JEE Advanced 2023 Paper 2 · Q13 · Kinetic Theory of Gases
One mole of an ideal gas undergoes two different cyclic processes $\alpha$ and $\beta$, as shown in the P-V diagrams. In cycle $\alpha$, processes $a, b, c, d$ are isobaric, isothermal, isobaric, and isochoric, respectively. In cycle $\beta$, processes $a', b', c', d'$ are isothermal, isochoric, isobaric, and isochoric, respectively. The total work done during cycle $\alpha$ is $W_\alpha$ and that during cycle $\beta$ is $W_\beta$. The ratio $W_\alpha/W_\beta$ is ___.
Reveal answer + step-by-step solution
Correct answer:2
Solution
Read corner pressures and volumes from the P-V diagrams.
Cycle $\alpha$ (steps $a,b,c,d$ = isobaric, isothermal, isobaric, isochoric): • $a$ at $P=4P_{0}$, $V_{0}\to 2V_{0}$: $W_{a}=4P_{0}V_{0}$. • $b$ isothermal ($PV=8P_{0}V_{0}$), $2V_{0}\to 4V_{0}$: $W_{b}=8P_{0}V_{0}\ln 2$. • $c$ at $P=2P_{0}$, $4V_{0}\to V_{0}$: $W_{c}=-6P_{0}V_{0}$. • $d$ isochoric: $W_{d}=0$. $$W_{\alpha} = 4P_{0}V_{0} + 8P_{0}V_{0}\ln 2 - 6P_{0}V_{0} = 2P_{0}V_{0}(4\ln 2 - 1).$$
Cycle $\beta$ (steps $a',b',c',d'$ = isothermal, isochoric, isobaric, isochoric): • $a'$ isothermal ($PV=4P_{0}V_{0}$), $V_{0}\to 2V_{0}$: $W_{a'}=4P_{0}V_{0}\ln 2$. • $b'$ isochoric: $0$. • $c'$ at $P=P_{0}$, $2V_{0}\to V_{0}$: $W_{c'}=-P_{0}V_{0}$. • $d'$ isochoric: $0$. $$W_{\beta} = 4P_{0}V_{0}\ln 2 - P_{0}V_{0} = P_{0}V_{0}(4\ln 2 - 1).$$
Ratio: $W_{\alpha}/W_{\beta} = 2$.
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