# Integration of Maxwell speed distribution function

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1. Mar 27, 2016

### theghost28

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. Mar 27, 2016

### Samy_A

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?

3. Mar 27, 2016

### theghost28

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.

4. Mar 27, 2016

### Ray Vickson

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?