The initial pressure and volume of an ideal gas are `P_0` and `V_0`. The final pressure of the gas when the gas is suddenly compresssed to volume `(V_0)/4` will be :
(Given `gamma` =ratio of specific heats at constant pressure and at constant volume)

Options

A. `P_0(4)^(gamma)`
B. `4P_0`
C. `P_0(4)^(1/gamma)`
D. `p_0`

Answer

To solve the problem, we need to apply the principles of an adiabatic process for an ideal gas. The relationship we will use is derived from the adiabatic condition, which states that for an ideal gas undergoing an adiabatic process, the following equation holds:

\[ P V^\gamma = \text{constant} \]

where \( P \) is the pressure, \( V \) is the volume, and \( \gamma \) (gamma) is the ratio of specific heats (C_p/C_v).

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
- The initial pressure of the gas is \( P_0 \).
- The initial volume of the gas is \( V_0 \).

2. **Identify Final Conditions**:
- The final volume after compression is \( \frac{V_0}{4} \).
- We need to find the final pressure \( P_2 \).

3. **Apply the Adiabatic Condition**:
- According to the adiabatic condition, we can write:
\[ P_0 V_0^\gamma = P_2 \left(\frac{V_0}{4}\right)^\gamma \]

4. **Rearranging the Equation**:
- Substitute \( V_2 = \frac{V_0}{4} \) into the equation:
\[ P_0 V_0^\gamma = P_2 \left(\frac{V_0}{4}\right)^\gamma \]

5. **Simplifying the Right Side**:
- The right side can be simplified as follows:
\[ P_0 V_0^\gamma = P_2 \left(\frac{V_0^\gamma}{4^\gamma}\right) \]
- This simplifies to:
\[ P_0 V_0^\gamma = \frac{P_2 V_0^\gamma}{4^\gamma} \]

6. **Canceling \( V_0^\gamma \)**:
- Since \( V_0^\gamma \) is common on both sides, we can cancel it out:
\[ P_0 = \frac{P_2}{4^\gamma} \]

7. **Solving for \( P_2 \)**:
- Rearranging gives:
\[ P_2 = P_0 \times 4^\gamma \]

8. **Final Answer**:
- The final pressure of the gas after compression is:
\[ P_2 = P_0 \times 4^\gamma \]

### Conclusion:
The final pressure of the gas when it is suddenly compressed to a volume of \( \frac{V_0}{4} \) is \( P_0 \times 4^\gamma \).