Question

No work is done during free expansion of an ideal gas:

• A. Only in reversible isothermal process
• B. Only in irreversible isothermal process
• C. Both in reversible and irreversible isothermal process
• D. Only in reversible adiabatic process but not in irreversible adiabatic process

Hint:

For free expansion of an ideal gas, remember:
- External pressure is zero, so work \( W = 0 \).
- For an ideal gas, isothermal processes have \( \Delta U = 0 \), so \( q = -W \).
- Distinguish between reversible (work done) and irreversible (e.g., free expansion, no work) processes.

Answer 1

The Correct Option is B

In thermodynamics, free expansion of an ideal gas refers to the process where the gas expands into a vacuum (e.g., an evacuated chamber) without doing work on the surroundings, as there is no external pressure opposing the expansion. Work done by a gas is given by \( W = -\int P_{\text{ext}} dV \), where \( P_{\text{ext}} \) is the external pressure. In free expansion, \( P_{\text{ext}} = 0 \), so \( W = 0 \), meaning no work is done.
Let’s analyze the options:
- Reversible isothermal process: In a reversible isothermal expansion of an ideal gas, the gas expands slowly against a gradually decreasing external pressure, maintaining equilibrium. Work is done, calculated as \( W = -nRT \ln\left(\frac{V_f}{V_i}\right) \), which is non-zero since the volume changes. Thus, work is done, ruling out option (A).
- Irreversible isothermal process: Free expansion is an example of an irreversible isothermal process for an ideal gas. Since \( P_{\text{ext}} = 0 \), no work is done (\( W = 0 \)). Additionally, for an ideal gas, the internal energy \( U \) depends only on temperature (\( \Delta U = 0 \) for isothermal processes). From the first law of thermodynamics, \( \Delta U = q + W \), so \( 0 = q + 0 \), implying \( q = 0 \). This makes free expansion a unique case where no work is done, supporting option (B).
- Both reversible and irreversible isothermal process: Since work is done in reversible isothermal processes, this option is incorrect, ruling out (C).
- Reversible adiabatic process: In a reversible adiabatic process, \( q = 0 \), and work is done as the gas expands, changing its temperature (\( \Delta U = W \)). Free expansion is not typically adiabatic because the temperature remains constant for an ideal gas in free expansion (isothermal). Even in an irreversible adiabatic process, work depends on external pressure, so this option is incorrect, ruling out (D).
Thus, no work is done only in an irreversible isothermal process like free expansion.