At what temperature is the r.m.s velocity of a hydrogen molecule equal to that of an oxygen molecule at 47^ \circ C?

Heat and Thermodynamics PYQ — JEE Main 2002 · Shift 0 · Q150 with Solution

View written solutionFree

Correct answer: D

Use the formula for rms speed

For a gas molecule, $v_{\text{rms}}=\sqrt{\frac{3RT}{M}}$ where $T$ is absolute temperature and $M$ is molar mass.

Condition for equal rms speeds

Given:

Hydrogen molecule: $\mathrm{H_2}$ , so $M_H=2$
Oxygen molecule: $\mathrm{O_2}$ , so $M_O=32$
Oxygen temperature: $47^\circ C = 320\,K$

Equal rms speeds means $\sqrt{\frac{3RT_H}{M_H}}=\sqrt{\frac{3RT_O}{M_O}}$ Squaring both sides, $\frac{T_H}{M_H}=\frac{T_O}{M_O}$ So, $T_H=T_O\cdot \frac{M_H}{M_O}$

Substitute values

$T_H=320\times \frac{2}{32}=320\times \frac{1}{16}=20\,K$

Match with options

The required temperature is $\boxed{20\,K}$ So the correct option is D.

Comparison with stored answer

Stored correct answer: D

Our derived answer: D

They match.

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