Which of the following is correct in terms of increasing work done for the same initial and final volume?

Options

A. Adiabatic `lt` Isothermal `lt`Isobaric
B. Isobaric `lt` Adiabatic `lt` Isothermal
C. Adiabatic `lt` Isobaric `lt` Isothermal
D. None of these

Answer

To determine the correct order of work done in different thermodynamic processes (isobaric, isothermal, and adiabatic) for the same initial and final volumes, we can analyze the work done in each process using a PV diagram.

### Step-by-Step Solution:

1. **Understanding Work Done in Thermodynamic Processes**:
The work done (W) in a thermodynamic process is represented by the area under the PV (Pressure-Volume) curve on a graph.

2. **Identifying the Processes**:
- **Isobaric Process**: This process occurs at constant pressure. On a PV diagram, it is represented as a horizontal line. The work done is calculated as:
\[ W = P \Delta V \]
- **Isothermal Process**: This process occurs at constant temperature. On a PV diagram, it is represented by a hyperbolic curve. The work done is given by:
\[ W = nRT \ln\left(\frac{V_f}{V_i}\right) \]
- **Adiabatic Process**: This process occurs without heat transfer. On a PV diagram, it is represented by a curve that is steeper than the isothermal curve. The work done can be calculated using:
\[ W = \frac{P_f V_f - P_i V_i}{\gamma - 1} \]
where \(\gamma\) is the heat capacity ratio.

3. **Sketching the PV Diagram**:
- Draw a PV diagram with the initial volume \(V_i\) on the left and the final volume \(V_f\) on the right.
- Draw a horizontal line for the isobaric process (constant pressure).
- Draw a hyperbolic curve for the isothermal process (constant temperature).
- Draw a steeper curve for the adiabatic process.

4. **Comparing the Areas**:
- The area under the isobaric line represents the maximum work done because it covers the entire horizontal distance at constant pressure.
- The area under the isothermal curve is less than that of the isobaric process but more than that of the adiabatic process.
- The area under the adiabatic curve is the smallest, as it has the steepest slope and thus covers the least area for the same change in volume.

5. **Conclusion**:
Based on the areas under the curves, we can conclude that:
- Work done in adiabatic process \(<\) Work done in isothermal process \(<\) Work done in isobaric process.

Thus, the correct order of increasing work done for the same initial and final volume is:
**Adiabatic < Isothermal < Isobaric**.