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. Adiabatic
B. Isobaric
C. Isochoric
D. Isothermal

Answer

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. The processes to consider are adiabatic, isobaric, isochoric, and isothermal.

### Step-by-Step Solution:

1. **Understanding Work Done on Gas**:
The work done on a gas during a process can be calculated using the area under the pressure-volume (P-V) curve. The work done (W) is given by the integral of pressure (P) with respect to volume (V):
\[ W = \int_{V_i}^{V_f} P \, dV \]
where \( V_i \) is the initial volume and \( V_f \) is the final volume.

2. **Analyzing Different Processes**:
- **Adiabatic Process**: In an adiabatic process, there is no heat exchange with the surroundings. The P-V curve is steeper compared to isothermal processes. The work done is significant but not the maximum.
- **Isobaric Process**: In an isobaric process, the pressure remains constant. The work done is given by:
\[ W = P \Delta V = P(V_f - V_i) \]
This process allows for a larger area under the curve, resulting in more work done.
- **Isochoric Process**: In an isochoric process, the volume remains constant, meaning no work is done (W = 0).
- **Isothermal Process**: In an isothermal process, the temperature remains constant. The work done is given by:
\[ W = nRT \ln\left(\frac{V_f}{V_i}\right) \]
While this process allows for work to be done, it is less than that of the isobaric process when compressing to half the volume.

3. **Comparing Work Done**:
- The work done in the isochoric process is zero.
- The work done in the isothermal process is less than that in the isobaric process.
- The adiabatic process does involve work but is still less than the isobaric process.

4. **Conclusion**:
The process that results in the maximum work done on the gas when compressed to half its initial volume is the **isobaric process**. This is because it allows for the greatest area under the P-V curve compared to the other processes.

### Final Answer:
The maximum work done on the gas occurs during the **isobaric process**.