`400 c c` volume of gas having `gamma=5/2` is suddenly compressed to `100 c c`. If the initial pressure is `P`, the final pressure will be

Options

A. `P//32`
B. 8P
C. 32P
D. 16P

Answer

To solve the problem of finding the final pressure of a gas that is suddenly compressed from a volume of 400 cc to 100 cc under adiabatic conditions, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Initial volume, \( V_1 = 400 \, \text{cc} \)
- Final volume, \( V_2 = 100 \, \text{cc} \)
- Initial pressure, \( P_1 = P \)
- Heat capacity ratio (gamma), \( \gamma = \frac{5}{2} \)

2. **Understand the Adiabatic Process**:
- In an adiabatic process, the relationship between pressure and volume is given by the equation:
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]

3. **Rearranging the Equation**:
- We need to find \( P_2 \). Rearranging the equation gives:
\[ P_2 = P_1 \left( \frac{V_1}{V_2} \right)^\gamma \]

4. **Substituting Known Values**:
- Substitute \( P_1 = P \), \( V_1 = 400 \, \text{cc} \), \( V_2 = 100 \, \text{cc} \), and \( \gamma = \frac{5}{2} \):
\[ P_2 = P \left( \frac{400}{100} \right)^{\frac{5}{2}} \]

5. **Calculating the Volume Ratio**:
- The volume ratio \( \frac{400}{100} = 4 \):
\[ P_2 = P \cdot 4^{\frac{5}{2}} \]

6. **Calculating \( 4^{\frac{5}{2}} \)**:
- \( 4^{\frac{5}{2}} = (2^2)^{\frac{5}{2}} = 2^{5} = 32 \):
\[ P_2 = 32P \]

7. **Final Result**:
- Therefore, the final pressure \( P_2 \) after the sudden compression is:
\[ P_2 = 32P \]

### Final Answer:
The final pressure will be \( 32P \).

---