Page:Reflections on the Motive Power of Heat.djvu/214

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52. The following proposition is proved by Carnot as a deduction from his general theorem regarding the specific heats of gases.

The excess of specific heat under a constant pressure above the specific heat at a constant volume, is the same for all gases at the same temperature and pressure.

53. To prove this proposition, and to determine an expression for the "excess" mentioned in its enunciation, let us suppose a unit of volume of a gas to be elevated in temperature by a small amount, τ. The quantity of heat required to do this will be Aτ, if A denote the specific heat at a constant volume. Let us next allow the gas to expand without going down in temperature, until its pressure becomes reduced to its primitive value. The expansion which will take place will be $$\tfrac{E\tau}{1 + Et}$$, if the temperature be denoted by t; and hence, by (8), the quantity of heat that must be supplied, to prevent any lowering of temperature, will be

$$\tfrac{E p_{_0} v_{_0}}{\mu} \cdot \tfrac{E\tau}{1 + Et}$$,  or   $$\tfrac{E^2 p}{\mu(1 + Et)^2}\tau$$.