Match List - I with List - II. Given $V_1$ and $V_2$ are initial and final volumes respectively.

List - I (Isothermal process)	List - II (Expression)
A. Reversible expansion	I. $q = 0$
B. Free expansion	II. $q = nRT \ln\dfrac{V_2}{V_1}$
C. Irreversible Compression	III. $w = -p_{ext}(V_1 - V_2)$
D. Cyclic reversible	IV. $\dfrac{q_{rev}}{T} = 0$

Choose the correct answer from the options given below :

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