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: Understanding the Concept:
The rms speed of gas molecules is \(v_{rms} = \sqrt{3RT/M}\), so \(v_{rms}\propto\sqrt{T}\). For an adiabatic change, \(TV^{\gamma - 1} = \text{constant}\).

Step 2: Find the temperature ratio.
If \(v_{rms}\) becomes \(\dfrac{1}{3}\) of its value, \(T\) becomes \(\dfrac{1}{9}\) of its value. So \(\dfrac{T_1}{T_2} = 9\).

Step 3: Use the adiabatic relation.
\[ T_1V_1^{\gamma-1} = T_2V_2^{\gamma-1} \Rightarrow \left(\frac{V_2}{V_1}\right)^{\gamma - 1} = \frac{T_1}{T_2} = 9 \]
With \(\gamma - 1 = 0.5\):
\[ \frac{V_2}{V_1} = 9^{1/0.5} = 9^2 = 81 \]

Step 4: Check the options.
A factor of 9 is the temperature ratio, not the volume ratio. 3 times and 27 times do not follow from the power 2.

Final Answer:
The gas must expand 81 times, option (B). \[ \boxed{81\text{ times}} \]