Two thermodynamic processes $A$ and $B$ are shown in the figure as straight lines through the origin on a $\log P$ versus $\log V$ graph. Line $A$ has slope $\gamma$ and line $B$ makes $45^\circ$ with the $\log V$ axis. The molar heat capacities for processes $A$ and $B$ are $C_A$ and $C_B$. The molar heat capacities at constant pressure and constant volume are $C_P$ and $C_V$ respectively. Choose the correct statement.
Answer: (A) $C_P > C_B > C_V$
A straight line of slope $s$ on the $\log P$–$\log V$ graph means $P \propto V^{s}$, i.e. a polytropic process $PV^{x} = $ constant with $x = -s$. Its molar heat capacity is
$$C = C_V + \frac{R}{1 - x}$$
Process B: $s = 1$, $x = -1$, so $C_B = C_V + \dfrac R2$. This lies between $C_V$ and $C_P = C_V + R$.
(Process A: $x = -\gamma$, $C_A = C_V + \dfrac{R}{1+\gamma}$, which also lies between them.)
Hence $C_P > C_B > C_V$.
Solution by Sreeraj P, M.Sc Physics