Molar Specific Heats: $C_p$ and $C_v$, and the Relation $C_p - C_v = R$

A gas's temperature can be raised by supplying heat under many different conditions, and the amount of heat needed to produce a given rise in temperature depends on which condition is chosen. The molar specific heat of a gas is defined as the heat required to raise the temperature of one mole of the gas by one kelvin, under some specified condition; because this quantity is different for different conditions, thermodynamics distinguishes two particular values that matter most for a gas.

$C_v$, the molar specific heat at constant volume, is the heat needed per mole per kelvin when the gas is heated in a rigid, sealed container so that its volume cannot change. Since the volume is fixed, the gas can do no work on its surroundings ($W=0$), so by the first law every joule of heat supplied goes directly and entirely into raising the gas's internal energy: $dQ = dU = C_v\,dT$ (for one mole).

$C_p$, the molar specific heat at constant pressure, is the heat needed per mole per kelvin when the gas is instead heated in a container that is free to expand against a constant external pressure (a cylinder with a freely-moving, weighted piston, for instance). Now, as the gas's temperature rises, it must also expand and do external work against that constant pressure, in addition to raising its internal energy by exactly the same amount as before (since, for an ideal gas, $\Delta U$ depends only on the change in temperature, regardless of how that change was brought about). Because some of the heat supplied at constant pressure is "used up" doing this extra work, more heat is needed, for the same rise in temperature, than at constant volume -- so $C_p$ is always greater than $C_v$ for any gas.

Deriving the exact relation. For one mole of an ideal gas heated at constant pressure, the first law gives $dQ = dU + P\,dV$. Since internal energy of an ideal gas depends only on temperature, $dU = C_v\,dT$ regardless of the process (this is the same $C_v$ as defined above, since $U$ does not "know" whether the process it is part of happens to be at constant pressure or constant volume). Differentiating the ideal gas equation of state for one mole, $PV = RT$, while holding $P$ fixed, gives $P\,dV = R\,dT$. Substituting both of these into the constant-pressure heat, and using the very definition $dQ = C_p\,dT$ for this process,

and since this must hold for every infinitesimal temperature change $dT$, the coefficients themselves must be equal:

$\boxed{C_p - C_v = R}$ …