Question

The initial pressure and volume of a gas is \( P \) and \( V \) respectively. First, by isothermal process, gas is expanded to volume \( 9V \), and then by adiabatic process its volume is compressed to \( V \). Then its final pressure is (Ratio of specific heat at constant pressure to constant volume \( \gamma = \frac{3}{2} \))

• A. 6P
• B. 27P
• C. 3P
• D. 9P

Hint:

In an isothermal process, the pressure and volume are inversely proportional, and in an adiabatic process, the relationship between pressure and volume is given by \( P V^\gamma = \text{constant} \).

Answer 1

The Correct Option is C

Step 1: Understand the isothermal process.
In the isothermal process, the temperature remains constant. The equation for an isothermal expansion is: \[ P_1 V_1 = P_2 V_2 \] Here, \( P_1 = P \), \( V_1 = V \), and \( V_2 = 9V \). After expansion: \[ P_2 = \frac{P V}{9V} = \frac{P}{9} \] Step 2: Apply the adiabatic process.
In an adiabatic process, the relationship between pressure and volume is given by: \[ P_2 V_2^\gamma = P_3 V_3^\gamma \] Here \( P_2 = \frac{P}{9} \), \( V_2 = 9V \), and \( V_3 = V \). Using \( \gamma = \frac{3}{2} \): \[ \frac{P}{9} (9V)^{3/2} = P_3 V^{3/2} \] Simplifying: \[ \frac{P}{9} \times 27V^{3/2} = P_3 V^{3/2} \] \[ P_3 = 3P \] Step 3: Conclusion.
Thus, the final pressure is \( 3P \), which corresponds to option (C).