P-V Diagrams & Cyclic Processes question from JEE Main 2024 (30 Jan, Shift 1)

Answer: (a)

1. A straight line on the log P - log V graph with slope $k$ means $P \propto V^{k}$, i.e. a polytropic process $PV^{n} =$ const with $n = -k$.
2. Molar heat capacity: $C = C_V + \frac{R}{1-n} = C_V + \frac{R}{1+k}$.
3. For $k > 0$: $C_V < C < C_V + R = C_P$, and a larger $k$ gives a smaller $C$.
4. Line A is steeper than line B ($k_B = \tan 45^\circ = 1$), so $C_A < C_B$.
5. So the answer is (a) $C_P > C_B > C_A > C_V$.