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If the rms speeds of helium and oxygen are equal, then the ratio of the temperatures of helium and oxygen is
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Verified Answer
The correct answer is:
$1: 8$
rms speed is given by
$\begin{aligned}
& \mathrm{V}_{\mathrm{rms}}=\sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}}} \\
& \mathrm{M}_{\mathrm{He}}=4 \mathrm{M}_{\mathrm{O}_2}=32 \\
& \sqrt{\frac{3 \mathrm{RT}_{\mathrm{He}}}{\mathrm{M}_{\mathrm{He}}}}=\sqrt{\frac{3 \mathrm{RT}_{\mathrm{O}_2}}{\mathrm{M}_{\mathrm{O}_2}}} \\
& \frac{\mathrm{T}_{\mathrm{He}}}{\mathrm{T}_{\mathrm{O}_2}}=\frac{\mathrm{M}_{\mathrm{He}}}{\mathrm{M}_{\mathrm{O}_2}}=\frac{4}{3^2}=\frac{1}{8}
\end{aligned}$
$\begin{aligned}
& \mathrm{V}_{\mathrm{rms}}=\sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}}} \\
& \mathrm{M}_{\mathrm{He}}=4 \mathrm{M}_{\mathrm{O}_2}=32 \\
& \sqrt{\frac{3 \mathrm{RT}_{\mathrm{He}}}{\mathrm{M}_{\mathrm{He}}}}=\sqrt{\frac{3 \mathrm{RT}_{\mathrm{O}_2}}{\mathrm{M}_{\mathrm{O}_2}}} \\
& \frac{\mathrm{T}_{\mathrm{He}}}{\mathrm{T}_{\mathrm{O}_2}}=\frac{\mathrm{M}_{\mathrm{He}}}{\mathrm{M}_{\mathrm{O}_2}}=\frac{4}{3^2}=\frac{1}{8}
\end{aligned}$
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