Question

A) $W_{\text{Rev.~}} = - \int P_{\text{gas~}}dV$

$$W_{\text{Rev. Isot. Exp.~}} = - nRTln\left\lbrack \frac{V_{f}}{V_{i}} \right\rbrack $$A → (II)

B Free expansion

$${W_{\text{irrev.~}} = - P_{\text{ext~}}\Delta V }{P_{\text{ext~}} = 0 }{W = 0 }$$B $\rightarrow$ (I)

C) Irreversible expansion

$${W_{\text{irrev.~}} = - P_{ext}\Delta V }{W_{\text{irrev.~}} = - P_{\text{ext~}}\left( V_{f} - V_{i} \right) }$$C → (III)

D Irreversible compression

$${W_{\text{irrev.~}} = - P_{ext}\Delta V }{W_{\text{irrev.~}} = - P_{\text{ext~}}\left( V_{i} - V_{f} \right)}$$