is as a check (45) upon results obtained in the ordinary manner. The student may observe that in the second method the shot falls suddenly in temperature, on account of the heat which it gives up to the calorimeter. This heat is subsequently restored to the mixture when the calorimeter is cooled to its original temperature ; hence in the first method no account need be taken of the thermal capacity of the calorimeter. Again in the fourth method, the cold water may at first rise rapidly in temperature on account of the heat im- parted to it from the calorimeter, but this heat is restored to the calorimeter when it is again raised by the mixture to its original temperature ; hence in the third method no account need be taken of the thermal capacity of the calorimeter. Instead of lead shot, copper or iron rivets may be employed with very slight modifications of the experi- ment. In the case however of solids which are soluble in water, we must substitute for water some other liquid of known specific heat in which the solids are insoluble (^[ 92, 1). The student may be guided in 182 CALORIMETRY. [Exp. 33. his choice of methods by obvious considerations of practical convenience as well as by the principles explained below in ^[ 95; but he should make at least one determination of specific heat of a solid by the method of mixture and reduce it as will be ex- plained in ^[ 98. ^[ 95. Comparison of Methods for the Determination of Specific Heat. The principal difficulty in the first method (^[ 94, I.), for the determination of specific heat, is to avoid a great loss of heat while the shot is being transferred from the heater to the calorimeter. In the second method (^[ 94, II.) there is no opportunity for a loss of heat on the part of the shot, since its temperature is determined by a series of observations within the calorimeter, from which its temperature at any point of time may be found (^[ 93, 2). The principal objection to the second method is the difficulty of determining accurately a series of temperatures in which rapid changes take place ; and the necessity of allowing for the thermal capacity of the calorimeter, which is always a more or less uncertain quantity, and bears a considerable proportion to the thermal capacity of the shot. The third method (^[ 94, III.) has the same prac- tical advantages and disadvantages as the first. The fourth method (^[ 94, IV.) is the one com- monly employed for the determination of specific heat. Since the temperature of the water is found when within the calorimeter, there is no opportunity (as in the other methods) for heat to be imparted to it in the act of pouring. There is however difficulty, 1 95.] SPECIFIC HEAT OF SOLIDS. 183 as in the second method, in determining accurately a temperature which is changing (^[ 92, 3), and still further difficulty in maintaining a uniform tempera- ture throughout the calorimeter with a quantity of water which only half fills it <^[ 92, 8). When the latter difficulty is avoided by using water at the tem- perature of the room, the mixture must have a tem- perature considerably above that of the room, and one therefore which is hard to determine (^[ 92, 3). The thermal capacity of the calorimeter must also, as in the second method, be taken into account. By comparing the results of the first and second methods, we are able to estimate the effect of the heat lost in pouring the shot into the calorimeter (see also ^[ 93, 4), and by comparing results of the third and fourtji methods, we are able to estimate the effect of the heat absorbed by the ice-cold water when it is poured from one vessel to another. This will be found to be small in comparison with the heat lost by shot at 100 under similar circumstances. The second method, in which the latter is eliminated, is therefore preferable to the fourth. In the first and third methods, the heat lost by the shot is partly offset by that imparted to the water. Since the former is greater than the latter, the third method is preferable to the first; because the longer exposure of the water may compensate for the more rapid cool- ing of the shot. The choice between the second and third methods will depend largely upon the compara- tive accuracy with which we can determine the heat given out by the calorimeter (^[ 87) and the heat lost 184 CALORIMETRY. [Exp. 34. by the shot flf 93, 4). The advantages of using in any case hot shot and cold water have been already stated flf 94, V.). EXPERIMENT XXXIV. SPECIFIC HEAT OF LIQUIDS. ^f 96. Determination of the Specific Heat of a Liquid by the Method of Mixture. The specific heat of a liquid may be determined either by mixing it me- chanically with water, or by bringing it in contact with a solid of known specific heat. The first method is the more direct, but cannot be employed with liquids which unite chemically with water, unless we know the amount of heat given out or absorbed by the reaction (see ^f 92, 1). Before deciding which method we shall employ, we therefore mix together the contents of two test-tubes, each at the temper- ature of the room, one containing water, the other the liquid in question. If no change of temperature is observed, the first method is adopted. If the tem- perature rises or falls, we must either make a sepa- rate experiment to determine accurately the amount of this rise or fall (see Exp. 35), or else adopt the indirect method, using a solid instead of water. I. The determination of the specific heat of an insoluble liquid by the method of mixture does not differ essentially from the case of a solid. A heavy oil may for instance be heated by the same apparatus (Fig. 79, Tf 94) employed for the shot, and mixed with 1T96.] SPECIFIC HEAT OF LIQUIDS. 185 ice-cold water, according to either of the methods described (^[ 94). Instead of shaking the mixture, a brass fan or stirrer (Fig. 50, ^[ 65) may be em- ployed. The objections to mixing hot water with a cold liquid are not nearly as strong as in the case of solids (^[ 94, V.) ; for though most liquids have a specific heat less than that of water, the differences are very much less. By pouring a comparatively small quan- tity of water at a temperature not exceeding 40 or 50 into a liquid at a mixture may be had not far from the temperature of the room. With liquids less dense than water this method is generally to be preferred (see ^[ 92, 6 and 8). The results may be reduced by the appropriate formula from ^[ 98. Attention has already been drawn (^[ 92, 1) to precautions against chemical action in the case of corrosive liquids, and in the case of volatile liquids against evaporation fl[ 92, 6) and combustion ([[ 83). II. In the case of liquids which mix with water, the ordinary methods of mixture cannot generally be employed, on account of the heat absorbed or devel- oped by solution or combination. It is necessary to find some substance, of known specific heat, upon which such a liquid exerts no thermal action. This substance is then mixed with the liquid by either of the methods of ^ 94. The data necessary for find- ing the specific heat of the liquid are as usual the weight of the two substances in question, the tem- perature of each before the experiment, and the resulting temperature of the mixture. 186 CALORIMETRY. [Exp. 34. The lead shot already employed (^[ 94) may be used to determine m this way the specific heat of alcohol, glycerine, saline solutions, etc. For corro- sive liquids, like nitric acid, glass beads (of specific heat about 0.19) may be similarly employed (see general formula, *f[ 98). Evidently this indirect method is more general than the ordinary method of mixture, since it can be applied to all liquids, whether soluble or insoluble in water. It has the advantage of eliminating almost completely the heat lost by the hot body between the heater and the calorimeter, since this loss is practically the same in the case of water as in the case of other liquids with which a comparison is made. ^[ 97. Peculiar Devices employed in Calorimetry. In the method of mixture (Exps. 33 and 34) a thermal equilibrium between two or more substances is estab- lished by bringing them in contact. It is not, how- ever, necessary that the two bodies should touch each other. The difficulties which arise from the mu- tual action of two substances may often be avoided by surrounding one of them with an envelope, through which, by the conduction of heat, an equalization of temperature takes place. If, for instance, a hot liquid contained in a glass bulb be surrounded by cold water, a certain quantity of heat will be given out. Having found by a separate experiment how much heat is derived from the bulb alone, we may calculate the specific heat of the liquid in the ordinary manner, that is, from the weights and changes of temperature involved (see general formula, ^[ 98). 1T97.] SPECIFIC HEAT OF LIQUIDS. 187 The liquid in question may be contained in an ordinary thermometer bulb. In this case its change of temperature may be inferred very accurately from its contraction, as shown by the fall of a column of liquid in the stem of the thermometer. It is neces- sary, of course, to make a careful comparison of a thermometer containing an unknown liquid with an ordinary mercurial thermometer (see ^f 92, 2). This method has obvious advantages in the case of costly liquids. On the other hand, when the supply of a fluid is unlimited, it is frequently advantageous to use an envelope in the form of a spiral tube, or coil, through which the fluid in question may be passed in a con- tinuous stream. We are thus enabled to bring a great volume of the fluid in thermal equilibrium with a small volume of water. This device is exceedingly important in the case of gases, since it would be otherwise impossible to bring enough gas in thermal equilibrium with a given quantity of water to af- fect the temperature of the water by a measurable amount. The weight of the gas employed is not measured directly, but is determined from its density (see Tf^T 44, 46) and from the volume employed. The volume is indicated by a gas-meter (aJ, Fig. 80) through which the gas is first passed. The gas is then raised to the temperature of 100 by passing it through a steam jacket, Id. Then it circulates through a coiled tube surrounded with water, and escapes from an orifice where its final temperature 188 CALORIMETRY. [Exp. 34. can be observed. From the thermal capacity and rise of temperature of the calorimeter, we may cal- culate the quantity of heat given out by a known quantity of gas in falling through a known number of degrees, and hence the specific heat of the gas. It is found that the specific heat of air at the con- stant pressure of one atmosphere is about 0.238, or a FIG. 80. little less than one fourth that of an equal weight of water. A much more difficult task consists in the determi- nation of the specific heat of a gas when confined to a constant volume. The following method is sug- gested. It depends upon the fact that a given electric current passing for a given time through a given con- ductor generates in that conductor a given quantity of heat. This quantity may be found by experiment (see Exp. 86), or calculated by the principles of 136. Let us suppose that a known quantity of heat is thus suddenly generated within a closed flask (Fig. 81) ; and that the increased pressure of the air is IT 97.] SPECIFIC HEAT OF GASES. 189 measured, as in If 80, by the rise of mercury in an open tube. Then the average temperature of the air within the flask can be calculated (see 76). We may therefore find the ther- mal capacity of a known volume or of a known weight, and hence the specific heat in question (about .169). It is found that the thermal ca- pacity of a cubic metre of air is about 219 units at and 76 cm. when prevented from expanding, as against 308 units when free to expand under a constant pressure. The thermal capacity of an equal volume of oxy- gen, of nitrogen, or of hydrogen is very nearly the same as that of air under similar conditions. Instead of using an electrical current to generate heat (as illustrated in Fig. 81), we may employ vari- ous other agents, as for instance the combustion, the solidification, the fusion, the condensation, or the vaporization of a known weight of a given substance, or the conversion through friction of a given amount of work into heat (see Exp. 70). If, for example, the combustion of a gram of coal heats a kilogram of water 8, and a kilogram of petroleum 16 ; or if 100 grams of ice cool these liquids 8 and 16 respec- tively ; the specific heats must be to each other as 2 to 1. The same inference would be drawn if the same quantity (100 grams) of steam which heats 1 kilogram of water 54 were found to heat 2 kilo- grams of petroleum by the same amount. The spe- 190 CALOEIMETRY. [Exp. 34. cific heats of different substances are to each other, in general, inversely as the changes of temperature pro- duced by a given cause, and also inversely as the weights affected. The determination of specific heat is evidently capable of as many modifications as there are different methods by which a definite quantity of heat may be generated or absorbed. Instead of using the pressure of air to measure its temperature, we may also employ its expansion ( 80) as in the air thermometer (If 74). The specific heat of air under a constant pressure might obviously be determined by an apparatus similar to that repre- sented in Fig. 81 ; hence, conversely, if this specific heat is known, we may measure quantities of heat by the expansion which they produce in air at a given pressure. It does not (as one might think) make any difference theoretically hoiv much air is heated ; because an increase in the quantity of air will be offset by a decrease in the temperature to which it will be raised by a given amount of heat ; and for the same reason it is indifferent whether a small portion of the air is heated a great deal, or whether a con- siderable portion is heated by a proportionately small amount. In this method of estimating heat it is not necessary to wait for an equilibrium of temperature. We hasten in fact to make our observations before an equilibrium is reached, so as to avoid loss of heat by contact of the air with the sides of the vessel in which it is contained. It has been calculated that one unit of heat should in all cases cause in a body of air at 76 cm. pressure an expansion of about 12 cubic centi- IT 97.] ICE CALORIMETER. 191 metres. Since, an expansion of less than 1 cubic milli- metre is easily detected, we have, in the air ther- mometer, a very delicate means of measuring small quantities of heat. 1 Instead of air, we may use any other fluid which has a regular rate of expansion to determine quanti- ties of heat. The principle above explained has been applied by Favre and Silbermann in the con- struction of their mercury calorimeter. 2 This is essen- tially a thermometer with a huge bulb. If even a small quantity of hot liquid be introduced into a cavity in this bulb, there will be a perceptible expan- sion of the mercury, by which we may measure the heat given out by the liquid in question ; for it has been found that 1 unit of heat always causes in a body of mercury an expansion of about 4 cubic millimetres. There are various other definite effects produced by a given quantity of heat, any one of which might theoretically be applied to the pur- poses of calorimetry. The only ap- plication of practical importance depends, however, upon the heat required for the fusion of ice (see Experiment 36). A rough form of ice calorimeter consists of a block of ice (Fig. 82) with a small cavity in which a hot body may be 1 The air thermometer has been used in the Jefferson Physical Laboratory to measure minute quantities of heat generated in a car- bon fibre by telephone currents. 2 See Ganot's Physics, 463. 192 CALORIMETRY. [Exr. 34. placed. A second block may be used as a cover. The water formed by the liquefaction of ice is gathered by a sponge, and weighed by the usual method of dif- erence. Since one unit of heat melts one-eightieth of a gram of ice, the quantity of heat given out by the body in falling to a temperature of can easily be calculated. In Bunsen's ice calorimeter, the quan- tity of ice melted is estimated by the change in volume of a mixture of ice and water. T[ 98. Calculation of Specific Heat in the Method of Mixture. If w l is the weight of the body, the specific heat of which ( x ) is to be determined, and t t the temperature of this body, reduced to the time of mixing ; if w 2 is the weight of the body the specific heat (&,) of which is known, and if t 2 is its tempera- ture, also reduced to the time of mixing ; if c is the thermal capacity of the calorimeter, t 3 its original tem- perature and, t the temperature of the mixture ; then if q is the quantity of heat lost by cooling, that is, absorbed by the air, etc., we have, by the principle of 90, the general formula, From this formula we may obtain the solution of all problems in the determination of specific heat by the method of mixture. In addition to ,, c, and q (which are known, or may be calculated), we require at least five data for a determination of specific heat ; namely, the two weights employed, w l and 0,, the two corresponding temperatures, ^ and ^, also the temperature, , of IT 98.] CALCULATIONS OF SPECIFIC HEAT. 193 the mixture. The original temperature, 8 , of the ca- lorimeter must also be determined, unless by the nature of the experiment it is known to agree with one of the other temperatures. When water is used s 2 = 1 ; hence we have, if the water used is colder than the mixture, or if the water is warmer than the mixture, If the temperature of the water is taken in the ca- lorimeter, so that 2 = 8 , we may combine the terms in the numerator, so that for cold water, (w 2 + c}(t * 8 ) + g *i - Wl (^ _ t) or for hot water, 0-9. TV If the original temperature of the calorimeter is the same as that of the mixture, the terms c (t 3 ) and c ( 8 t) disappear from I. and II. respectively ; hence, for cold water, _ t ~i . v - and for hot water, 194 CALORIMETRY. [Exp. 35. If, finally, the temperature of the mixture is the same as that of the room, there is no loss of heat by cooling ( 89), that is, q = ; hence the term q disappears from all the formulas. We have therefore in the simplest possible case, when the calorimeter is at the temperature of the room both before and after the experiment, if cold water is used, s = ^t-t 2 ) yiL and if hot water is used, _W 2 (t 2 -f) VIII. The calculation of the thermal capacity of the calorimeter (c) is explained in ^[ 86 and 91 ; that of the heat lost (q) in ^ 93, 3. The correction of the temperatures t l and t 2 to the time of mixing may be done either by graphical or by analytical methods (t 93, 1 and 2). EXPERIMENT XXXV. HEAT OF SOLUTION. ^[ 99. Determination of Latent Heat of Solution. When a solid dissolves in a liquid, or when two liquids mix together, there is almost always a rise or fall of temperature. This is due probably to a a molecular re-arrangement which takes place. The object of this experiment is to find how much heat is given out or absorbed, as the case may be, by one f 99.] LATENT HEAT OF SOLUTION. 195 gram of a given substance when mixed with or dis- solved in water. I. LIQUIDS. When equal volumes of alcohol and water are mixed together (see ^[ 96) a rise of tem- perature may be observed. To measure this rise accurately, a calorimeter is to be weighed empty, and re-weighed with a quantity of alcohol which fills it half-full, and which is at a temperature, accurately observed, not far from that of the room. An equal volume of water, heated or cooled if necessary so as to have exactly the same temperature, is then mixed with the alcohol in the calorimeter, and the resulting temperature accurately determined by a series of ob- servations (^[ 92, 10). The weight of water is also to be found (see ^[ 92, 5). If the thermal capacity of the calorimeter and the specific heat of the liquid are both known, the latent heat of solution may be calculated by formula II., ^[ 100. It is better, however, to repeat the experiment with water at a much lower temperature, which must be determined (see [[ 92, 10) by a series of observations. The object aimed at is to offset in this way the heat due to mixture. When alcohol in a calorimeter at the temperature of the room is mixed with an equal volume of water, which is cooler than it by the right number of degrees, scarcely any rise or fall of temper- ature will be observed in the calorimeter. In this case a single observation will suffice. Let us suppose, for example, that equal volumes of alcohol and water rise 8 when mixed at the same temperature, but that if the water is 9 cooler than 196 CALOKIMETKY. [Exp.35. the alcohol, the rise is 2. Then since 9 in the water makes a difference of 8 2, or 6, in the mixture, 12 in the water would make a difference of 8 in the mixture. It follows that the alcohol could be mixed with an equal volume of water 12 below it in temperature without being warmed or cooled by the process. It would be well to test the accuracy of such a conclusion by a third experiment. When the desired difference of temperature has been found, either by experiment or by calculation, the latent heat of mix- ing is easily computed. We multiply the weight of water by its rise of temperature to find the number of units of heat received, and divide by the weight of alcohol to find the amount given out by one gram ; or we may use formula III., ^[ 100. The experiment may be varied by using different liquids, or by mixing a given liquid with water in different proportions. II. SOLIDS. When ammonic nitrate is dissolved in water a fall of temperature is observed. The amount of this fall may be determined as in the case of alco- hol ; but in order that the solid may be readily dis- solved, it is better to use only one part of the salt in nine of water. To ensure rapid solution, the salt should be pulverized. In the first experiment the salt, the water, and the calorimeter should all start at the temperature of the room. The fall of temperature of the water may require a thermometer divided yito tenths of degrees for its accurate determination. The use of a stirrer is very important (^f 65, 5). f 100.] LATENT HEAT OF SOLUTION. 197 The experiment may now be repeated with water somewhat warmer than before, with a view to mak- ing the resulting temperature agree with that of the room. The water should, however, be placed first in the calorimeter, in order that the temperature of the latter may be accurately determined. A series of observations must be taken (^[ 92, 10). The salt is finally added, and the fall of temperature accurately measured. If the water has been heated too much or too little, the experiment may be repeated until the mixture agrees in temperature with the room ; or the desired temperature of the water may be calcu- lated by the same process of reasoning as was em- ployed in I. In calculating the latent heat of solution by this method, the thermal capacity of the calori- meter must be taken into account, since part of the heat absorbed by the salt is supplied by the calori- meter. In other respects the reduction is the same as in I. (see also formula IV., ^[ 100). If, for instance, 10 grams of salt cool 90 grams of water contained in a calorimeter with a thermal ca- pacity equal to 10 units, from 22 to 20, that is 2, we have (90 -j- 10) X 2 = 200 units of heat given out. Since 10 grams of the salt absorb 200 units, each gram must require 20 units of heat ; hence the latent heat of solution is 20. The latent heat in question varies slightly according to the strength of the solu- tion formed. ^f 100. Calculation of the Latent Heat of Solution. If Wi is the weight of the substance whose latent heat of solution, l is to be determined, * x its specific 198 CALORIMETKY. [Exp. 35. heat, and t v its original temperature; if w 2 is the weight of the solvent, s 2 its specific heat, and t 2 its original temperature ; if c is the thermal capacity, t s the original and t the final temperature of the calori- meter (hence also of the mixture), then the quan- tities of heat absorbed are, (1) w v s t (t ^) in raising the temperature of the substance dissolved ; (2) w 2 s 2 (t 2 ) in raising the temperature of the solvent ; and (3) c (t t s ) in raising the tempera- ture of the calorimeter and (4) w l ^ in the act of solution. Hence, by the principle of 90, neglecting the heat lost by cooling. This gives for the latent heat of mixing with water, which we consider positive if heat is absorbed, but negative if (as is usually the case when two liquids are mixed) heat is given out, 1 since s 2 = 1, and since ti and t s are the same (the temperature of the liquid being determined in the calorimeter), If the experiment is varied so that t = t 1 then we have simply _^(*-*Q nL W l If, however, the temperature of the water is found within the calorimeter, so that t 2 = t m the substance 1 The same formula may be used to determine the heat of combi- nation, only that the sign must be reversed (see 1T 106). T 101.] LATENT HEAT OF LIQUEFACTION. 199 dissolved being as before unchanged in temperature, we have for the latent heat of solution, which we call positive when heat is absorbed, the formula w l EXPERIMENT XXXVI. LATENT HEAT OF LIQUEFACTION. ^J 101. Determination of the Latent Heat of Water. Latent heats of liquefaction are determined in essentially the same manner as latent heats of solu- tion (Exp. 35, II.). Instead, however, of dissolving a solid in a fluid, the solid is simply melted by the fluid. Knowing the weights, specific heats, and changes of temperature of the substances in question, we may calculate by the general formula (^[ 100, I.) the heat required to melt one gram of the solid ; or, in other words, its latent heat of liquefaction. It is evident that the liquid must exert no solvent action on the solid, otherwise we should have to allow for heat of solution (see Exp. 35). It is also necessary that the mixture be at a higher tempera- ture than the solid, else the solid will not melt. It is well that the solid should start at its melting-point, since otherwise we must allow for the heat necessary to raise it to the temperature in question. A consid- erable time must generally be allowed for the process of melting ; to shorten this time as much as possible, 200 CALORIMETKY. [Exp. 36. the mixture should be vigorously stirred. Observa- tions of temperature should be taken from time to time (^[ 92, 8) during the process. When ice is the solid employed, difficulty will be found in obtaining sufficiently small pieces free from water. The ice should be cracked into fragments weighing a few grams each, which are then to be wrapped up in cotton-waste and weighed. Any moisture formed by the melting of the ice should be absorbed by the waste. The calorimeter is weighed empty, and re-weighed when about half-full of warm water. The tempera- ture of the water should be about 50, and is deter- mined by a series of observations (^[ 92, 10) ; then ice is added until the calorimeter is nearly full. The ice should be handled by means of a portion of the cotton waste which surrounds it, and each fragment should be wiped as dry as possible before placing it in the calorimeter. The time occupied by this pro- cess and by the fusion of the ice should be noted (^[ 92, 9). The resulting temperature of the water must be accurately determined. The quantity of ice used should be found both by re-weighing the cotton waste and by re-weighing the calorimeter (^[ 92, 5). ^| 102. Calculation of the Latent Heat of Water. If w l is the weight of ice employed, ^ its original temperature (that is, 0) and s l its specific heat in the liquid state (that is, 1) ; if w 2 is the weight of water employed, its temperature reduced to the time of mixing (^[ 93), and s 2 its specific heat (that is 1) ; if c is the thermal capacity of the calorimeter calculated If 102.] LATENT HEAT OP LIQUEFACTION. 201 as in Tf 91, t 3 its original temperature (the same as 2 ), and t the temperature of the mixture ; we have, sub- stituting these values in formula II., ^[ 100, From the numerator of this fraction should be sub- tracted a correction expressing the number of units of heat lost by the warm water while the ice is being melted. Since the water begins at a temperature 2 , and ends at a temperature t, its average temperature is I ( 2 -|- ), nearly. Subtracting the temperature of the room, we have, approximately, the average excess of temperature. Multiplying as in ^[ 93 (3), by the number of minutes required to melt the ice, and also by the heat lost per minute when the temperature is 1 above that of the room (see ^[87), we have the correction in question. Evidently, if the average temperature of the water is the same as that of the room, no correction for cooling need be made. The truth of the formula for the latent heat of water may be seen by the following considerations : Since w z grams of water and the equivalent of c grams of water (in the brass and other materials composing the calorimeter) are cooled from 2 -to t, the heat lost by the hot bodies amounts to (w z -\- c) X ( 2 units. Subtracting from this the correction for cooling, we have a remainder which must repre- sent the heat absorbed by the cold bodies ; that is, the ice and the water formed by its liquefaction. Now w l grams of ice form w l grams of water at ; 202 CALORIMETRY. [Exp. 37. and to raise this to t requires w l X t units of heat. Subtracting this from the previous remainder, we have, therefore, the heat required to melt w l grams of ice. Finally, dividing by w^ we have the heat required to melt 1 gram, or the latent heat in question. EXPERIMENT XXXVII. LATENT HEAT OF VAPORIZATION. ^[ 103. Determination of the Latent Heat of Steam. There are many points of resemblance between the determination of the latent heat of vaporization and that of the latent heat of liquefaction (Exp. 36). Instead of melting a solid in a liquid, a vapor is con- densed in a liquid. From the weights, specific heats, and changes of temperature in question, latent heats of vaporization may be calculated by the same general formula (^[ 100, I.) as latent heats of liquefaction. The vapor must evidently have no chemical affinity for the liquid. The liquid must be at lower tem- perature than the vapor, in order that the latter may be condensed. The vapor should start as nearly as possible at its temperature of condensation, otherwise an allowance must be made for the heat given out in reaching this temperature. Care must, however, be taken that the vapor is freed from particles of liquid formed by its condensation, before it passes into the calorimeter. IF 103.] LATENT HEAT OF VAPORIZATION. 203 When steam is used, it is passed from a generator (a, Fig. 83) through a trap (6), where nearly all its moisture is deposited. It will be seen in the diagram that the exit tube is completely sur- rounded, either by steam or by cork, until it reaches the calorimeter. If, therefore, this tube is well heated by a cur- rent of steam before the experiment, there is no reason why any condensation should take place within it. The calorimeter is weighed when empty, and re- weighed with a quantity of water sufficient nearly to fill the inner cup, and as cold as possible. The temperature of this water is determined by a series of observations at intervals of one minute (^[ 92, 10) ; then the current of steam issuing from the trap is turned suddenly into the water. The water is stirred vigorously by twisting the stem of a thermometer to which a stirrer is attached. When the temperature of the water has risen as much above that of the room as it was below it before the admission of steam, the trap is taken away from the calorimeter, and the resulting temperature determined by another series of observations. The time used in heating the water to the required temperature should be as small as possible, to avoid errors due to gain or loss of 204 CALOKIMETRY. [Exp 37 heat ; but if the average temperature agrees with that of the room, no correction for cooling need be applied (see ^[ 102). The weight of steam con- densed is found by re-weighing the calorimeter, and the temperature of this steam determined by an ob- servation of the barometer (see ^[ 69, II.). T[ 104. Calculation of the Latent Heat of Steam. If w^ is the weight of steam condensed, * x the specific heat of the liquid formed by its condensation (that is, I), 1 and j its original temperature (let us say 100, but see Table 14) ; if w 2 is the weight of water, s 2 its specific heat (that is, 1) and t z its original tempera- ture ; if c is the thermal capacity of the calorimeter, t s its original temperature (the same as 2 ), and t the temperature of the mixture ; we have, substituting these values in the general formula (^[ 100, I.), Q 2 + c) (t-t,) -w, (100-Q To the numerator of this fraction should be added the heat (if any) lost in cooling, since this is also at the expense of the steam. The formula may also be established by a process of reasoning similar to that used in ^[ 102. To raise the equivalent of w 2 + c grams of water (t 2 ) de- grees requires (w, + c) X (t 2 ) units of heat. Part of this was furnished by the iv l grams of water at 100 (nearly) in cooling to t. This part is clearly w^ (100 t). Subtracting this from the total heat 1 The specific heat of water varies from 1.000 at to 1.013 at 100, having a mean value of about 1.005. t 105 ] HEAT OF COMBINATION. 205 received by the water, we have that given up to it by w^ grams of steam in the act of condensation ; hence, dividing by w 19 we have the heat given out by one gram of steam at 100 when condensed into water at 100 ; that is, the latent heat in question. EXPERIMENT XXXVIII. HEAT OP COMBINATION. ^[ 105. Determination of Heats of Combination. The same method, essentially, is employed for the determination of heats of combination as for heats of solution (Experiment 35) ; the only difference being that the solvent has a chemical affinity for the sub- stance dissolved. From the weights, specific heats, and changes of temperature of the materials involved, the heat of combination may be calculated by the general formula (1 100, I.). Heats of combination are, however, called positive when the result of mix- ture is to raise the temperature of the constituents. (1) ZINC AND NITRIC ACID. A gram of pure zinc filings is to be dissolved in at least fifty times its weight of dilute nitric acid. The student should de- termine by a preliminary experiment what strength of acid may be required to ensure rapid solution without danger of accident from excessive efferves- cence. This will depend largely upon the fineness of the zinc. When " zinc dust " is used, very dilute acids must be employed. The zinc dust should be 206 CALORIMETRY. [Exp. 38. poured into the acid, not the acid on the zinc dust. The inner cup of the calorimeter (Fig. 71, ^[ 85) should be replaced by one of glass (^[ 92, 1), the thermal capacity of which must be calculated as in 1 91. The glass cup is then nearly filled (f 92, 8) with the dilute acid at a temperature below that of the room. This temperature must not, however, be so low as to arrest the chemical action. The process of solution may be greatly accelerated by the use of a platinum-stirrer ; l but a brass stirrer coated with asphaltum may be employed (see ^[ 92, 1). The quantity of dilute acid used must be found by weigh- ing the calorimeter with and without it ; and the rise of temperature of this acid must be determined by a series of observations of temperature (^[ 92, 10) both before and after the experiment. It is well also to re-weigh the calorimeter after the experiment, to guard against any loss of material (^[ 92, 5). The loss of weight due to the escape of nitric oxide gas will hardly be detected. (2) ZINC OXIDE AND NITRIC ACID. The experi- ment is now to be repeated with a quantity of zinc oxide which would be formed by the combustion of 1 gram of zinc. This quantity is 1.25 g., very nearly. The same weight and strength of acid are to be used as before (1) ; but the temperature should be very little below that of the room. 1 Currents of electricity generated by the contact of platinum and zinc assist the chemical action. It is, indeed, stated by some author- ities that in the absence of such currents perfectly pure zinc is not attacked by dilute acids. 1 106.] HEAT OF COMBINATION. 207 The density of the acid used should be determined roughly as in ^[ 40. From the results of this experiment the student is to calculate (as in H 106, below) the number of units of heat given out by 1 gram of zinc in uniting with an excess of dilute nitric acid, also what part of this heat is due to its uniting with the oxygen of the acid. The heat of combination of zinc with nitric acid will be found to have an important bearing upon problems relating to electric batteries in which zinc is the dissolving element and nitric acid the oxidizing agent ( 145). ^| 106. Calculations relating to Heat of Combina- tion. It is necessary, in general, to find the spe- cific heat of the liquid used for a determination of heats of combination (see Experiment 34). The specific heats of certain solutions, amongst them nitric acid, may be found, when their densities are known, by Table 30. In calculating the thermal capacity of a calorimeter, the specific heat of the glass composing the inner cup may be taken as 0.19. If w v is the weight of zinc employed, ^ its specific heat (.09,")), ^ its original temperature if w 2 is the weight of acid employed, s 2 its specific heat (from Table 30), and t z its original temperature reduced (see ^[ 93, 2) to the time of solution ; if c is the thermal capacity of the calorimeter, t s its original temperature (the same as 2 ) and t the temperature of the mixture, we have for the heat of combination h (substituting h for I in the general formula of ^[ 100, 208 CALORIMETRY. [Exp. 38. and changing signs, since h would be negative if heat were absorbed), (w 9 s 2 + e) ( O + w, s, (tt.) If, as in the experiment, a comparatively large quan- tity of acid is employed, the second term of the nu- merator may be neglected. When, moreover, 1 grain of zinc is used, w 1 = 1, and we have, h=(w 2 s 2 + c) (t 2 ), nearly. II. The truth of the last formula is sufficiently evi- dent, since s 2 is the thermal capacity of 1 gram of the acid, w 2 s 2 must be that of tv 2 grams ; and this added to the thermal capacity (c) of the calorimeter must represent (neglecting the 1 gram of zinc) the total thermal capacity. In the formula (II.) the total thermal capacity is simply multiplied by the number of degrees rise in temperature. This must give the number of units of heat developed by the combination of the zinc with the acid. The heat of combination of zinc oxide may be calculated by formula I. To find the heat given out by a quantity of zinc oxide (1.25 grams, nearly) which contains 1 gram of metallic zinc, this heat of combination must be multiplied by 1.25. The same result may be obtained directly by formula II. if, as in the experiment described, we have employed 1.25 grams of zinc oxide. The chemical reaction which takes place when zinc is dissolved in nitric acid may be divided theo- T106.] HEAT OF COMBINATION. 209 retically into two stages : first, the combination of 1 gram of zinc with oxygen, which is obtained by the decomposition of a part of the nitric acid, 1 thus : Zinc. Oxygen. Zinc oxide. Zn + O = Zn O ; (1) and, second, the combination of the 1.25 grams of zinc oxide thus formed with more of the nitric acid to form zinc nitrate, thus : Zinc oxide. Nitric acid. Zinc nitrate. Water ZnO + N 2 O 5 -H 2 O = ZnON 2 O 5 + H 2 O. (2) We have already found the heat developed by the process as a whole. We have also found the heat developed in the second stage of the process, namely, the union of 1.25 grams of zinc oxide with nitric acid. The difference between these two quantities of heat must (by the principle of the conservation of energy) be equal to the heat developed by 1 gram of zinc in combining with oxygen extracted from nitric acid. If, for example, 1 gram of zinc dissolving in 100 grams of nitric acid of a certain strength gives out 1 Nitric acid, thus deprived of its oxygen, may be reduced to ni- trous acid, nitric oxide (gas), or even to ammonic nitrate. The reac- tions are as follows : 2 Zn + 3 N 2 5 H 2 O = 2 ZnO N 2 O 5 + 2 H 2 O + N 2 3 H 2 O (nitrous acid). 3 Zn + 4 N 2 5 H 2 = 3 ZnO N 2 O 6 + 4 H 2 O + 2 NO (nitric oxide). 4 Zn + 5 N. 2 5 H 2 = 4 ZnO N 2 5 + 3 H,O + ( H 4 N) (NO 3 ) (ammo- nic nitrate). Nitrous acid may be formed by the reduction of strong nitric acid. The presence of nitric oxide gas may usually be recognized by the red fumes which are generated when nitric acid is reduced. Ammonic nitrate is formed only in very weak solutions (Wurtz, Chimie Moderne, p. 169). 210 CALORIMETRY. [Exp. 38. 1,500 units of heat, while an equivalent (1.25 grams) of zinc oxide gives out only 400 units of heat, it is evident that 1500400, or 1100, units of heat are due to the combination of 1 gram of zinc with the oxygen of the acid. Tf 107. Heat of Combustion. We have seen in the last section how we may find indirectly the amount of heat given out by a gram of a given material when it combines with the oxygen of an acid. This heat varies greatly according to the difficulty of extracting the oxygen in question. If, for instance, as in sul- phuric acid, the oxygen must be taken away from hydrogen, for which it has a great affinity, nearly three fourths of the energy will be spent in decom- posing the acid. In the case of nitric acid, less diffi- culty is encountered ; since nitric acid is more readily decomposed (see footnote, ^[ 106). Even, however, in the case of chromic acid, in which the oxygen approaches very nearly its condition in the free state, the heat of combination with oxygen will differ somewhat from the result which we should obtain by burning a metal in oxygen gas. The heat given out by one gram of a substance when burned in oxygen is called its heat of combustion in oxygen. It may be determined directly by an appa- ratus shown in Fig. 84. The substance in question is placed in a deflagrating spoon, f, contained in a FIG. 84. ff 107.] HEAT OF COMBUSTION. 211 water-tight chamber, h ; oxygen (or air) is admitted to this chamber by the tube a, and the gaseous pro- ducts of combustion, if any, escape through the spiral tube gfc. The whole system of tubes is surrounded by water, contained in a calorimeter of the ordinary sort. When the temperature of the water has been observed, the substance is ignited by a current of electricity. From the rise of temperature and the thermal capacities of the calorimeter and its contents, the heat of combustion is calculated. To determine the heat of combustion of a gas with this apparatus, a third tube must be added to supply the gas. A much simpler device consists, however, of a small metallic cone soldered into the bottom of a calorimeter. The cone ends above in a spiral tube, surrounded by water. A gas jet burned beneath this cone will give up nearly all of its heat to the water. The quantity of gas used is measured by a gas-meter. The determination of heats of combustion in general is an exceedingly difficult problem, but the ambitious student may be encouraged to attempt a rough deter- mination of the heat of combustion of coal-gas or alcohol with a simple apparatus like the one de- scribed. 212 RADIATION. [Exp. 39. EXPERIMENT XXXIX. RADIATION OF HEAT. ^[ 108. The Fyroheliometer. A simple form of pyro- heliometer (jrvp, fire, heat; jjXios, sun; perpov, meas- ure), or instrument for measuring the heat radiated by the sun, consists of a hollow tin box (Fig. 85) filled with water. One of the outer surfaces of the box is blackened, so as to absorb most of the heat which falls upon 1 IG. 85. it. This surface is turned per- pendicularly to the rays, the intensity of which is to be measured. The temperature of the water is ob- served by a thermometer passing through a hole in the side of the box. The number of heat units ab- sorbed is calculated from the rise of temperature and thermal capacity of the vessel and its contents, as in other experiments in calorimetry. An allowance for cooling is made by watching the thermometer when the instrument is in shadow. It is found in this way that the solar radiation may amount to nearly 2 units of heat per minute on each square centimetre of surface. The p3 r roheliometer may also be used to measure the heat radiated by a candle, or any other source of heat ; or it may be employed simply to compare two sources with each other. In all such experiments it is obvious that the distance of a given source of heat [ 109.] EFFECT OF DISTANCE. 213 must be taken into account. It will be found, for instance, that the heat radiated by an ordinary can- dle-flame at a distance of about 2 cm. may be as intense as the sun's heat. At the distance of a deci- metre, the heat from the candle could hardly be detected by a pyroheliometer. ^[ 109. Application of the Law of Inverse Squares. When a person stands midway between two sources of heat which are equal in every respect, he feels of course equal intensities of radiation. If, however, one of these sources is much more powerful than the other, he must approach the smaller of the two in order that the warmth from both may seem to be the same. Let the power of the first source be x, and the distance from it a ; let the power of the second source be y, and the distance from it b ; then accord- ing to the law of inverse squares ( 94) the effects of the two sources will be proportional to x -f- a 2 and to y -5- 6 2 , respectively. If the two effects are equal, it follows that x -T- a 2 = y -r- b 2 ; or x : y : : a 2 : b 2 . It thus appears that the powers of any two sources of radiant heat are to each other directly as the squares of the distances at which they produce equal effects. The same reasoning may be applied to two sources of light, to two sources of sound, or to any two sources of radiant energy, the effect of which dimin- ishes as the square of the distance increases. We have, accordingly, a principle by which we may compare any two sources of energy of the same 214 RADIATION. [Exp. 39. kind ; namely to find two distances, a and 6, at which equal effects are produced. To test the equality of two effects with any degree of precision, it is necessary to employ a "differential" instrument of some sort ; that is, an instrument which is constructed especially to indicate the difference between two effects. The instrument must be so delicate that in the absence of any indication, we may assume that the two effects are equal. The methods for the comparison of two sources of heat about to be described, will be found to belong to the general class known as " null methods " ( 42). ^[ 110. The Differential Thermometer and the Ther- mopile. I. A differential thermometer, useful for the comparison of two sources of radiant heat, may be constructed as follows: two cylindrical metallic boxes, d and e, about 10 cm. in diameter, and 1 cm. deep, are made out of the thinnest brass, and fastened by a layer of wax to the support bh. The glass FlG> 86> U-tube or gauge, fg, contains a little colored liquid, and is attached by rubber couplings to the boxes d and e, so that the system may be air-tight. The outer faces of the boxes, d and e, are coated with lampblack, to absorb heat ; the sides may be covered with wool to prevent loss of heat. The two conical shields, a and : N". IV. This shows that the rates of vibration of different harmonics are proportional to the number of vibrating segments in the wire. It has been stated that the ratio between two rates of vibration, n' and n", determines the interval be- tween the two notes to which they correspond. The ordinary musical scale consists of a series of notes whose rates of vibration, whether high or low, are always relatively proportional to the following num- bers set beneath their names : DO RE MI FA SOL LA SI DO 24 27 30 32 36 40 45 48 The interval between the first and third note of this series is called a "third;" between the first and 278 MUSICAL INTERVALS. [Exp. 50. fourth, a " fourth," etc. The first two are said to be one tone apart ; the last two, one semitone apart. The most common musical 'intervals may be arranged as follows, according to the simplicity of the ratios which they involve when reduced to their lowest terms : Name. Ratio. Name. Ratio. Name. Ratio. Unison . . . 1:1 Fourth . . . 4:3 Minor Third .6:5 Octave . . . 2:1 Sixtli ....5:3 Whole Tone .9:8 Fifth ....3:2 Third ....5:4 Semitone . . 16 : 15 The sum of two or more intervals is always repre- sented by the product of the ratios in question ; thus, when we say that two notes are an octave and a fifth apart, we mean that the higher makes one and one half times as many vibrations per second as the octave of the lower note ; or, again, twice as many vibrations as a note a " fifth " above the lower note ; that is, in either case, three times as many vibrations as the lower ^ote itself. In the same way an inter- val of two octaves corresponds to the ratio 4 : 1 between the rates of vibration ; an interval of three octaves corresponds to the ratio 8 : 1, etc. It is a fact to be noted that the musical intervals involving the simplest ratios are the most agreeable to the ear. END OF PART FIRST. 4098 A 000938035 3 UNIVEI LU .KS UBRARY