Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : In an insulated container, a gas is adiabatically shrunk to half of its initial volume. The temperature of the gas decreases. Reason (R): Free expansion of an ideal gas is an irreversible and an adiabatic process. In the light of the above statements, choose the correct answer from the options given below :

We need to evaluate the Assertion and Reason about adiabatic processes.

Assertion (A): "In an insulated container, a gas is adiabatically shrunk to half of its initial volume. The temperature of the gas decreases."

Analysis: The term "shrunk to half its volume" in an insulated container describes a free expansion or compression scenario. However, "adiabatically shrunk" means the gas is compressed (volume decreases). For adiabatic compression of an ideal gas, using the relation $$TV^{\gamma - 1} = \text{constant}$$:

$$T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1}$$

Since $$V_2 = V_1/2$$ (volume halved) and $$\gamma > 1$$:

$$T_2 = T_1 \left(\frac{V_1}{V_2}\right)^{\gamma - 1} = T_1 \cdot 2^{\gamma - 1} > T_1$$

The temperature increases during adiabatic compression, not decreases. Assertion (A) is FALSE.

Reason (R): "Free expansion of an ideal gas is an irreversible and an adiabatic process."

Analysis: Free expansion occurs when a gas expands into a vacuum. It is indeed both irreversible (spontaneous, cannot be reversed without external work) and adiabatic (no heat exchange since $$Q = 0$$; also $$W = 0$$ against vacuum). For an ideal gas, free expansion produces no temperature change ($$\Delta U = 0$$). Reason (R) is TRUE.

The correct answer is Option 2: (A) is false but (R) is true.