A gas is compressed isothermally to half its initial volume. The same gas is compressed separately through an adiabatic process untill its volume is again reduced to half. Then

Options

A. Compressing the gas through adiabatic process will require more work to be done
B. Compressing the gas isothermally or adiabatically will required the same amount of work
C. Which of the case(Whehter compression through isothermal of through adiabatic process) requires more work will depends upon the atomicity of the gas
D. Compressing the gas isothermally will require more work to be done

Explanation

To solve the problem of comparing the work done during isothermal and adiabatic compression of a gas to half its initial volume, we can follow these steps:

### Step 1: Understand the Processes
- **Isothermal Process**: The temperature of the gas remains constant during the compression. According to the ideal gas law, \( PV = nRT \), where \( T \) is constant.
- **Adiabatic Process**: No heat is exchanged with the surroundings. The relationship between pressure and volume in an adiabatic process is given by \( PV^\gamma = \text{constant} \), where \( \gamma \) is the heat capacity ratio (\( C_p/C_v \)).

### Step 2: Work Done in Isothermal Compression
- The work done \( W \) during isothermal compression from volume \( V_1 \) to \( V_2 \) (where \( V_2 = \frac{V_1}{2} \)) can be calculated using the formula:
\[ W_{\text{isothermal}} = nRT \ln\left(\frac{V_1}{V_2}\right) = nRT \ln(2) \]

### Step 3: Work Done in Adiabatic Compression
- The work done during adiabatic compression can be derived from the first law of thermodynamics or can be calculated using the formula:
\[ W_{\text{adiabatic}} = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1} \]
However, it is more straightforward to use the relationship for the work done in an adiabatic process, which can also be expressed as:
\[ W_{\text{adiabatic}} = \frac{nC_v(T_1 - T_2)}{\gamma - 1} \]
where \( T_1 \) and \( T_2 \) are the initial and final temperatures, respectively.

### Step 4: Compare Work Done
- To compare the two works, we can analyze the shapes of the PV diagrams for both processes. In the PV diagram:
- The area under the curve represents the work done.
- The isothermal curve is less steep than the adiabatic curve, indicating that for the same change in volume, the area (and thus work done) under the adiabatic curve is greater than that under the isothermal curve.

### Conclusion
- Since the area under the adiabatic curve is greater than that under the isothermal curve, we conclude that:
\[ W_{\text{adiabatic}} > W_{\text{isothermal}} \]
- Therefore, compressing the gas through an adiabatic process requires more work than compressing it isothermally.

### Final Answer
- **The work done during adiabatic compression is greater than that during isothermal compression.**