Vardaan Institute - Level 3 Solutions

An ideal gas undergoes a polytropic process $P V^n = \text{Constant}$. Find the molar heat capacity $C$ of the gas in terms of $C_v$, $R$, and $n$.

Sol: $dQ = dU + dW = n C_v dT + \frac{n R dT}{1-n} \implies C = C_v + \frac{R}{1-n}$.

One mole of monoatomic gas goes through a cyclic process $A \to B \to C \to A$ on a $P-V$ diagram consisting of constant volume heating $A(P_0,V_0) \to B(2P_0,V_0)$, isobaric expansion $B \to C(2P_0,2V_0)$, and linear process $C \to A$. Find net work done and efficiency of cycle.

Sol: Area of right triangle $W_{net} = \frac{1}{2} (2V_0 - V_0)(2P_0 - P_0) = \frac{1}{2} P_0 V_0$. Heat added $Q_{in} = Q_{AB} + Q_{BC} = 1.5 P_0 V_0 + 2.5(2 P_0 V_0) = 6.5 P_0 V_0$. Efficiency $\eta = \frac{0.5 P_0 V_0}{6.5 P_0 V_0} = \frac{1}{13} \approx 7.69\%$.

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