A monoatomic gas is compressed adiabatically to `(1)/(4)^(th)` of its original volume , the final pressure of gas in terms of initial pressure P is

Options

A. 7.08 P
B. 8.08 P
C. 9.08 P
D. 10.08 P

Explanation

To solve the problem of finding the final pressure of a monoatomic gas compressed adiabatically to \( \frac{1}{4} \) of its original volume, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
Let the initial pressure be \( P \) and the initial volume be \( V \). The final volume after compression is given by:
\[ V_2 = \frac{1}{4} V \]

2. **Use the Adiabatic Condition**:
For an adiabatic process, the relationship between pressure and volume is given by:
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]
where \( \gamma \) (gamma) is the heat capacity ratio. For a monoatomic gas, \( \gamma = \frac{5}{3} \).

3. **Substitute Known Values**:
Substitute the known values into the adiabatic condition:
\[ P V^\gamma = P_2 \left(\frac{1}{4} V\right)^\gamma \]

4. **Simplify the Equation**:
Rearranging the equation gives:
\[ P_2 = P \frac{V^\gamma}{\left(\frac{1}{4} V\right)^\gamma} \]
This can be simplified further:
\[ P_2 = P \frac{V^\gamma}{\left(\frac{1}{4^\gamma} V^\gamma\right)} = P \cdot 4^\gamma \]

5. **Substitute the Value of Gamma**:
Now substitute \( \gamma = \frac{5}{3} \):
\[ P_2 = P \cdot 4^{\frac{5}{3}} \]

6. **Calculate \( 4^{\frac{5}{3}} \)**:
To compute \( 4^{\frac{5}{3}} \):
\[ 4^{\frac{5}{3}} = (2^2)^{\frac{5}{3}} = 2^{\frac{10}{3}} = 2^{3.33} \approx 10.0794 \]

7. **Final Expression for Pressure**:
Therefore, the final pressure \( P_2 \) can be expressed as:
\[ P_2 \approx 10.08 P \]

### Conclusion:
The final pressure of the gas in terms of the initial pressure \( P \) is approximately:
\[ P_2 \approx 10.08 P \]