574 HEAT thus denote the cooling effect per atmo of differential pressure, is a function of the temperature, and is independent of the whole pres sure. "With this notation (36) becomes .dv JK- t- v = 0. dt n This is a linear equation in z with a second member, if for a moment we put 2 = log t. Integrating it, and replacing t, we find, as the complete integral, 61. Expansions of different gases, pressure constant, calculated from the Joule and Thomson experiment. We have from equation (37) ^^ji+JKfe j_ r odt j _ (38) _ 1 ~~7^n For each of the gases experimented on, except hydrogen, was found to vary nearly in the inverse ratio of f 2 . Putting then = A( /) 2 , we find Hence, for these gases at pressures from to 5 or 6 atmos, (38) becomes v - v _ t - This shows that the " proper mean cooling effect " (M in the table of 58) is A, -)A=-769A, which differs so little from 1 + 1-3663 the arithmetic mean of the cooling effects at and 100 C., that if we had simply taken the arithmetic mean for each of the other gases, as for want of knowing better we took it for hydrogen, the difference in the result would have been barely perceptible. 62. Modifying (39) or (38) to suit any two temperatures, t, t , we have m denoting the proper mean cooling effect per atmo in the Joule and Thomson experiment (to be reckoned as negative in the case of hydrogen or any other gas, if there is any other, in which the ex periment shows a heating effect). This "proper mean "may be taken as the arithmetic mean of the values for t and t?, unless t - 1 considerably exceeds 100. To reduce (40) to numbers, let V be the volume of unit mass of the gas when at the temperature of melting ice, and under one atmo of pressure. Eegnault (vol. ii. p. 303) finds that the value of K/V is within 1 per cent, the same for oxygen, nitrogen, and hydrogen as for common air. He also (vol. ii. pp. 224-226) finds K to be the same for common air at from 1 to 12 atmos ; for hydrogen, 1 to 9 ; for carbonic acid, 1 to 37. No doubt similar constancy would be found for oxygen and nitrogen. Hence, as above ( 56), for common air we still have JK/nV = 0126, and thus (40) becomes , , . __ ( for common air, v-v t-t f - V .io t - jo f ox ygcn, hydrogen, and v. nitrogen (41). If in this formula we take t and tf for the temperatures of 100 C. and C., (v-v )j v becomes 273^1 or 3662 ; and we therefore find -0126m ..... (42), J E=-3662fl + V which agrees with (33) above. 63. The values given by Joule and Thomson s experi ment for m are -0 039 for hydrogen, + 0-208 for air, + 0-253 for oxygen, and + 249 for nitrogen. From these and from the previous results for carbonic acid ( 58) we have the following table for calculating the expansions from C. to 100 C. of the gases named : Name of Gas. Expansion under constant pressure (=E). Hydrogen -3662 (1 - 00049 VJv )} Common air 3662(1 + 0026 V /0 ) I Oxygen 3662(1 + 0-0032 V n /i> ) }-. . Nitrogen 3662(1 + 0-0031 V /t> ) Carbonic acid 8662 (1 + 0163 V /0 )J (43). These formuke must be exceedingly near the truth for all pressures from to G atmos, because within this range the thermal effects in the Joule and Thomson experiment were very approximately in simple proportion to the differences of pressure on the two sides of the plug. The following table of results calculated from (43) for several pressures of from to 6 atmos is interesting as showing such different expansions for the different cases, determined by thermodynamic theory from ReguauU s measurements of specific heats and Joule and Thomson s of their particular thermal effect, with absolutely no direct measurement of expansion except the one for common air at one atmo, shown as the third entry of column 5 in the table. The other five entries of column 5 show a fair amount of agreement between our theoretical results and the only direct measurements by Regnault. More of direct measurement, to allow a more extensive comparison, is very desirable. Ratio of Bulk at C. to Bulk Ratio of Den sity at C. to Density Expansion, pressure constant, from to 100 C. supposing supposing Pressure were Pressure Name of Gas. 1 atmo at the same tempera ture. were 1 atmo at the same temperature. According to According to direct experi v g Io. theory. ments by Vo Vo Regnault. 1
3662 1 1 3660 36613 Hydrogen 4 1/3 3 3657 1 1/3-35 3-35 3656 36616 I 1/6 6 3651 f =
3662 1 1 3672 36706 Common Air
1/3
3 3691 1/3-38 3-38 3694 36954 I 1/6 6 3719 (
3662 Oxygen 1 J 3 1 3 3674 "3697 U/6 6 3732 f
3662 Nitrogen I 1 1 3 3673 3696 I 1/6 6 3730 f
3662 1 1 3721 37099 Carbonic Acid
1/3
3 3841 1/3-316 3-316 3859 38455 I 1/6 6 4019 ... j 64. We are now quite prepared to make a practical working thermometer directly adapted to show temperature on the absolute thermodynamic scale through the whole range of temperature, from the lowest attainable by any means to the highest for which glass remains solid. It is to be remarked that our investigation of 51, and all the deduced formula? and relative calculations, are absolutely independent of the approximate fulfilment of Boyle s law by the gases to which we have applied them, and are equally applicable without any approach to fulfilment of Boyle s law ; also that the only experimental data on which are founded our special numerical conclusions of 59 to 63 are Regnault s measurements of specific heats under constant pressures, and Joule and Thomson s measurements of the thermal effect of forcing through a porous plug. From these experimental data alone we see by formula (38) of 61 how to graduate a constant-pressure gas thermo meter so that it shall show temperature on the absolute thermodynamic scale. Hence, notwithstanding the diffi culty ( 24 above) which Regnault found in the thermo- metric use of air or other gases on the system of constant
pressure, and his practical preference for the constant-volume
