In his article on Heat Thomson gave two definitions of the scale of absolute temperature. One is that stated on p. [126] above, namely, that the temperature of the source and refrigerator are in the ratio of the heat taken in from the source to the heat given to the refrigerator, when the engine describes a Carnot cycle consisting of two isothermal and two adiabatic changes.
The other definition is better adapted for general use, as it applies to any cycle whatever which is reversible. Let the working substance expand under constant pressure by an amount dv (AB' in Fig. [12]), and let heat H be given to the substance at the same time. The external work done is pdv. Thomson called pdv ⁄ H the work ratio. Now let the temperature be raised by dT without giving heat to the substance or taking heat from it, and let the corresponding pressure rise be dp; and call dp ⁄ p the pressure ratio. The temperature ratio dT ⁄ T is equal to the product of the work ratio and the pressure ratio, that is,
This is clearly true; for dvdp is the area of a cycle like AB'C'D, represented in Fig. [12], for which an amount of heat H is taken in, though not in this case strictly at one temperature. And clearly, since in Fig. [12] the change from B' to B is adiabatic, H is the heat which would have to be taken in for the isothermal change AB in the Carnot cycle ABCD, which has the same area as AB'C'D. Thus the efficiency of the cycle is dvdp ⁄ H, and this by the former definition is dT ⁄ T.
Or we may regard the matter thus:—The amount of heat H which corresponds to an infinitesimal expansion dv may be used in equation (A) whether the expansion is isothermal or not, if we take T as the average temperature of the expansion. Hence we have dp ⁄ dT = H ⁄ (dv.T), that is, dT ⁄ T = dpdv ⁄ H. The theorem on p. [128] is obtained by what is virtually this process.
Comparison of Absolute Scale with Scale of Air Thermometer
The comparison which Joule and Thomson carried out of the absolute thermodynamic scale with the scale of the constant pressure gas thermometer has already been referred to, and it has been shown that the two scales would exactly agree, that is, absolute temperature would be simply proportional to the volume of the gas in a gas thermometer kept at the temperature to be measured, if the internal energy of the gas were not altered by an alteration of volume without alteration of temperature, that is, if the de − ∂e of p. [107] above were zero. Joule tested whether this was the case by immersing two vessels, connected by a tube which could be opened or closed by a stopcock, in the water of a calorimeter, ascertaining the temperature with a very sensitive thermometer, and then allowing air which had already been compressed into one of the vessels to flow into the other, which was initially empty. It was found that no alteration of temperature of the water of the calorimeter that could be observed was produced. But the volume of the air had been doubled by the process, and if any sensible alteration of internal energy had taken place it would have shown itself by an elevation or a lowering of the temperature of the water, according as the energy had been diminished or increased.
Thomson suggested that the gas to be examined should be forced through a pipe ending in a fine nozzle, or, preferably, through a plug of porous material placed in a pipe along which the gas was forced by a pump, and observations made of the temperature in the steady stream on both sides of the plug. The experiments were carried out with a plug of compressed cotton-wool held between two metal disks pierced with holes, in a tube of boxwood surrounded also by cotton-wool, and placed in a bath of water closely surrounding the supply pipe. This was of metal, and formed the end of a long spiral all immersed in the bath. Thus the temperature of the gas approaching the plug was kept at a uniform temperature determined by a delicate thermometer; another thermometer gave the temperature in the steady stream beyond the plug.
In the case of hydrogen the experiments showed a slight heating effect of passage through the plug; air, oxygen, nitrogen and carbonic acid were cooled by the passage.
The theory of the matter is set forth in the original papers, and in a very elegant manner in the article on Heat. The result of the analysis shows that if ∂w be the positive or negative work-value of the heat which will convert one gramme of the gas after passage to its original temperature; and T be absolute temperature, and v volume of a gramme of the gas at pressure p, and the difference of pressure on the two sides of the plug be dp, the equation which holds is