If we suppose the difference of temperatures between source and refrigerator to be finite, T − T', say, then since T is the temperature of the source, we have for the efficiency (T − T') ⁄ T. If the heat taken in be H, the heat rejected is HT' ⁄ T, so that the heat received by the engine is to the heat rejected by it in the ratio of T' to T. Thus, as was done by Thomson, we may define the temperatures of the source and refrigerator as proportional to the heat taken in from the source and the heat rejected to the refrigerator by a perfect engine, working between those temperatures. The scale may be made to have 100 degrees between the temperature of melting ice and the boiling point, as already explained. We shall return to the comparison of this scale with that of the air thermometer. At present we consider some of the thermodynamic relations of the properties of bodies arrived at by Thomson.
First we take the working substance of the engine as consisting of matter in two states or phases; for example, ice and water, or water and saturated steam. Let us apply equation (A) to this case. If v1, v2 be the volume of unit of mass in the first and second states respectively, the isothermal expansion of the first part of the cycle will take place in consequence of the conversion of a mass dm from the first state to the second. Thus dv, the change of volume, is dm (v2 − v1). Also if L be the latent heat of the substance in the second state, e.g. the latent heat of water, Mdv = Ldm; so that M (v2 − v1) = L. If dp be the step of pressure corresponding to the step dT of temperature, equation (A) becomes
In the case of coexistence of the liquid and solid phases, this gives us the very remarkable result that a change of pressure dp will raise or lower the temperature of coexistence of the two phases, that is, the melting point of the solid, by the difference of temperature, dT, according as v2 is greater or less than v1 Thus a substance like water, which expands in freezing, so that v2 − v1 is negative, has its freezing point lowered by increase of pressure and raised by diminution of pressure. This is the result predicted by Professor James Thomson and verified experimentally by his brother (p. [113] above). On the other hand, a substance like paraffin wax, which contracts in solidifying, would have its melting point raised by increase of pressure and lowered by a diminution of pressure.
The same conclusions would be applicable when the phases are liquid and vapour of the same substance, if there were any case in which v2 − v1 is negative. As it is we see, what is well known to be the case, that the temperature of equilibrium of a liquid with its vapour is raised by increase of pressure.
Another important result of equation (B), as applied to the liquid and vapour phases of a substance, is the information which it gives as to the density of the saturated vapour. When the two phases coexist the pressure is a function of the temperature only. Hence if the relation of pressure to temperature is known, dp ⁄ dT can be calculated, or obtained graphically from a curve; and the volume v2 per unit mass of the vapour will be given in terms of dp ⁄ dT, the temperature T, and the volume v per unit mass of the liquid. The density of saturated steam at different temperatures is very difficult to measure experimentally with any approach of accuracy: but so far as experiment goes equation (B) is confirmed. The theory here given is fully confirmed by other results, and equation (B) is available for the calculation of v2 for any substance for which the relation between p and T is known. It is thus that the density of saturated steam can best be found.
We can obtain another important result for the case of the working substance in two phases from equation (B). The relation is
where c and h are the specific heats of the substance in the two phases respectively, and L is the latent heat of the second phase at absolute temperature T.