If the rms speeds of helium and oxygen are equal, then the ratio of the temperatures of helium and oxygen is

If the rms speeds of helium and oxygen are equal, then the ratio of the temperatures of helium and oxygen is
  1. $1: 8$
  2. $2: 1$
  3. $1: 4$
  4. $4: 1$

Solution

rms speed is given by $\begin{aligned} & \mathrm{V}_{\mathrm{rms}}=\sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}}} \\ & \mathrm{M}_{\mathrm{He}}=4 \mathrm{M}_{\mathrm{O}_2}=32 \\ & \sqrt{\frac{3 \mathrm{RT}_{\mathrm{He}}}{\mathrm{M}_{\mathrm{He}}}}=\sqrt{\frac{3 \mathrm{RT}_{\mathrm{O}_2}}{\mathrm{M}_{\mathrm{O}_2}}} \\ & \frac{\mathrm{T}_{\mathrm{He}}}{\mathrm{T}_{\mathrm{O}_2}}=\frac{\mathrm{M}_{\mathrm{He}}}{\mathrm{M}_{\mathrm{O}_2}}=\frac{4}{3^2}=\frac{1}{8} \end{aligned}$

Asked in: AP EAMCET 2023 (17 May Shift 1)

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