A mass of gas is first expanded isothermally and then compressed adiabatically to its original volume. What further simplest operation must be performed on the gas to restore it to its original state?

Options

A. (a) An isobaric cooling to bring its temperature to initial value
B. (b) An isochoric cooling to bring its pressure to its initial value
C. (c) An isothermal process to take its pressure to its initial value
D. (d) An isochoric heating to bring its temperature to initial value

Explanation

To solve the problem, we need to analyze the processes that the gas undergoes and determine what operation is required to restore it to its original state.

### Step-by-Step Solution:

1. **Understanding the Initial State**:
- Let the initial state of the gas be represented by the point (P1, V1), where P1 is the initial pressure and V1 is the initial volume.

2. **First Process - Isothermal Expansion**:
- The gas is expanded isothermally. In this process, the temperature remains constant while the volume increases. This can be represented on a PV diagram as a curve moving from point (P1, V1) to a new point (P2, V2), where V2 > V1 and P2 < P1.

3. **Second Process - Adiabatic Compression**:
- After the isothermal expansion, the gas is then compressed adiabatically back to its original volume V1. An adiabatic process is characterized by no heat exchange with the surroundings, and it typically results in a steeper curve on the PV diagram. The gas moves from point (P2, V2) back to point (P3, V1), where the pressure P3 is higher than P2.

4. **Restoring to Original State**:
- To restore the gas to its original state (P1, V1), we need to adjust the pressure while keeping the volume constant (V1). This means that we need to perform an operation at constant volume.

5. **Determining the Required Process**:
- Since we need to decrease the pressure from P3 to P1 while keeping the volume constant, the process that accomplishes this is an isochoric process (constant volume).
- According to the ideal gas law (PV = nRT), if the volume is constant and we want to decrease the pressure, the temperature must also decrease. This indicates that the gas must undergo isochoric cooling.

6. **Conclusion**:
- Therefore, the simplest operation that must be performed on the gas to restore it to its original state is **isochoric cooling** to bring its pressure back to P1.

### Final Answer:
The required operation is **isochoric cooling to bring its pressure to its initial value**.

---