At which temperature does the r.m.s. velocity of a hydrogen molec… — Solution | JEE Physics

MCQEasyJEE 2024Molecular Speeds (rms, Average, Most Probable)

At which temperature does the r.m.s. velocity of a hydrogen molecule equal that of an oxygen molecule at $47^\circ \text{C}$ ?

A
$80 \, \text{K}$
B
$-73 \, \text{K}$
C
$4 \, \text{K}$
D
$20 \, \text{K}$

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: The r.m.s. velocity of hydrogen equals that of oxygen at $47^\circ \text{C}$ .

Find: The temperature of hydrogen.

Using the formula for root mean square velocity,

v_{\text{rms}} = \sqrt{\frac{3RT}{M}}

For equal r.m.s. velocities of hydrogen and oxygen,

\sqrt{\frac{3RT_{\mathrm{H}_2}}{M_{\mathrm{H}_2}}} = \sqrt{\frac{3RT_{\mathrm{O}_2}}{M_{\mathrm{O}_2}}}

Detailed Working

Squaring both sides,

\frac{3RT_{\mathrm{H}_2}}{M_{\mathrm{H}_2}} = \frac{3RT_{\mathrm{O}_2}}{M_{\mathrm{O}_2}}

Cancelling $3R$ ,

T_{\mathrm{H}_2} = T_{\mathrm{O}_2} \times \frac{M_{\mathrm{H}_2}}{M_{\mathrm{O}_2}}

Now convert the oxygen temperature to Kelvin:

T_{\mathrm{O}_2} = 47 + 273 = 320 \, \text{K}

Using $M_{\mathrm{H}_2} = 2$ and $M_{\mathrm{O}_2} = 32$ ,

T_{\mathrm{H}_2} = 320 \times \frac{2}{32} = 20 \, \text{K}

Therefore, the temperature is $20 \, \text{K}$ and the correct option is D. The solution states option C, but its working clearly gives $20 \, \text{K}$ , which matches option D.

Common mistakes

Using the Celsius temperature directly in the r.m.s. formula is incorrect because gas-kinetic formulas require absolute temperature. Convert $47^\circ \text{C}$ to $320 \, \text{K}$ first.
Taking the mass ratio inversely is a common error. Since $v_{\text{rms}} \propto \sqrt{T/M}$ , equal velocities give $T \propto M$ here, so use $\frac{M_{\mathrm{H}_2}}{M_{\mathrm{O}_2}}$ , not the reverse.
Confusing molecular mass with atomic mass leads to wrong substitution. Hydrogen gas is $\mathrm{H}_2$ with mass $2$ , and oxygen gas is $\mathrm{O}_2$ with mass $32$ .

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Answer

Step-by-step solution

Common mistakes

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