Question

An ideal gas (\(γ = 1.5\)) is expanded adiabatically. To reduce the root mean square velocity of molecules two times, the gas should be expanded

• A. \(20\) times
• B. \(16\) times
• C. \(12\) times
• D. \(8\) times

Hint:

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

Answer 1

The Correct Option is B

Step 1: Use pressure and volume
For an adiabatic change, $PV^\gamma = $ constant, and $v_{rms}^2 \propto T \propto PV$.

Step 2: Chain
$PV \propto V^{1 - \gamma} = V^{-0.5}$. So $v_{rms}^2 \propto V^{-0.5}$ and $v_{rms} \propto V^{-0.25}$.

Step 3: Halve the speed
$V^{-0.25}$ must become $\frac{1}{2}$, so $V^{0.25} = 2$ and $V = 2^4 = 16$ times the original.

Step 4: Result
The gas must be expanded to 16 times its volume.

Final Answer:
The gas is expanded 16 times. This is option (B). \[ \boxed{\text{(B) }16\ \text{times}} \]