An ideal gas has initial volume V and pressure p. In doubling its volume the minimum work done will be in the process (of the given processes)

Options

A. (a) isobaric process
B. (b) isothermal process
C. (c) adiabatic process
D. (d) same in all given processes

Explanation

To determine the process in which the minimum work is done when an ideal gas doubles its volume from an initial volume \( V \) and pressure \( P \), we can analyze the work done in different thermodynamic processes: isobaric, isothermal, and adiabatic.

### Step-by-Step Solution:

1. **Understanding Work Done in Thermodynamic Processes**:
The work done \( W \) by a gas during expansion or compression can be calculated as the area under the PV curve. The formula for work done in a process can vary depending on the type of process.

2. **Isobaric Process**:
- In an isobaric process, the pressure remains constant.
- The work done is given by:
\[ W = P \Delta V = P (V_f - V_i) \]
- Here, \( V_f = 2V \) (final volume) and \( V_i = V \) (initial volume), so:
\[ W = P (2V - V) = PV \]

3. **Isothermal Process**:
- In an isothermal process, the temperature remains constant.
- The work done can be calculated using:
\[ W = nRT \ln \left( \frac{V_f}{V_i} \right) \]
- Assuming one mole of gas (\( n = 1 \)), and using the ideal gas law, we can express \( R \) and \( T \) in terms of \( P \) and \( V \):
\[ W = PV \ln(2) \]
- This work done is greater than that in the isobaric process since \( \ln(2) > 1 \).

4. **Adiabatic Process**:
- In an adiabatic process, there is no heat exchange with the surroundings.
- The work done can be derived from the first law of thermodynamics and is given by:
\[ W = \frac{P_i V_i - P_f V_f}{\gamma - 1} \]
- For an ideal gas undergoing adiabatic expansion, the work done is less than that in both isobaric and isothermal processes.

5. **Comparison of Work Done**:
- From our calculations:
- Isobaric: \( W = PV \)
- Isothermal: \( W = PV \ln(2) \)
- Adiabatic: \( W < PV \) (exact value depends on specific heat ratio \( \gamma \))
- The adiabatic process results in the least amount of work done when doubling the volume.

### Conclusion:
The minimum work done in doubling the volume of an ideal gas occurs during the **adiabatic process**.