Question

An ideal gas having pressure \(P\), volume \(V\) and temperature \(T\) is expanded isothermally, to a volume \(3V\) and final pressure \(P_I\). The same gas is expanded adiabatically to a volume \(3V\), the final pressure being \(P_A\). The ratio \(\frac{P_A}{P_I}\) is \((\frac{C_P}{C_V} = γ)\)

• A. \(3γ\)
• B. \(3^γ\)
• C. \(3^{(1-γ)}\)
• D. \(3^{(γ-1)}\)

Hint:

Isothermal: PV constant. Adiabatic: PV to the gamma constant.

Answer 1

The Correct Option is C

Step 1: Approach
Use logarithms to compare the two final pressures.

Step 2: Pressures
$P_I=P\cdot3^{-1}$ and $P_A=P\cdot3^{-\gamma}$.

Step 3: Divide
$\dfrac{P_A}{P_I}=3^{-\gamma+1}=3^{1-\gamma}$.

Step 4: Answer
Option (C).

Final Answer:
The adiabatic pressure is P over 3 to the gamma and the isothermal one is P over 3, giving 3^(1 - gamma), option (C). \[ \boxed{3^{(1-\gamma)}} \]