A monoatomic gas at pressure `P_(1)` and volume `V_(1)` is compressed adiabatically to `1/8th` of its original volume. What is the final pressure of gas.

Options

A. `P_(1)`
B. `16 P_(1)`
C. `32 P_(1)`
D. `64 P_(1)`

Answer

To solve the problem of finding the final pressure of a monoatomic gas that is compressed adiabatically to 1/8th of its original volume, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**:
In an adiabatic process, the relationship between pressure (P), volume (V), and the adiabatic index (γ) is given by the equation:
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]
where \( \gamma \) for a monoatomic gas is \( \frac{5}{3} \).

2. **Define Initial and Final Conditions**:
- Initial pressure: \( P_1 \)
- Initial volume: \( V_1 \)
- Final volume: \( V_2 = \frac{V_1}{8} \)

3. **Substitute the Values into the Equation**:
We can substitute the known values into the adiabatic equation:
\[ P_1 V_1^{\frac{5}{3}} = P_2 \left(\frac{V_1}{8}\right)^{\frac{5}{3}} \]

4. **Simplify the Equation**:
Rewrite the equation:
\[ P_1 V_1^{\frac{5}{3}} = P_2 \left(\frac{V_1^{\frac{5}{3}}}{8^{\frac{5}{3}}}\right) \]
This simplifies to:
\[ P_1 = P_2 \frac{1}{8^{\frac{5}{3}}} \]

5. **Rearranging for Final Pressure \( P_2 \)**:
Rearranging gives:
\[ P_2 = P_1 \cdot 8^{\frac{5}{3}} \]

6. **Calculate \( 8^{\frac{5}{3}} \)**:
Since \( 8 = 2^3 \), we have:
\[ 8^{\frac{5}{3}} = (2^3)^{\frac{5}{3}} = 2^5 = 32 \]

7. **Final Expression for \( P_2 \)**:
Thus, substituting back, we get:
\[ P_2 = 32 P_1 \]

### Conclusion:
The final pressure of the gas after being compressed adiabatically to 1/8th of its original volume is:
\[ \boxed{32 P_1} \]