Two samples A and B of a gas are initially at the same temperature and pressure are compressed from volume `V_1` to V/2 (A isothermally and B adiabatically). The final pressure is

Options

A. A is greater than that of B
B. A is equal that of B
C. A is less than that of B
D. A is twice that of B

Answer

To solve the problem of finding the final pressures of two gas samples A and B after compression, we will use the principles of isothermal and adiabatic processes.

### Step-by-Step Solution:

**Step 1: Analyze Sample A (Isothermal Compression)**
For gas A, which is compressed isothermally from volume \( V_1 = V \) to \( V_2 = \frac{V}{2} \), we can use the ideal gas law for isothermal processes. The relationship is given by:

\[ P_1 V_1 = P_2 V_2 \]

Where:
- \( P_1 \) is the initial pressure (P),
- \( V_1 \) is the initial volume (V),
- \( P_2 \) is the final pressure,
- \( V_2 \) is the final volume \(\left(\frac{V}{2}\right)\).

Substituting the known values into the equation:

\[ P \cdot V = P_2 \cdot \frac{V}{2} \]

**Step 2: Solve for \( P_2 \) (Final Pressure for A)**
Rearranging the equation to solve for \( P_2 \):

\[ P_2 = \frac{P \cdot V}{\frac{V}{2}} = \frac{P \cdot V \cdot 2}{V} = 2P \]

Thus, the final pressure for gas A is:

\[ P_2 = 2P \]

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**Step 3: Analyze Sample B (Adiabatic Compression)**
For gas B, which is compressed adiabatically from volume \( V_1 = V \) to \( V_2 = \frac{V}{2} \), we use the adiabatic relation:

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

Where \( \gamma \) (gamma) is the heat capacity ratio (specific heat at constant pressure to specific heat at constant volume).

Substituting the values:

\[ P \cdot V^\gamma = P_2 \cdot \left(\frac{V}{2}\right)^\gamma \]

**Step 4: Solve for \( P_2 \) (Final Pressure for B)**
Rearranging the equation to find \( P_2 \):

\[ P_2 = P \cdot \frac{V^\gamma}{\left(\frac{V}{2}\right)^\gamma} = P \cdot \frac{V^\gamma}{\frac{V^\gamma}{2^\gamma}} = P \cdot 2^\gamma \]

Thus, the final pressure for gas B is:

\[ P_2 = P \cdot 2^\gamma \]

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**Step 5: Compare Final Pressures**
Now we compare the final pressures of A and B:

- Final pressure of A: \( P_A = 2P \)
- Final pressure of B: \( P_B = P \cdot 2^\gamma \)

Since \( \gamma > 1 \), it follows that \( 2^\gamma > 2 \). Therefore:

\[ P_B > P_A \]

This indicates that the final pressure of gas B (adiabatically compressed) is greater than that of gas A (isothermally compressed).

### Conclusion:
The final pressure of gas A is less than that of gas B.