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:

Express each initial pressure in terms of the common final pressure using the isothermal, adiabatic and isobaric relations.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Let the initial volume be \(V\) and the final volume \(2V\). Let the common final pressure be \(P_f\).

Step 2: Isothermal X
\(P_XV=P_f(2V)\), so \(P_X=2P_f\).

Step 3: Adiabatic Y
\(P_YV^{\gamma}=P_f(2V)^{\gamma}\), so \(P_Y=2^{\gamma}P_f=2^{3/2}P_f=2\sqrt2\,P_f\).

Step 4: Isobaric Z
The pressure is constant, so \(P_Z=P_f\).

Step 5: Ratio
\(P_X:P_Y:P_Z=2:2\sqrt2:1\), the second option.

Final Answer:
The ratio of initial pressures is \(2:2\sqrt2:1\). \[ \boxed{\text{(B) }2:2\sqrt{2}:1} \]