The temperature at which r.m.s. velocity of hydrogen molecules is 4.5 times that of an oxygen molecule at…

The temperature at which r.m.s. velocity of hydrogen molecules is 4.5 times that of an oxygen molecule at $47^{\circ} \mathrm{C}$ is (Molecular weight of hydrogen and oxygen molecules are 2 and 32 respectively)
  1. $47^{\circ} \mathrm{C}$
  2. $132^{\circ} \mathrm{C}$
  3. $320^{\circ} \mathrm{C}$
  4. $405^{\circ} \mathrm{C}$

Solution

We know, $\begin{aligned} & \mathrm{V}_{\mathrm{rms}}=\sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}}} \\ & \text {Given } \mathrm{V}_{\mathrm{H}_{\mathrm{mms}}}=4.5 \mathrm{~V}_{\mathrm{O}_{\mathrm{rms}}} \\ & \sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}}}=4.5 \sqrt{\frac{3 \mathrm{R}(320)}{32}} \\ & \sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}}}=4.5 \sqrt{30 \mathrm{R}} \\ & 3 \mathrm{RT}=20.25 \times 30 \mathrm{R} \times 2 \\ & \therefore \quad \mathrm{~T}=20.25 \times 2 \times 10=405 \mathrm{~K}=132^{\circ} \mathrm{C} \end{aligned}$

Asked in: MHT CET 2024 (02 May Shift 2)

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