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Integrating Maxwell-Boltzmann speed distribution

by zebala
Tags: boltzmann, integral, maxwell
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zebala
#1
Dec6-10, 07:54 AM
P: 2
1. The problem statement, all variables and given/known data

Let P(v) represent the Maxwell-Boltzmann speed distribution. Basically what it comes down to is that I have to find the definite integral (0,inf) of P(v)*v^2 and get vrms from this.

2. Relevant equations

We are given the definite integral from 0 to inf for the function (x^4)*exp(-x^2), which is (3/8)*Sqrt(Pi).

3. The attempt at a solution

I first move all the constants in front of the integral and then the integral simplifies to (v^4)*exp(-mv^2/kT). However, I have no idea what to do with the constants stuck in the exponent. How would I continue from here?

Thank you beforehand.
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phyzguy
#2
Dec6-10, 08:04 AM
P: 2,179
You have to make a substitution. Set:
x=v*sqrt(m/(kT)). (Except I think it's really x=v*sqrt(m/(2kT)) and you forgot the 2. E= 1/2 mv^2).
zebala
#3
Dec6-10, 08:47 AM
P: 2
Thanks, that seems to be a working solution! Do you know what the answer should be? 3kT/m? For some reason I get Sqrt(9kT/2m).

EDIT: Forgot to replace dv with the corresponding operator for x, dx*[(m/2kT)^(-1/2)]. Got it now!


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