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: Understand the concept
The rms speed is \(v_{rms} = \sqrt{\dfrac{3RT}{M}}\), so \(v_{rms} \propto \sqrt{T}\). Halving the speed means \(T\) falls to one fourth.

Step 2: Temperature ratio
\(\dfrac{v_2}{v_1} = \dfrac{1}{2}\) gives \(\dfrac{T_2}{T_1} = \dfrac{1}{4}\).

Step 3: Adiabatic relation
For an adiabatic process \(TV^{\gamma - 1} = \text{constant}\), so
\[ \frac{V_2}{V_1} = \left(\frac{T_1}{T_2}\right)^{\frac{1}{\gamma - 1}} = 4^{\frac{1}{0.5}} = 4^2 \]

Step 4: Result
\(V_2 = 16V_1\), so the gas must be expanded 16 times, option (B). Options (A), (C) and (D) are not powers of 4.

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