A mono-atomic ideal gas is compressed form volume `V` to `V//2` through various process. For which of the following processes final pressure will be maximum:

Options

A. isobaric
B. isothermal
C. adiabatic
D. `PV^(2) =` constant

Explanation

To determine which process results in the maximum final pressure when a monoatomic ideal gas is compressed from volume \( V \) to \( \frac{V}{2} \), we will analyze four different processes: isobaric, isothermal, adiabatic, and a process where \( PV^2 \) is constant.

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
- Let the initial pressure be \( P_i = P_0 \).
- The initial volume is \( V_i = V \).
- The final volume is \( V_f = \frac{V}{2} \).

2. **Isobaric Process**:
- In an isobaric process, the pressure remains constant.
- Therefore, the final pressure \( P_f \) is:
\[ P_f = P_i = P_0 \]

3. **Isothermal Process**:
- In an isothermal process, the temperature remains constant.
- Using the ideal gas law \( PV = nRT \), we have:
\[ P_i V_i = P_f V_f \]
- Substituting the values:
\[ P_0 V = P_f \left(\frac{V}{2}\right) \]
- Rearranging gives:
\[ P_f = 2 P_0 \]

4. **Adiabatic Process**:
- For an adiabatic process, the relationship between pressure and volume is given by:
\[ P V^\gamma = \text{constant} \]
- For a monoatomic gas, \( \gamma = \frac{5}{3} \).
- Thus, we have:
\[ P_i V_i^\gamma = P_f V_f^\gamma \]
- Substituting the values:
\[ P_0 V^{\frac{5}{3}} = P_f \left(\frac{V}{2}\right)^{\frac{5}{3}} \]
- Rearranging gives:
\[ P_f = P_0 \left(\frac{V}{\frac{V}{2}}\right)^{\frac{5}{3}} = P_0 \left(2\right)^{\frac{5}{3}} = P_0 \cdot 2^{\frac{5}{3}} \]

5. **Process where \( PV^2 \) is Constant**:
- For this process, we have:
\[ P_i V_i^2 = P_f V_f^2 \]
- Substituting the values:
\[ P_0 V^2 = P_f \left(\frac{V}{2}\right)^2 \]
- Rearranging gives:
\[ P_f = P_0 \cdot 4 \]

6. **Comparison of Final Pressures**:
- From the calculations:
- Isobaric: \( P_f = P_0 \)
- Isothermal: \( P_f = 2 P_0 \)
- Adiabatic: \( P_f = P_0 \cdot 2^{\frac{5}{3}} \approx 2.519 P_0 \)
- \( PV^2 \) constant: \( P_f = 4 P_0 \)

- Comparing these values:
- \( P_0 < 2 P_0 < 2.519 P_0 < 4 P_0 \)

7. **Conclusion**:
- The maximum final pressure occurs in the process where \( PV^2 \) is constant, yielding \( P_f = 4 P_0 \).

### Final Answer:
The final pressure will be maximum for the process where \( PV^2 \) is constant.