Which is correct about isothermal expansion of an ideal gas?

\(W_{rev} > W_{irr}\)

The correct answer is option 3. \(W_{rev} > W_{irr}\).

Understanding Work Done in Isothermal Expansion of an Ideal Gas

During an isothermal expansion of an ideal gas, the temperature remains constant (isothermal) while the volume increases. This process can be carried out reversibly or irreversibly, impacting the amount of work done.

Reversible vs. Irreversible Expansion

Reversible Expansion: This process is slow and controlled. The system (gas) can continuously exchange heat with the surroundings to maintain a constant temperature. As the gas expands, the opposing pressure from the surroundings can be adjusted to ensure a smooth and controlled expansion. This maximizes the work done on the surroundings.

Imagine using a piston with a very heavy weight on top. You slowly add more weights (increasing pressure) as the gas expands, mimicking a constant opposing pressure.

Irreversible Expansion: This process is rapid. There's insufficient time for heat exchange with the surroundings. The gas expands against a pressure that's not perfectly adjusted, leading to a less efficient expansion. This results in less work done on the surroundings compared to the reversible case.

Imagine quickly removing the lid from a container filled with gas. The gas expands rapidly against the surrounding atmosphere, not a perfectly adjusted pressure.

Why \(W_{rev} > W_{irr}\)

Given the differences between reversible and irreversible expansion, here's why the work done in a reversible process \((W_{rev})\) is greater than the work done in an irreversible process \((W_{irr})\):

Continuous Heat Transfer: In a reversible expansion, the system can exchange heat with the surroundings to maintain constant temperature. This allows the gas to expand against a constant opposing pressure, maximizing the work done on the surroundings (positive \(W_{rev}\)).

Optimal Pressure Adjustment: The opposing pressure from the surroundings in a reversible expansion is continuously adjusted to match the gas's pressure as it expands. This minimizes any energy wasted overcoming a higher opposing pressure, leading to more efficient work extraction.

Friction and Inefficiencies: In an irreversible expansion, the rapid process can introduce friction and inefficiencies. The gas might expand against a higher pressure initially, wasting energy, and then against a lower pressure later, not extracting as much work. This results in a lower overall work done \((W_{irr})\).

In essence: The controlled nature of a reversible expansion allows for optimal heat exchange and pressure adjustment, maximizing the work done on the surroundings. Conversely, the rapid and uncontrolled nature of an irreversible expansion leads to less efficient work extraction. This translates to \(W_{rev}\) being greater than \(W_{irr}\).

Additional Information:

The amount of heat transferred \((q)\) also varies depending on the control of the process. A reversible expansion allows for more efficient heat exchange, potentially requiring less heat input \((q_{rev})\) compared to an irreversible expansion \((q_{irr})\).