Kinetic Theory Speed distribution (general form)

[Robsta]

Homework Statement

A molecule has a velocity v and speed v. I've worked out (and understand) that the number of molecules in a gas with speeds between v and v+dv and moving at angles between Ө and Ө+dӨ to any chosen axis is: (1/2)nf(v)dvsin(Ө)dӨ The internet verifies this. f(v) is the speed distribution (often Maxwellian but this question is general).

Here's the question. If u is a cartesian component of v (eg v_x), show by integration that:
<u> = 0,
<u^2>=(1/3)<v^2>
<mod(u)> = (1/2)<mod(v)>

Homework Equations

The question hints that <v> is the integral from 0 to infinity of vf(v)dv divided by the same integral of f(v)dv and think I understand why. But I'm not sure how to put it in terms of u?

The Attempt at a Solution

Intuitively <u> = 0 because there are the same number of molecules going "left" as there are going "right". But I can't show it rigorously using the equation above. I've found a couple of cases where f(v) is the maxwellian distribution but this question specifically asks you to solve it generally. I keep getting myself in a muddle with the infinite limits as well.

The question is attached as a picture. I've taken photos as well of some jottings. Thanks in advance to anybody even just for some hints.

 

[TSny]

Hello, and welcome to PF!

Suppose that you have some property, g, of a molecule that depends on the speed v and angle θ that the velocity of the molecule makes to the z axis. So, g = g(v,θ). How would you find the average of g(v,θ) over all the molecules of the gas? That is, how would you find <g(v,θ)>? What integral (or integrals) would you need to perform?

 

[Robsta]

I suppose I'd have to integrate g with respect to every possible velocity dv and angle dθ?
Attached to this is a solution that I've tried to each part of the question, but I've fudged part i and parts ii and iii are wrong and I how now idea why :(

 

[Robsta]

I've spoken to a lot of people about this and they've all tried to help but none of them really seem to understand the question. Specifically what exactly is mod(v) when v is already the modulus of a vector??

 

[TSny]

You have the right idea. When doing <v_z>, how would you express v_z as a function of v and θ? The θ part of v_z will contribute to the integrand for the integration over θ.

 

[Robsta]

Yep, v_z = v*sin (theta) and the sin goes into the sin integral and the v goes into the v integral. For part a you integrate sin squared and for part b you integrate sin cubed. I've found this so far to give the wrong answer though. I think it's all in the file I uploaded above, can you see it?

 

[TSny]

Yep, v_z = v*sin (theta) ...

Oops, check that.

 

[Robsta]

Incredible. You're right, v_z = v cos(theta). I've just solved all three parts. I'm so grateful to you for doing this, I'm going back to uni after some time off and this is part of my first hand in. I really want to make a good impression and because of you I can. I'll make sure I spend some time on this site and pay it forward. Attached is the full answer for anybody that might stumble across this in the future.

Thanks again.

 

[TSny]

Good work!