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 :

<p>Let's analyze both statements step by step.</p> <p><p>Assertion (A): "In an insulated container, a gas is adiabatically shrunk to half of its initial volume. The temperature of the gas decreases."</p> <p> • In an adiabatic process, no heat is exchanged with the surroundings. </p> <p> • For a reversible adiabatic (compression) process, the relationship between temperature and volume is given by </p> <p>  $T V^{\gamma-1} = \text{constant},$ </p> <p>  where $\gamma$ is the heat capacity ratio (greater than 1). </p> <p> • If the volume is reduced from $V$ to $V/2$, then </p> <p>  $$T_{\text{final}} = T_{\text{initial}} \left(\frac{V}{V/2}\right)^{\gamma-1} = T_{\text{initial}}\,(2)^{\gamma-1}.$$ </p> <p> • Since $(2)^{\gamma-1} > 1,$ the final temperature is higher than the initial temperature. </p> <p> • Therefore, the assertion that the temperature decreases is incorrect.</p></p> <p><p>Reason (R): "Free expansion of an ideal gas is an irreversible and an adiabatic process."</p> <p> • In a free expansion, the gas expands into a vacuum without doing any work on the surroundings. </p> <p> • Since the container is insulated, no heat is exchanged, making it adiabatic. </p> <p> • However, the process is inherently irreversible due to the spontaneous expansion and mixing of states. </p> <p> • Thus, this statement is true.</p></p> <p>Since Assertion (A) is false and Reason (R) is true (but the reason does not explain the false assertion about temperature change during compression), the correct answer is:</p> <p>Option C </p> <p>(A) is false but (R) is true.</p>