Match List - I with List - II. Given V_1 and V_2 are initial and...

For an isothermal reversible expansion, the change in internal energy $\Delta U = 0$. From the first law of thermodynamics, $q = -w$. The work done is $w = -nRT \ln\left(\dfrac{V_2}{V_1}\right)$, which gives $q = nRT \ln\left(\dfrac{V_2}{V_1}\right)$. Thus, A matches II.

For free expansion, the external pressure is zero ($p_{ext} = 0$), so the work done $w = 0$. Since the process is isothermal, $\Delta U = 0$. Therefore, the heat exchanged $q = 0$. Thus, B matches I.

For an irreversible compression, the work done against a constant external pressure is given by $w = -p_{ext}\Delta V$. This corresponds to the expression in III. Thus, C matches III.

For a cyclic reversible process, the initial and final states are the same, so the change in entropy $\Delta S = 0$. Since $\Delta S = \dfrac{q_{rev}}{T}$, we get $\dfrac{q_{rev}}{T} = 0$. Thus, D matches IV.

The correct matching is A-II, B-I, C-III, D-IV.

Answer: A-II, B-I, C-III, D-IV