An ideal gas is compressed to half its initial volume by means of several peocesses. Which of the process results in the maximum work done on the gas ?

Options

A. Isothermal
B. Adiabatic
C. Isobaric
D. Isochoric

Explanation

To determine which process results in the maximum work done on an ideal gas when it is compressed to half its initial volume, we can analyze the work done in different thermodynamic processes: isobaric, isothermal, adiabatic, and isochoric.

### Step-by-Step Solution:

1. **Identify Initial and Final Volumes**:
- Let the initial volume of the gas be \( V_1 \).
- The final volume after compression is \( V_2 = \frac{V_1}{2} \).

2. **Understand the Different Processes**:
- **Isobaric Process**: The pressure remains constant while the volume changes.
- **Isothermal Process**: The temperature remains constant while the volume changes.
- **Adiabatic Process**: No heat is exchanged with the surroundings during the volume change.
- **Isochoric Process**: The volume remains constant, so no work is done (Work = 0).

3. **Work Done in Each Process**:
- **Isochoric**: Work done \( W = 0 \) (no volume change).
- **Isobaric**: Work done \( W = P \Delta V = P(V_2 - V_1) \). Since \( V_2 < V_1 \), this work will be negative (work done on the gas).
- **Isothermal**: The work done can be calculated using the formula \( W = nRT \ln\left(\frac{V_1}{V_2}\right) \). This will also be negative.
- **Adiabatic**: The work done can be calculated using the relationship between pressure and volume. The work done is generally greater than in isothermal and isobaric processes due to the nature of the adiabatic curve being steeper.

4. **Comparison of Work Done**:
- The area under the curve in a PV diagram represents the work done. The steeper the curve, the more work is done.
- In the PV diagram:
- The area under the isochoric process is zero.
- The area under the isobaric process is less than that under the isothermal process.
- The area under the adiabatic process is the largest compared to the other processes.

5. **Conclusion**:
- Since the adiabatic process has the largest area under the curve in the PV diagram, it results in the maximum work done on the gas when compressed to half its initial volume.

### Final Answer:
The process that results in the maximum work done on the gas is the **Adiabatic Process**.