A sample of gas expands from volume `V_(1)` to `V_(2)`. The amount of work done by the gas is greatest when the expansion is

Options

A. adiabatic
B. isobaric
C. isothermal
D. Equal in all above cases

Answer

To determine when the work done by a gas during expansion is greatest, we need to analyze the different thermodynamic processes: isobaric, isothermal, and adiabatic.

### Step-by-Step Solution:

1. **Understanding Work Done in Thermodynamics**:
The work done by a gas during expansion can be expressed mathematically as:
\[ W = \int_{V_1}^{V_2} P \, dV \]
where \( W \) is the work done, \( P \) is the pressure, and \( V_1 \) and \( V_2 \) are the initial and final volumes, respectively.

2. **Identifying Different Processes**:
- **Isobaric Process**: The pressure remains constant during the expansion. The work done is given by:
\[ W = P(V_2 - V_1) \]
- **Isothermal Process**: The temperature remains constant. The pressure changes according to the ideal gas law, and the work done is:
\[ W = nRT \ln\left(\frac{V_2}{V_1}\right) \]
- **Adiabatic Process**: No heat is exchanged with the surroundings. The work done is less than in the isothermal process because the pressure decreases more rapidly than in the isothermal case.

3. **Comparing Work Done**:
- In the **isobaric process**, since the pressure is constant, the work done is directly proportional to the change in volume, which can be quite significant.
- In the **isothermal process**, while the work done can be substantial, it is dependent on the logarithmic relationship and may not be as high as in the isobaric case for the same volume change.
- In the **adiabatic process**, the work done is the least because the pressure drops significantly as the gas expands.

4. **Conclusion**:
The work done by the gas is greatest during an **isobaric expansion** because the pressure remains constant, allowing for maximum work output for a given change in volume.

### Final Answer:
The amount of work done by the gas is greatest when the expansion is **isobaric**.