The molar heat capacity of a gas in a process $P{V}^2=\text{constant}$ is given by what expression, where $R$ is the gas constant and $\gamma$ is the adiabatic index?

Solution

1. For a polytropic process defined by $P{V}^x=\text{constant}$, the molar heat capacity $C$ is given by $C=C_v+\frac{R}{1-x}$.
2. The molar heat capacity at constant volume is $C_v=\frac{R}{\gamma -1}$ and the given process $P{V}^2=\text{constant}$ implies $x=2$.
3. Substituting these values into the formula, we get $C=\frac{R}{\gamma -1}+\frac{R}{1-2}$.
4. Simplifying the expression yields $C=\frac{R}{\gamma -1}-R$.
5. Hence the answer is (B).