Initial pressure and volume of a gas are P and V respectively. First it is expanded isothermally to volume 4 V and then compressed adiabatically to volume V. The final pressure of gas will be (given `gamma=(3)/(2)`)

Options

A. `1P`
B. `2P`
C. `4P`
D. `8P`

Explanation

To solve the problem, we need to follow the steps of the gas processes: isothermal expansion and adiabatic compression.

### Step 1: Isothermal Expansion
Initially, the gas has pressure \( P \) and volume \( V \). The gas is expanded isothermally to a volume of \( 4V \). According to the ideal gas law for isothermal processes, we have:

\[ P_1 V_1 = P_2 V_2 \]

Where:
- \( P_1 = P \) (initial pressure)
- \( V_1 = V \) (initial volume)
- \( P_2 \) is the pressure after isothermal expansion
- \( V_2 = 4V \) (final volume after expansion)

Substituting the values, we get:

\[ P \cdot V = P_2 \cdot 4V \]

Dividing both sides by \( V \):

\[ P = 4P_2 \]

Thus, we can express \( P_2 \):

\[ P_2 = \frac{P}{4} \]

### Step 2: Adiabatic Compression
Next, the gas is compressed adiabatically to a volume of \( V \). For an adiabatic process, we use the relation:

\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]

Where:
- \( P_1 = P_2 = \frac{P}{4} \) (pressure after isothermal expansion)
- \( V_1 = 4V \) (volume after isothermal expansion)
- \( P_3 \) is the final pressure after adiabatic compression
- \( V_2 = V \) (final volume after compression)
- \( \gamma = \frac{3}{2} \)

Substituting the values into the adiabatic equation:

\[ \left(\frac{P}{4}\right) (4V)^{\frac{3}{2}} = P_3 V^{\frac{3}{2}} \]

Calculating \( (4V)^{\frac{3}{2}} \):

\[ (4V)^{\frac{3}{2}} = 4^{\frac{3}{2}} V^{\frac{3}{2}} = 8 V^{\frac{3}{2}} \]

Now substituting back into the equation:

\[ \left(\frac{P}{4}\right) \cdot 8 V^{\frac{3}{2}} = P_3 V^{\frac{3}{2}} \]

This simplifies to:

\[ 2P V^{\frac{3}{2}} = P_3 V^{\frac{3}{2}} \]

Dividing both sides by \( V^{\frac{3}{2}} \):

\[ P_3 = 2P \]

### Final Result
The final pressure \( P_3 \) of the gas after the adiabatic compression is:

\[ P_3 = 2P \]