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

Choose the correct answer from the options given below :

The thermodynamic expressions for each process can be obtained using the first law of thermodynamics and the properties of ideal gases.

For a reversible isothermal expansion of an ideal gas, the temperature remains constant, so the change in internal energy is zero.

$$\Delta U=0.$$

From the first law,

$$\Delta U=q+w,$$

which gives

$$q=-w.$$

The work done in a reversible isothermal expansion is

$$w=-nRT\ln\left(\frac{V_2}{V_1}\right).$$

Hence,

$$q=nRT\ln\left(\frac{V_2}{V_1}\right),$$

so A matches II.

In free expansion, the gas expands against zero external pressure ((P_{\text{ext}}=0)). Therefore,

$$w=0.$$

Since the process is isothermal for an ideal gas,

$$\Delta U=0,$$

and from the first law,

$$q=0.$$

Hence, B matches I.

For an irreversible compression against a constant external pressure, the work done is given by

$$w=-P_{\text{ext}}(V_2-V_1).$$

Thus, C matches III.

In a cyclic reversible process, the system returns to its initial state. Since entropy is a state function, the net change in entropy over one complete cycle is zero.

$$\Delta S=\oint\frac{dq_{\text{rev}}}{T}=0.$$

Hence, D matches IV.

Therefore, the correct matching is

• A — II
• B — I
• C — III
• D — IV

Hence, the correct answer is A-II, B-I, C-III, D-IV.