The molar specific heats of an ideal gas at constant pressure and volume are denoted by $C_P$ and $C_V$, respe

Question

The molar specific heats of an ideal gas at constant pressure and volume are denoted by $C_P$ and $C_V$, respectively. If $\gamma = \frac{C_P}{C_V}$ and $R$ is the universal gas constant, then $C_V$ is equal to:

Accepted Answer

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:
$$\gamma C_V - C_V = R$$
$$C_V(\gamma - 1) = R$$
$$C_V = \frac{R}{\gamma - 1}$$

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