Question

An ideal gas expands adiabatically (\(γ = 1.5\)). To reduce the r.m.s. velocity of the molecules 3 times the gas has to be expanded

• A. 9 times
• B. 81 times
• C. 3 times
• D. 27 times

Hint:

The rms speed varies as the square root of temperature, and an adiabatic process obeys T V^(gamma-1) = constant.

Answer 1

The Correct Option is B

Step 1: Use pressure instead.
Another form is $TV^{\gamma-1} = $ constant, with $\gamma - 1 = 1/2$, so $V \propto T^{-1/(\gamma - 1)} = T^{-2}$.

Step 2: Put the temperature change.
The temperature falls to 1/9, so the volume grows by $(9)^2 = 81$.

Step 3: Sanity check.
A larger drop in temperature requires a larger expansion. Because $\gamma$ is small, the temperature falls slowly with volume, so a large ratio like 81 is reasonable.

Final Answer:
Option (B). \[ \boxed{81\text{ times}} \]