Like

Report

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

$$ \overline{v} = \sqrt{\frac{8RT}{\pi M}} $$

$\sqrt{\frac{8 R T}{\pi M}}$

Integration Techniques

You must be signed in to discuss.

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

Boston College

for this problem. I'm r unti are cast in the numbers and it's on a reliable is way. What we need to do is to compute this improper in general. Here we use you substitution first that you is ICO too. On over. I'm comes with square over to Artie. So do you Is Nico too? I'm over r you TV Our TV DVD now. Just improper anti girl and sickle too. And you go from there on to infinity on DH here we square is equal to you. Oh, you tams. Who are you? The u taps to art he over I'm this is we're square on the way with the way Izzy Coto Artie hams. Archy. Over um r e v o r on you and hey to just powers that they need to negative you. You This is the road to casting. Number is who are you? You are square o r on school on the limit. It goes to infinity integral from zero to a you terms you connect you, you, you Andi, we compute this definitely into girl first. And since he come to here we used my third of the integration by parts. This's you goto integral from zero to make you a yes you. I think if you to make you and we use interview zero from pain. So here we use integration My purse. So this's equal to you Hams. You connect your view. It's negative from there are a then minus the girl from zero to a negative toe Makes you you You Andi, this is Echo two. Make it a into negative, eh? Class Negative, you connective Hugh. From zero to a this since they go to negative, eh? To connect you a minus. It makes you a thank you Want on what? They goes to infinity? Each one eighty vagos toa hero on dhe ate hams It connect away also goes to zero. Here we can use a peon has rules. This is equal to the limit. A cause to infinity a hour. Ito, eh? He was a PT Astra. Or so This is the limit. One over here too. A ghost too. And it is it. Is this so here? The answer is one now way. Bar is vehicle too for over motive times. It's the times. Yeah. I'm over to Artie. Time's over. Who are you on terms tool are we square over. Um square simplified. This result we have This's Iko too tough it he over high.