Consider two containers A and B containing identical gases at the same pressure, volume and temperature. The gas in container A is compressed to half of its original volume isothermally while the gas in container B is compressed to half of its original value adiabatically. The ratio of final pressure of gas in B to that of gas in A is

Options

A. `2^(gamma -1)`
B. `(1)/(2)^(gamma-1)`
C. `(1)/(1-gamma)^(2)`
D. `(1)/(gamma-1)^(2)`

Explanation

To solve the problem, we need to find the final pressures of the gases in containers A and B after they undergo isothermal and adiabatic compression, respectively, and then calculate the ratio of these final pressures.

### Step 1: Determine the final pressure for gas in container A (isothermal compression)

1. **Initial conditions**: Let the initial pressure, volume, and temperature of the gas in container A be \( P_1 \), \( V_1 \), and \( T \) respectively.
2. **Isothermal process**: For an isothermal process, we can use the ideal gas law, which states that \( P_1 V_1 = P_2 V_2 \).
3. **Final volume**: The gas in container A is compressed to half its original volume, so \( V_2 = \frac{V_1}{2} \).
4. **Final pressure calculation**:
\[ P_2^A = \frac{P_1 V_1}{V_2} = \frac{P_1 V_1}{\frac{V_1}{2}} = 2P_1 \]
Thus, the final pressure for gas A is \( P_2^A = 2P_1 \).

### Step 2: Determine the final pressure for gas in container B (adiabatic compression)

1. **Adiabatic process**: For an adiabatic process, the relationship between pressure and volume is given by:
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]
where \( \gamma \) is the heat capacity ratio (specific heat at constant pressure to specific heat at constant volume).
2. **Final volume**: The gas in container B is also compressed to half its original volume, so \( V_2 = \frac{V_1}{2} \).
3. **Final pressure calculation**:
\[ P_2^B = P_1 \left( \frac{V_1}{V_2} \right)^\gamma = P_1 \left( \frac{V_1}{\frac{V_1}{2}} \right)^\gamma = P_1 (2)^\gamma \]
Thus, the final pressure for gas B is \( P_2^B = P_1 \cdot 2^\gamma \).

### Step 3: Calculate the ratio of final pressures

1. **Ratio of final pressures**:
\[ \text{Ratio} = \frac{P_2^B}{P_2^A} = \frac{P_1 \cdot 2^\gamma}{2P_1} \]
Here, \( P_1 \) cancels out:
\[ \text{Ratio} = \frac{2^\gamma}{2} = 2^{\gamma - 1} \]

### Conclusion

The ratio of the final pressure of gas in container B to that of gas in container A is:
\[ \frac{P_2^B}{P_2^A} = 2^{\gamma - 1} \]