A monoatomic gas `(gamma=5//3)` is suddenly compressed to `(1//8)` of its volume adiabatically then the pressure of the gas will change to

Options

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

Answer

To solve the problem step by step, we will use the principles of adiabatic processes for an ideal gas.

### Step 1: Understand the given information
We have a monoatomic gas with a heat capacity ratio (gamma) of \( \gamma = \frac{5}{3} \). The gas is compressed adiabatically to \( \frac{1}{8} \) of its initial volume.

### Step 2: Define the initial and final states
Let:
- Initial pressure = \( P_1 \)
- Initial volume = \( V_1 = V \)
- Final volume = \( V_2 = \frac{V}{8} \)
- Final pressure = \( P_2 \)

### Step 3: Use the adiabatic condition
For an adiabatic process, the relationship between pressure and volume is given by:
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]
Substituting the values of \( V_1 \) and \( V_2 \):
\[ P_1 V^\gamma = P_2 \left(\frac{V}{8}\right)^\gamma \]

### Step 4: Rearranging the equation
We can rearrange this equation to find the ratio of the final pressure to the initial pressure:
\[ P_2 = P_1 \frac{V^\gamma}{\left(\frac{V}{8}\right)^\gamma} \]
This simplifies to:
\[ P_2 = P_1 \frac{V^\gamma}{\frac{V^\gamma}{8^\gamma}} = P_1 \cdot 8^\gamma \]

### Step 5: Substitute the value of gamma
Now substituting \( \gamma = \frac{5}{3} \):
\[ P_2 = P_1 \cdot 8^{\frac{5}{3}} \]

### Step 6: Calculate \( 8^{\frac{5}{3}} \)
Calculating \( 8^{\frac{5}{3}} \):
\[ 8^{\frac{5}{3}} = (2^3)^{\frac{5}{3}} = 2^5 = 32 \]

### Step 7: Final ratio of pressures
Thus, the ratio of the final pressure to the initial pressure is:
\[ \frac{P_2}{P_1} = 32 \]

### Conclusion
The pressure of the gas after adiabatic compression will change to \( 32 \) times the initial pressure.