Problem 62 Hard Difficulty

The average speed of molecules in an ideal gas is
$$ \overline{v} = \frac{4}{\sqrt{\pi}} \left (\frac{M}{2RT} \right)^{\frac{3}{2}} \int_0^\infty v^3 e^{\frac{-Mv^2}{(2RT)}}\ dv $$
where $ M $ is the molecular weight of the gas, $ R $ is the gas constant, $ T $ is the gas temperature, and $ v $ is the molecular speed. Show that
$$ \overline{v} = \sqrt{\frac{8RT}{\pi M}} $$

Answer

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Video Transcript

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