Question:
An ideal gas is expanded adiabatically. How many times has the gas to be expanded to reduce the r.m.s. speed of molecules 3 times ? (\(γ = 1.5\))
An ideal gas is expanded adiabatically. How many times has the gas to be expanded to reduce the r.m.s. speed of molecules 3 times ? (\(γ = 1.5\))
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v_rms is proportional to sqrt(T), and for adiabatic change T V^(gamma-1) is constant.
Updated On: Oct 1, 2026
- \(32\) times
- \(16\) times
- \(8\) times
- \(81\) times
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept
The rms speed is \(v_{rms} = \sqrt{3RT/M}\), so \(v_{rms} \propto \sqrt T\). In an adiabatic process \(TV^{\gamma - 1} = \text{constant}\).
Step 2: Temperature ratio
If the rms speed falls three times, the temperature falls nine times: \(T_2 = T_1/9\).
Step 3: Volume ratio
\[ \frac{V_2}{V_1} = \left(\frac{T_1}{T_2}\right)^{\frac{1}{\gamma - 1}} = 9^{\frac{1}{0.5}} = 9^2 = 81 \]
The gas must expand 81 times, option (D).
Final Answer:
The gas must expand 81 times, option (D). \[ \boxed{81} \]
The rms speed is \(v_{rms} = \sqrt{3RT/M}\), so \(v_{rms} \propto \sqrt T\). In an adiabatic process \(TV^{\gamma - 1} = \text{constant}\).
Step 2: Temperature ratio
If the rms speed falls three times, the temperature falls nine times: \(T_2 = T_1/9\).
Step 3: Volume ratio
\[ \frac{V_2}{V_1} = \left(\frac{T_1}{T_2}\right)^{\frac{1}{\gamma - 1}} = 9^{\frac{1}{0.5}} = 9^2 = 81 \]
The gas must expand 81 times, option (D).
Final Answer:
The gas must expand 81 times, option (D). \[ \boxed{81} \]
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