A monatomic gas is suddenly compressed to `(1/8)`th of its initial volume adiabatically. The ratio of its final pressure to the initial pressure is (Given, the ratio of the specific heats of the given gas to be 5/3)

Options

A. `32`
B. `40//3`
C. `24//5`
D. `8`

Explanation

To solve the problem of finding the ratio of the final pressure to the initial pressure of a monatomic gas that is suddenly compressed to \( \frac{1}{8} \) of its initial volume adiabatically, we can follow these steps:

### Step 1: Understand the Adiabatic Process
In an adiabatic process for an ideal gas, the relationship between pressure and volume is given by the equation:
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]
where \( \gamma \) (gamma) is the ratio of specific heats, \( \frac{C_p}{C_v} \). For a monatomic gas, \( \gamma = \frac{5}{3} \).

### Step 2: Set Up the Volume Ratio
Given that the gas is compressed to \( \frac{1}{8} \) of its initial volume, we can express the volumes as:
\[ V_2 = \frac{1}{8} V_1 \]
Thus, the ratio of the volumes is:
\[ \frac{V_1}{V_2} = \frac{V_1}{\frac{1}{8} V_1} = 8 \]

### Step 3: Substitute into the Adiabatic Equation
Now we substitute the volume ratio into the adiabatic equation:
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]
Rearranging gives us:
\[ \frac{P_2}{P_1} = \left(\frac{V_1}{V_2}\right)^\gamma \]

### Step 4: Calculate the Pressure Ratio
Substituting the volume ratio and the value of \( \gamma \):
\[ \frac{P_2}{P_1} = \left(8\right)^{\frac{5}{3}} \]

### Step 5: Simplify the Expression
Calculating \( 8^{\frac{5}{3}} \):
\[ 8 = 2^3 \implies 8^{\frac{5}{3}} = (2^3)^{\frac{5}{3}} = 2^{5} = 32 \]

### Conclusion
Thus, the ratio of the final pressure to the initial pressure is:
\[ \frac{P_2}{P_1} = 32 \]

### Final Answer
The ratio of the final pressure to the initial pressure is \( 32 \).

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