Question

An ideal gas with pressure P, volume V and temperature T is expanded isothermally to volume 2V and a final pressure \(P_i\). The same gas is expanded adiabatically to a volume 2V then the final pressure is \(P_a\). In terms of \(γ\), the ratio of the two specific heats for the gas, the ratio \(P_i/P_a\) is

• A. \(2^γ\)
• B. \(2^{1-γ}\)
• C. \(2^{γ-1}\)
• D. \(2γ\)

Hint:

Isothermal: PV constant. Adiabatic: P V^gamma constant. Compare the two final pressures at volume 2V.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The gas starts at \((P, V)\) and expands to \(2V\) by two different processes. In the isothermal case, temperature stays fixed. In the adiabatic case, no heat flows in or out.

Step 2: Key Formula or Approach:
1. Isothermal: \(PV = P_i(2V)\).
2. Adiabatic: \(PV^\gamma = P_a(2V)^\gamma\).

Step 3: Detailed Explanation:
Isothermal final pressure:
\[ P_i = \frac P2 \]
Adiabatic final pressure:
\[ P_a = P\left(\frac{V}{2V}\right)^\gamma = \frac{P}{2^\gamma} \]
Ratio:
\[ \frac{P_i}{P_a} = \frac{P/2}{P/2^\gamma} = 2^{\gamma - 1} \]
Since \(\gamma > 1\), this ratio is above 1, which means the isothermal pressure is higher. That is right, because the adiabatic expansion also cools the gas. Option (B) is the inverse of the correct ratio, and option (A) forgets the factor \(\frac12\) in \(P_i\).

Final Answer:
\(P_i/P_a = 2^{\gamma-1}\), option (C). \[ \boxed{2^{\gamma-1} \text{ (C)}} \]