For an ideal gas, the relationship between the molar specific heat at constant pressure ($C_P$) and the molar specific heat at constant volume ($C_V$) is given by Mayer's relation : $C_P - C_V = R$ where $R$ is the universal gas constant.

The adiabatic index (or heat capacity ratio) $\gamma$ is defined as: $\gamma = \frac{C_P}{C_V} \implies C_P = \gamma C_V$

Substituting this into Mayer's relation: $\gamma C_V - C_V = R$ $C_V(\gamma - 1) = R$ $C_V = \frac{R}{\gamma - 1}$