Result

Molar heat capacities

C_{V} = \frac{f}{2} R, C_{P} = C_{V} + R

Statement

Molar heat capacity at constant volume, set by the number of degrees of freedom f; at constant pressure it is larger by R.

Symbols and units

$C_{V}, C_{P}$	molar heat capacity at constant volume, pressure	J/(mol·K)
$f$	degrees of freedom	—
$R$	gas constant	8.314 J/(mol·K)

When it applies

An ideal gas; C_P − C_V = R specifically assumes ideal-gas behaviour.

Don't use when

The relation C_P − C_V = R assumes ideal-gas behavior specifically — it does not hold for a real gas or a liquid/solid.

Derivation

Equipartition gives the internal energy per mole directly in terms of f, and differentiating with respect to temperature gives C_V:

C_{V} = \frac{f}{2} R

At constant pressure, some added heat goes into expansion work rather than raising temperature, adding exactly R to the heat capacity:

C_{P} = C_{V} + R

Watch out for

C_P is always greater than C_V — the extra R accounts for work done during expansion at constant pressure, which constant-volume heating never has to supply.