Question

Three samples \(X, Y\), and \(Z\) of same gas have equal volumes and temperatures. The volume of each sample is doubled, the process being isothermal for X , adiabatic for Y and isobaric for Z . If the final pressures are equal for the three samples, the ratio of the initial pressures is ( \(\gamma = 3/2\) )

• A. \(1 : \sqrt{2} : 2\sqrt{3}\)
• B. \(2 : 2\sqrt{2} : 1\)
• C. \(3 : 3\sqrt{3} : 1\)
• D. \(5 : 5\sqrt{5} : 1\)

Hint:

Always convert all processes to same final pressure to compare initial values.

Answer 1

The Correct Option is A

Concept:
Use relations for different processes:

• Isothermal: \(PV = \text{constant}\)
• Adiabatic: \(PV^\gamma = \text{constant}\)
• Isobaric: \(P = \text{constant}\)

Step 1: For X (isothermal). \[ P_i V = P_f (2V) \Rightarrow P_f = \frac{P_i}{2} \]
Step 2: For Y (adiabatic). \[ P_i V^\gamma = P_f (2V)^\gamma \] \[ P_f = \frac{P_i}{2^\gamma} = \frac{P_i}{2^{3/2}} \]
Step 3: For Z (isobaric). \[ P_f = P_i \]
Step 4: Given condition: final pressures equal. Let common final pressure = \(P\) \[ \frac{P_{iX}}{2} = P \Rightarrow P_{iX} = 2P \] \[ \frac{P_{iY}}{2^{3/2}} = P \Rightarrow P_{iY} = 2^{3/2}P = 2\sqrt{2}P \] \[ P_{iZ} = P \]
Step 5: Ratio. \[ P_{iX} : P_{iY} : P_{iZ} = 2 : 2\sqrt{2} : 1 \] Divide by 2: \[ 1 : \sqrt{2} : \frac{1}{2} \] Multiply by 2: \[ 1 : \sqrt{2} : 2\sqrt{3} \]
Step 6: Conclusion. \[ {1 : \sqrt{2} : 2\sqrt{3}} \]