Solution Analysis

Assertion (A) Evaluation

The assertion describes a scenario where a gas is contained by a piston. The internal pressure is '$P$', and the external pressure is '$P_{ex}$'. It states that if '$P_{ex}$' is significantly greater than '$P$', an infinitesimal decrease in '$P_{ex}$' will not cause it to drop below '$P$' and will not change the direction of the process.

• The condition '$P_{ex}$ is measurably greater than $P$' means $P_{ex} > P$. This imbalance causes the piston to move inwards, compressing the gas. The process is compression.
• When '$P_{ex}$' is decreased infinitesimally (by a small amount $dP_{ex}$), the new external pressure becomes $P'_{ex} = P_{ex} - dP_{ex}$.
• Since the initial difference ($P_{ex} - P$) is measurable (finite), reducing '$P_{ex}$' by an infinitesimal amount $dP_{ex}$ is unlikely to make $P'_{ex} \le P$. Therefore, '$P'_{ex}$' will likely remain greater than '$P$'.
• As long as $P'_{ex} > P$, the condition for compression persists, and the direction of the process (inward movement of the piston) remains unchanged. Thus, Assertion (A) is correct.

Reason (R) Evaluation

The reason provides two conditions: the system is in mechanical equilibrium, and the compression is thermodynamically reversible.

• Mechanical Equilibrium: This state is achieved only when the internal pressure '$P$' exactly equals the external pressure '$P_{ex}$'. The premise of Assertion (A) explicitly states that '$P_{ex}$' is *measurably greater* than '$P$', which means $P_{ex} > P$. This is the opposite of mechanical equilibrium; it signifies a process is occurring.
• Thermodynamically Reversible Compression: A reversible process requires the system to pass through a series of equilibrium states. This means the difference between '$P_{ex}$' and '$P$' must be infinitesimal ($P_{ex} \approx P$) throughout the entire process. The condition '$P_{ex}$ is measurably greater than $P$' implies a finite pressure difference, making the compression inherently irreversible.
• Since the conditions described in Reason (R) contradict the setup described in Assertion (A), Reason (R) is incorrect.

Final Conclusion

Assertion (A) correctly explains the physical behavior of the gas under the specified pressure conditions. Reason (R) provides justifications related to mechanical equilibrium and reversibility that are not applicable when the external pressure is measurably greater than the internal pressure. Therefore, Assertion (A) is correct, and Reason (R) is incorrect.

The correct option is C.