Page:Reflections on the Motive Power of Heat.djvu/211

Rh 48. In some very remarkable researches made by Mr. Joule upon the heat developed by the compression of air, the quantity of heat produced in different experiments has been ascertained with reference to the amount of work spent in the operation. To compare the results which he has obtained with the indications of theory, let us determine the amount of work necessary actually to produce the compression considered above.

49. In the first place, to compress the gas from the volume v + dv to v, the work required is pdv, or, since

$$pv = p_{_0} v_{_0} (1 + Et), $$ $$p_{_0} v_{_0} (1 + Et) \tfrac{dv}{v}. $$ Hence, if we denote by W the total amount of work necessary to produce the compression from V to V', we obtain, by integration,

$$W = p_{_0} v_{_0} (1 + Et) \log \tfrac{V}{V'}. $$

Comparing this with the expression above, we find

50. Hence we infer that— (1) The amount of work necessary to produce a unit of heat by the compression of a gas is the same for all gases at the same temperature;