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[63] 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 _A_τ, 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
(_E_τ)/(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
(_Ep_{0}v_{0}_)/(μ) . (_E_τ)/(1 + _Et_), or (_E^2p_)/(μ(1 + _Et_)^2)τ.