One mole of an ideal gas undergoes two different cyclic processes I and II, as shown in the $P-V$ diagrams below. In cycle I, processes $a, b, c$ and $d$ are isobaric, isothermal, isobaric and isochoric, respectively. In cycle II, processes $a^{\prime}, b^{\prime}, c^{\prime}$ and $d^{\prime}$ are isothermal, isochoric, isobaric and isochoric, respectively. The total work done during cycle I is $W_{I}$ and that during cycle II is $W_{II}$. The ratio $W_{I} / W_{II}$ is ____.

JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · Heat & Thermodynamics

One mole of an ideal gas undergoes two different cyclic processes I and II, as shown in the

P - V

diagrams below. In cycle I, processes

a, b, c

and

d

are isobaric, isothermal, isobaric and isochoric, respectively. In cycle II, processes

a^{'}, b^{'}, c^{'}

and

d^{'}

are isothermal, isochoric, isobaric and isochoric, respectively. The total work done during cycle I is

W_{I}

and that during cycle II is

W_{I I}

. The ratio

W_{I} / W_{I I}

is ____.

Answer: 2

For Cycle I:

Process a (isobaric expansion): $W_{a} = P_{0} (2 V_{0} - V_{0}) = P_{0} V_{0}$
Process b (isothermal): $W_{b} = n R T ln (2) = 2 P_{0} V_{0} ln (2)$
Process c (isobaric compression): $W_{c} = - 2 P_{0} (2 V_{0} - V_{0}) = - 2 P_{0} V_{0}$
Process d (isochoric): $W_{d} = 0$

Total:

W_{I} = P_{0} V_{0} + 2 P_{0} V_{0} ln (2) - 2 P_{0} V_{0} = 2 P_{0} V_{0} ln (2) - P_{0} V_{0}

For Cycle II:

Total:

W_{I I} = P_{0} V_{0} ln (2) - P_{0} V_{0}

Ratio:

\frac{W _{I}}{W _{I I}} = \frac{2 P _{0} V _{0} l n ( 2 ) - P _{0} V _{0}}{P _{0} V _{0} l n ( 2 ) - P _{0} V _{0}} = 2

JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · Integer answer