hw05 - hw05 ChE 132C due Feb 16 2011 1 The Boltzmann...

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hw05 ChE 132C due Feb. 16 , 2011 1) The Boltzmann distribution gives the probability density for the velocity of a molecule as 2 ( ) exp[ /(2 )] EQ B f vC m v k T  where m is the mass, k B is Boltzmann’s constant, and T is temperature. [answers given for parts (a), (b), and (c) for review] (a) What value of C EQ that normalizes the distribution ? [this is a bit difficult if you try to do the integral – use the form of a Gaussian to help you!] (b) What is the standard deviation in terms of m, k B , and T ? (c) What is the average absolute velocity, E[| v |] = <| v |> = ? (d) The velocities in different orthogonal directions are independent, so the joint distribution for velocity in three dimensional space is 32 2 2 ( , , ) exp[ ( ) /(2 )] xyz E Q x y z B fvvv C mv v v kT which is the product of separate distributions for v x , v y , and v z . Derive the Maxwell-Boltzmann distribution, i.e. the marginal distribution f S ( s ) for the speed s = (v x 2 +v y 2 +v z 2 ) 1/2 . You may find it helpful to change to spherical coordinates in
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hw05 - hw05 ChE 132C due Feb 16 2011 1 The Boltzmann...

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