Question

The temperature at which r.m.s. velocity of hydrogen molecules is 4.5 times that of an oxygen molecule at \(47^{\circ}\) C is (molecular weight of hydrogen and oxygen are 2 and 32 respectively)

• A. \(47^{\circ}\) C
• B. \(132^{\circ}\) C
• C. \(320^{\circ}\) C
• D. \(405^{\circ}\) C

Hint:

In an adiabatic change T V^(gamma-1) is constant and the r.m.s. speed goes as the square root of T.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
\(v_{rms}\propto\sqrt T\). To reduce \(v_{rms}\) by a factor of 3, the temperature must reduce by \(3^2 = 9\).

Step 2: Adiabatic relation:
\(TV^{\gamma - 1} = \text{constant}\). With \(\gamma = 1.5\), \(\gamma - 1 = 0.5\):
\[ \frac{T_1}{T_2} = \left(\frac{V_2}{V_1}\right)^{0.5} \]

Step 3: Solve:
\(9 = \left(\frac{V_2}{V_1}\right)^{1/2}\), so \(\frac{V_2}{V_1} = 81\). The gas must expand 81 times.

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