Integration of Maxwell speed distribution function

[theghost28]

Homework Statement

Show the steps needed to obtain the equation for average molecular speed, c_avg=√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.

Homework Equations

In the problem statement

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*e^x2 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?

 

[Samy_A]

Homework Statement

Show the steps needed to obtain the equation for average molecular speed, c_avg=√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.

Homework Equations

In the problem statement

The Attempt at a Solution

View attachment 98052

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*e^x2 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?

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?

 

[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.

 

[Ray Vickson]

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.

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?