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: Write both pressures as fractions of P:
Isothermal doubling of volume halves the pressure: $P_i = P\cdot2^{-1}$.
Adiabatic doubling reduces pressure by the factor $2^{-\gamma}$: $P_a = P\cdot2^{-\gamma}$.

Step 2: Divide:
$\frac{P_i}{P_a} = 2^{-1}\cdot2^{\gamma} = 2^{\gamma-1}$.

Step 3: Physical check:
For a monatomic gas, $\gamma = \frac53$, and the ratio is $2^{2/3} \approx 1.59$. The isothermal curve lies above the adiabatic curve, as expected, because the adiabatic expansion loses temperature.

Final Answer:
Option (C). \[ \boxed{2^{\gamma-1} \text{ (C)}} \]