An ideal gas at pressure $P$ and volume $V$ is expanded to volume$ 2V.$ Column $I$ represents the thermodynamic processes used during expansion. Column $II$ represents the work during these processes in the random order.:
| Column $I$ | Column $II$ |
| $(p)$ isobaric | $(x)$ $\frac{{PV(1 - {2^{1 - \gamma }})}}{{\gamma - 1}}$ |
| $(q)$ isothermal | $(y)$ $PV$ |
| $(r)$ adiabatic | (z) $PV\,\iota n\,2$ |
The correct matching of column $I$ and column $II$ is given by
Diffcult
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$\mathrm{V}_{\mathrm{i}}=\mathrm{V} \quad \mathrm{V}_{\mathrm{f}}=2 \mathrm{V}$
$P_{i}=P$
$(P) \rightarrow(y)$ isobaric process
$\mathrm{W}=\mathrm{P} \Delta \mathrm{V}=\mathrm{PV}$
$(q) \rightarrow(z)$ isothermal
$\mathrm{W}=\mathrm{nRT} \ln \frac{\mathrm{V}_{\mathrm{f}}}{\mathrm{V}_{\mathrm{i}}}=\mathrm{PV} \ln 2$
$(r) \rightarrow(x)$ Adiabatic
$P_{f}=\left(\frac{V_{i}}{V_{f}}\right)^{\gamma} P_{i}=2^{\gamma} P$
$W=\frac{P_{f} V_{f}-P_{i} V_{i}}{1-\gamma}=\frac{\left(2^{-\gamma} P\right)(2 V)-P V}{1-\gamma}$
$\Rightarrow w=\frac{P V\left(1-2^{1-\gamma}\right)}{\gamma-1}$
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