Question

A perfect gas undergoes a cyclic process ABCA. A→B: Isothermal expansion ($V_1 \to 2V_1$). B→C: Isobaric compression to $V_1$. C→A: Isochoric change to $P_1$. Total work done is :

• A. 0
• B. nRTln 2
• C. nRT (ln 2 + 1/2)
• D. nRT (ln 2 - 1/2)

Hint:

Work done in a cycle is the area enclosed by the $P-V$ loop. Clockwise cycles represent positive work done by the gas.

Answer 1

The Correct Option is D

Step 1: $W_{AB}$ (Isothermal) $= nRT \ln(V_2/V_1) = nRT \ln 2$.
Step 2: $W_{BC}$ (Isobaric) $= P_2(V_1 - V_2) = P_2(V_1 - 2V_1) = -P_2V_1$.
Step 3: In A→B, $P_1V_1 = P_2(2V_1) \Rightarrow P_2V_1 = \frac{1}{2}P_1V_1 = \frac{1}{2}nRT$.
Step 4: $W_{BC} = -nRT/2$.
Step 5: $W_{CA}$ (Isochoric) $= 0$.
Step 6: $W_{total} = nRT \ln 2 - \frac{1}{2}nRT = nRT(\ln 2 - 1/2)$.