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

Question

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 $\frac{W_{I}}{W_{I I}}$ is _______. [2023]

Smart Achievers · Accepted Answer

(2)

From P-V diagram

Work done during cycle I

$d$

$= 4 P_{0} (2 V_{0} - V_{0}) + n R T \ln (\frac{4 V_{0}}{2 V_{0}}) + 2 P_{0} (V_{0} - 4 V_{0}) + 0$

$= 4 P_{0} V_{0} + n R (\frac{8 P_{0} V_{0}}{n R}) \ln 2 - 6 P_{0} V_{0}$

$= 8 P_{0} V_{0} \ln 2 - 2 P_{0} V_{0}$

Work done during cycle II

$W_{I I} = W_{a}' + W_{b}' + W_{c}' + W_{d}'$

$= n R T \ln (\frac{2 V_{0}}{V_{0}}) + 0 + P_{0} (V_{0} - 2 V_{0}) + 0$

$= n R (\frac{4 P_{0} V_{0}}{n R}) \ln 2 - P_{0} V_{0}$ $= 4 P_{0} V_{0} \ln 2 - P_{0} V_{0}$

$∴$ $\frac{W_{I}}{W_{I I}} = \frac{8 P_{0} V_{0} \ln 2 - 2 P_{0} V_{0}}{4 P_{0} V_{0} \ln 2 - P_{0} V_{0}} = 2$

Solution

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