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Rh Bousfield's observations did not extend beyond 80", owing to the difficulty of excessive evaporation with an open calorimeter. According to his curves, the corresponding values of the specific heat appear to be approaching a maximum at 8oC., a little lower than that shown by Liidin's curve. The value of the specific heat at 8pC., according to Liidin's formula, is 1-0184 ' n terms of the specific heat at 2OC. taken as unity, and exceeds the value given by the continuous electric method by 1-55%. This looks alarm- ing at first sight, but the method of comparison in terms of the actual specific heat, though commonly adopted, is really unfair, because the quantity actually observed in Liidin's method is the total heat, which shows a difference of only 0-31 calorie according to the above table at 8oC. Dieterici's observations at looC., where they were most reliable, differ by only 0-14% from the con- tinuous electrical method, and he does not claim an order of accu- racy greater than o-i % for the ice-calorimeter.

According to Bousfield's experiments, the absolute value of the mechanical equivalent of the calorie at 2OC. is 4-1752 joules. He attributes the discrepancy between this value and the value 4-180 given by the continuous electric method to the uncertainty in the electromotive force of the Clark standard cell, the value of which was commonly taken in 1900 as being 1-4342 volts at I5C., which has since proved to be erroneous. Thus Wolff and Waters (U.S.A. Bureau of Standards, vol. 4, p. 64, 1907) found the value 1-4333 for Clark cells of the type employed in the continuous electric method, which would exactly account for the discrepancy. It should be observed, however, that the electromotive force of the actual cells employed was determined at the time with a specially designed elertrodynamometer (Phil. Trans., A, 1902, p. 81), and was found to be 1-4334 volts at I5C., which was used in place of the legal value 1-4342 volts in deducing the absolute value 4-180 joules for the equivalent of the calorie at 2OC. The same electrodynamometer was employed 15 years later by Prof. Norman Shaw (Phil. Trans., A, 1914, vol. 214, pp. 147-198) without any modification, in deter- mining the electromotive force of the Weston cell. In the course of this work he verified the constants of the coils and the theory of the instrument with a very high order of accuracy, so that there can be little doubt that the value 1-4334 found for the Clark cells at I5C. was substantially correct.

Continuous Mixture Method. Since the number of separate deter- minations of the specific heat of water at points between 50 and ioo c C. by the continuous electric method was only 12, and since these were made under conditions of exceptional difficulty, and differed most widely from the values found by Liidin and Bous- field, it was felt to be desirable to confirm the variation in this region by an entirely independent method of equal accuracy. The continuous mixture method (Bakerian Lecture, Phil. Trans., A, 1912, vol. 212, pp. 1-32) was devised for this purpose, and consisted in passing a steady current of water, initially at iooC., through an interchanger, in which it gave up a large part of its heat to a cur- rent of cold water initially at 25C., emerging at a temperature in the neighbourhood of 70 C., without having actually mixed with the cold current. The same current was then cooled to an accu- rately regulated temperature in the neighbourhood of 25C., and reentered the interchanger as the cold current. The point of the method is that the circulation is continuous, so that the water equivalent of the interchanger is not required, and that the hot and cold currents are the same, so that the quantity of the current divides out of the equation (except in the small term representing the exter- nal heat-loss) and need not be determined with an accuracy greater than I %, since the external heat-loss can easily be reduced to a small fraction of I % of the heat-exchange between the currents. The actual temperatures h and t 2 of the hot current on entering and leaving the interchanger, and those of the cold current, / 3 and ti, were observed with platinum thermometers to o-ooi C. If s' is the mean specific heat of the hot current between h and fe, and s" that of the cold current between t 3 and t t, we have the equation,

s'fo-fe) =*" (/-/,)+X/M,

where X is the external heat-loss in gram-calories per second, ar ! M the value of the water current in grams per second. The heat- loss was determined, as in the continuous electric method, by vary- ing the flow M while keeping the temperatures the same. In a large number of trials it was found that the ratio of s' to s" agreed with the value 1-0050 given by the continuous electric method, but disagreed materially with the value given by Liidin's formula. It was concluded that the discrepancy from Liidin's formula was probably to be attributed to the unavoidable errors of his method, due to losses by evaporation at temperatures above 50, and to the uncertainties of zero and stem-exposure which cannot be elim- inated in the employment of mercury thermometers.

Formulae for the Specific Heat of Water. It is usual to employ an empirical formula of the type, s = i+at+bt 2 -\-ct 3 +etc., which is familiar and convenient for the application of the method of least squares to the results of observation. The formulae most often quoted for water are those of Liidin and Dieterici, which are as follows in terms of the calorie at 2OC. :

Dieterici, s = 1-0013 0-0104(^/100) +0-0208 (J/ioo) 2 Liidin, s = i 0-07668(1/100) +o-i96(//ioo) 2 0-116 0-00025 0-040 0-030

The probable errors of the coefficients, as given by Liidin, are shown in the line below his formula. The formula of Dieterici repre- sents his observations satisfactorily from 50 to 3OOC., but does not apply to the variation near the freezing point, which cannot be represented satisfactorily by this type of formula without an additional term. The formula of Liidin is fairly accurate between o and 25", but appears to give results about I % too high between 6p and 9OC. It is also inconvenient in practice, because the coeffi- cients are large and of opposite signs, giving the small variation required as a difference between relatively large terms. In the pre- liminary reduction of the results of the continuous electric method (B. A. Rep., 1899) it appeared that a formula of this type would be unsuitable, and the observations were accordingly represented by three simple formulae for different ranges of temperature between o and 2OOC., as given in the previous article (see 5.66). These have since been combined into a single equivalent formula, which is more convenient for several purposes.

. . . (i)

The value of the constant is adjusted to make s = i when < = 2O. The other terms are small and positive, and can be calculated with sufficient accuracy for all possible purposes by means of a lo-in. slide rule. This formula agrees very closely with the table pre- viously given, but represents a later and more accurate reduction. It is of no theoretical significance, and cannot safely be extrapolated much above looC., but still agrees very closely with Regnault's observations at l6oC. Above looC. it is better to use the thermo- dynamical formula (see 27.903) suggested by McF. Gray, which agrees very closely with experiment from 40 to iooC., but does not represent the increase of specific heat with fall of temperature near the freezing point. Gray's formula was re-defined by Callendar as representing the change of total heat of water under saturation pressure, and then agrees very closely with the observations of Dieterici at high temperatures, when corrected to give the change of total heat in place of the intrinsic energy. It has a simple theo- retical foundation, and greatly simplifies the thermodynamical rela- tions between liquid and vapour. There is good reason to believe (Callendar, Properties of Steam, pp. 160, 196) that it continues to hold satisfactorily right up to the critical point, where the specific heat becomes infinite.

By experiments on the supercooled liquid, Prof. H. T. Barnes has shown that the increase of specific heat with fall of temperature con- tinues to follow the same curve above and below the freezing point. By very accurate experiments on mercury, using the continuous electric method, he has shown that a diminution of the specific heat with rise of temperature occurs as in the case of water, but persists up to a minimum at I4OC. It appears probable that a similar

Ehenomenon would be found for all liquids at low vapour pressures, ut it is masked in the case of volatile liquids by the opposite effect of the vapour-molecules, as represented by the thermodynamical formula. The diminution of the specific heat of water was attributed by H. A. Rowland to the presence of a small proportion of solid. molecules in the liquid near the freezing point. The rapid increase of the specific heat of a solid as the fusing-point is approached may similarly be attributed to the presence of a small but rapidly increas- ing proportion of liquid molecules in the solid. The proportion required in either case, to explain the diminution of hardness and rigidity of the solid, or the anomalous expansion of water near the freezing point, is small, but cannot be calculated with certainty on account of our imperfect knowledge of molecular forces and dimen- sions. Such a theory would be difficult to verify in any case by experiment for the liquid and solid molecules. On the other hand, the latent heat of the vapour-molecules in the liquid, according to the thermodynamical formula, is simply that of a volume of satu- rated vapour, equal to that of the liquid, and easily calculated.

Specific Heat of Gases and Vapours. The continuous electric method was first applied in the case of steam (see 27.901) and gave results near 100 corroborating Regnault's value at higher tempera- tures. The same method was applied to air and COz by W. F. G. Swann (Phil. Trans., A, 1910, vol. 210, p. 199), who found results from 2 to 5% higher than those of Regnault. Swann's formula has since been verified by Holborn and Jakob (Zeit. Ver.Deut., Ing., 58, p. 1429, 1914) and it is now generally recognized that this method is the most accurate for the determination of the specific heat of any fluid at constant pressure. Swann's values for air at 20 and iooC. were closely consistent with those of Joly at constant volume (see 5.67), and gave a ratio of specific heats very nearly equal to 1-40, as required by the kinetic theory for a diatomic gas. They also showed a very small increase with temperature at the rate of only one-half of I % for looC. His values for COz veri- fied with improved accuracy the rapid increase with temperature found by Regnault and Wiedemann for this gas, which amounted to 12% for 100. This increase of specific heat was not accounted for on the kinetic theory, which required that all the degrees of freedom of a gas molecule should be equally excited, and should contribute constant terms to the specific heat. The apparent dis- crepancy was explained (B. A. Rep., 1908, p. 340) by supposing that a natural frequency of the gas-molecule would be excited by radia- tion in direct proportion to the intensity of the corresponding fre- v quency at each temperature. It was shown that a natural frequency