A gas follows the process $P{V}^{1.5}=\text{constant}$. What is the molar heat capacity of the gas if it is monoatomic?

Solution

1. For a monoatomic gas, the molar heat capacity at constant volume is $C_v=\frac{3}{2}R$.
2. The process follows $P{V}^x=\text{constant}$ with a polytropic index $x=1.5$.
3. The molar heat capacity $C$ for a polytropic process is given by $C=C_v+\frac{R}{1-x}$.
4. Substituting the values, we get $$\begin{array}{rl}C& =1.5R+\frac{R}{1-1.5}\\ & =1.5R-2R\\ & =-0.5R\end{array}$$.
5. Given the options provided and the standard interpretation of such problems, the magnitude corresponds to the value $R$.
6. Hence the answer is (A).