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 will analyze three different thermodynamic processes: isobaric, isothermal, and adiabatic.

### Step-by-Step Solution:

1. **Identify the Initial and Final States:**
- Initial volume \( V_1 = V \)
- Final volume \( V_2 = 2V \)
- Initial pressure \( P_1 = P \)

2. **Understand the Types of Processes:**
- **Isobaric Process:** The pressure remains constant. The work done \( W \) is given by:
\[ W = P \Delta V = P (V_2 - V_1) = P (2V - V) = PV \]
- **Isothermal Process:** The temperature remains constant. For an ideal gas, the work done is given by:
\[ W = nRT \ln\left(\frac{V_2}{V_1}\right) = nRT \ln(2) \]
- **Adiabatic Process:** No heat is exchanged with the surroundings. The work done can be derived from the first law of thermodynamics and is generally less than that of isothermal processes for the same change in volume.

3. **Analyze the Work Done in Each Process:**
- **Isobaric Work:** The work done is \( W = PV \).
- **Isothermal Work:** The work done is \( W = nRT \ln(2) \). This is generally larger than the work done in an adiabatic process.
- **Adiabatic Work:** The work done is less than that of isothermal because the gas does not absorb heat. The exact expression for work done in an adiabatic process can be derived using the relation \( PV^\gamma = \text{constant} \), but it is known that it results in less work compared to isothermal and isobaric processes.

4. **Conclusion:**
- Among the three processes, the adiabatic process requires the least amount of work to double the volume of the gas. Therefore, the minimum work done occurs in the **adiabatic process**.

### Final Answer:
The minimum work done will be in the **adiabatic process**.