Question

Two gases \(A\) and \(B\) are initially at same pressure, volume and temperature. If \(A\) is compressed isothermally and \(B\) adiabatically to half of the initial volume, then the final pressure of \(A\)

• A. greater than the final pressure of \(B\)
• B. equal to the final pressure of \(B\)
• C. less than the final pressure of \(B\)
• D. twice the final pressure of \(B\)

Hint:

Adiabatic compression increases pressure more rapidly than isothermal compression because temperature also rises during compression.

Answer 1

The Correct Option is C

Step 1: Compare the two processes.
Both gases start the same and are squeezed to half the volume. Gas $A$ is squeezed slowly so its temperature stays fixed (isothermal). Gas $B$ is squeezed with no heat escaping (adiabatic), so it heats up.

Step 2: Pressure for the isothermal gas A.
For a fixed temperature, $PV$ stays constant. \[ P V = P_A' \left(\frac{V}{2}\right) \] So \[ P_A' = 2P \]
Step 3: Pressure for the adiabatic gas B.
Here $PV^\gamma$ stays constant, where $\gamma > 1$. \[ P V^\gamma = P_B' \left(\frac{V}{2}\right)^\gamma \] So \[ P_B' = P\,2^\gamma \]
Step 4: Compare the two factors.
Since $\gamma$ is bigger than $1$, the factor $2^\gamma$ is bigger than $2$.
Step 5: Decide which pressure is larger.
Because $2^\gamma > 2$, gas $B$ ends up at higher pressure than gas $A$. \[ P_B' > P_A' \]
Step 6: State the conclusion.
So the final pressure of $A$ is less than that of $B$. \[ \boxed{P_A' < P_B'} \]