HEAT 573 where E denotes the expansion of hydrogen, pressure constant, from to 100 C. in terms of its volume at 0, that is to say, E = 1ioozl2 (31). ~u Let V denote what the volume would be at C. if the pressure were n instead of the actual pressure p. We have 039JK/nV (32). Regnault finds (Experiences, vol. ii. p. 122) that the value of K/nV for hydrogen agrees within J per cent, with its value for common air; and for common air he finds K= "238. Thus with 423 5 for the value of J in metres ( 9 above) we find JK = 100 79 metres. And Regnault s observations on the density of air give for nV (or the height of the homogeneous atmosphere at C. ) 7990 metres. Hence for common air, and therefore also for hydro gen, JK/nV = 0126 ; and thus (32) becomes with c= - 00049 for hydrogen. For this gas expanding under constant pressure of one atmo Regnault found (Experiences, vol. i. p. 80) E= -36613, which gives -^ = 273-13. Hence (33), with that is to sa}- : 57. We conclude from Regnault s observations on the expansion of hydrogen from to 100 C. under a con- stant pressure of one atmo, and from the small heating effect discovered in Joule and Thomson s experiments on the forcing of hydrogen through a porous plug, that the absolute temperature of melting ice is 273 00, if the unit or degree of absolute temperature is so chosen as to make the difference one hundred between the temperatures of melting ice and of water with steam at one atmo of pres- sure. 58. An almost identical number for that most import- ant physical constant, the absolute temperature of melting ice, is obtained from observations on common air, and a not very different number from observations on carbonic acid, the only two gases besides hydrogen for which Regnault (Experiences, vol. i. p. 90) measured the expansion under constant pressure, and for which Joule and Thomson made their experiment on the thermal effect of passage through a porous plug. For each of these two gases the thermal effect observed was a lowering of temperature, and was found to vary at different temperatures very nearly in the inverse proportion of the square of the temperature C., by mercury thermometer, with 273 added. Hence nearly enough for use in the small term of the denominator of (26) we have, for air and carbonic acid, where t denotes as before absolute temperature, and A the amount of the cooling effect per atmo of difference of pressures, on the two sides of the plug, at the temperature of melting ice. The values of A found for common air and carbonic acid are -275 and 1-388. Regnault (Experiences, vol. ii. p. 126) finds JK/ITV greater for carbonic acid than for common air in the ratio of 1 -39 to 1 on the average of temperatures from to 210. But he found also that the specific heat of carbonic acid varies greatly with the temperature ; and, taking the mean of the values which he finds for it at and 100, p. 130, as the proper mean for our present purpose, we find for JK/nV , a value 1 29 times its value for common air. From these experimental results we find by the mathematical process below (61) still the same approximate formula (3 3), but with c = + -0026 for common air and c= + 0163 for carbonic acid. At constant pressure of one atmo Regnault s measurements gave E = -36706 for common air, and E = -3710 for car bonic acid; and dividing 100 by these decimals we find respectively 272 44 and 269-5. The corrections on these numbers by formula (33) to give the absolute temperature of freezing are accordingly + 70 and +4 -4, and the corresponding estimates for the required absolute tempera ture are 273 1 4 and 273 9. Bringing together the results in the three cases, we see them conveniently in the following table : Proper Name of Gas. Expan sion at one atmo according to Reg nault. mean 1 cool ing-effect of forcing through porous plug per atmo according to recked esti mate of absolute tempera ture of melting Correction calculated from cooling- effect. 100 JK X M . Resulting estimate of absolute tempera ture of melting E. Joule and E *nv Thomson. to- M. E Hydrogen 36613 -0 039 273-13 -0 13 273-00 Air 36706 + 208 272-44 + 070 273-14 Carbonic acid 37100 + 1 105 269-5 + 4 -4 273-9 1 Investigated in 61 below. The close agreement of the results from hydrogen and common air is very satisfactory, and it is interesting to see it brought about with so large a correction calculated from the Joule and Thomson effect. It is also interesting to see the sevenfold larger correction of nearly 5 bringing so nearly the same result from the 1 per cent, larger expan sion of carbonic acid. The ^ per cent, discrepance which remains between the results from carbonic acid and from hydrogen is not satisfactory, and requires explanation, particularly when we remark that, of five measurements by Regnault (Experiences, vol. i. p. 84) of the expansion of carbonic acid under constant pressure of one atmo, all lie within _L per cent, of the mean number 3710 which y / he has given, and we have taken, as his result. Notwithstanding that the Joule and Thomson correction is so much greater for common air than for hydrogen, the result from common air is probably the most trustworthy of the three, because both Regnault s experiments and Joule and Thomson s were probably more accurate for air than for either of the other two gases. The true result to one place of decimals may therefore be considered as most probably being 273*1, but the probability that it is nearer 273 1 than 273 is scarcely enough to make it worth while to use in any ordinary thermodynamic calculations any other number than 273, which is exactly that found from hydrogen. 59. The real meaning of our result 273 1 for the absolute temperature of melting ice, expressed without any choice of degrees or units for temperature, is that the ratio of the temperature at which vapour of water has a pres sure of one atmo to the temperature at which ice melts is 373-1/273-1. Still another way of saying the same thing, this time eliminating all numerical reckoning of tempera ture, is (see THERMODYNAMICS) as follows : FOR EVERY HUNDRED UNITS OF HEAT CONVERTED INTO Determi- WORK BY A PERFECT THERMODYNAMIC ENGINE, 373 1 ARE nation of TAKEN FROM THE SOURCE, AND 273 1 REJECTED TO THE dut jT f REFRIGERATOR, IF THE TEMPERATURE OF THE SOURCE BE engine THAT AT WHICH STEAM OF WATER HAS A PRESSURE OF with ONE ATMO, AND THE TEMPERATURE OF THE REFRIGERATOR source THAT AT WHICH ICE MELTS. and refri " 60. Integration of differential equation (26), 56, between volume ft those and absolute temperature for a gas, derived from the Joule and tempera- Tliomson experiment. turea Returning to 56 we may write equation (26) as follows : ^--JK|- (36). dt Sp For each of the five gases experimented on, namely, common air, oxygen, nitrogen, carbonic acid, hydrogen, the experiment showed that, for all pressures up to five or six atmos, St/Sp was sensibly independent of the pressure, but that it varied very considerably
with the temperature. Hence, if we put $t/8p = e/n, 0, which will