Question

When a gas undergoes a reversible adiabatic expansion, its entropy remains constant even though the volume increases. Explain how this can be consistent with the microscopic interpretation of entropy developed in Section 13.2. (Hint: Consider what happens to the distribution of velocities in the gas.)

Short answer

The detailed explanation is as follows in steps.

Even when the volume of gas grows due to reversible adiabatic expansion, its entropy remains constant since the temperature falls, resulting in a decrease in the velocity of the molecules. The molecules have greater space but less velocity as the volume is raised. Because these effects cancel each other out, the number of microstates and hence the entropy remain constant.

Step-by-step solution

Concept of adiabatic expansion

The adiabatic expansion of a perfect gas is reversible. When a perfect gas expands adiabatically, the temperature of the working gas should fall: work is done but no heat enters the system, the internal energy declines, and the temperature of the working gas falls as well.

Explanation

Even when the volume of gas grows due to reversible adiabatic expansion, its entropy remains constant since the temperature falls, resulting in a decrease in the velocity of the molecules. The molecules have greater space but less velocity as the volume is raised. Because these effects cancel each other out, the number of microstates and hence the entropy remain constant.