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Homework Help: Integration of Maxwell speed distribution function

  1. Mar 27, 2016 #1
    1. The problem statement, all variables and given/known data
    Show the steps needed to obtain the equation for average molecular speed, cavg=√8RT/πM from the integral (from negative infinity to infinity) ∫v*f(v)dv where f(v) is the Maxwell distribution of speeds function f(v)=4π*(M/2πRT)1.5v2e-Mv2/2RT

    M is the molar mass of the particle in kg/mol, R is the gas constant (8.314), v is particle velocity, e is the natural number and T is temperature in Kelvin.

    2. Relevant equations
    In the problem statement

    3. The attempt at a solution

    I changed v from the problem to x for simplicity since I'm used to using v for integration by parts. I'm fairly sure this solution is correct, as I've googled what the integral of x3*ex2 is and others have obtained this as well. The problem is when I evaluate that expression from negative infinity to infinity, I get zero. I've gone over my math multiple times, is there something I'm missing?
    Last edited: Mar 27, 2016
  2. jcsd
  3. Mar 27, 2016 #2


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    When you integrate the odd function vf(v) from -∞ to +∞, obviously you get 0.
    Are you sure about the integration limits?

    Maybe consider the following question: under what circumstances is speed negative?
  4. Mar 27, 2016 #3
    I'm sure about the integration limits given in the assignment although I agree they don't really make sense. I guess I'll use 0 to infinity and make a note of it.
  5. Mar 27, 2016 #4

    Ray Vickson

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    Your calculation looks at average velocity, which is zero in this case. Average speed is different.

    The first thing you need to figure out is whether the given ##f(v)## is a probability density of velocity or of speed. Can you see how to do that?
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