# Maxwell Speed Distribution

kashe

## Homework Statement

Confirm that the mean speed of molecules of molar mass M at a temperature T is equal to (8RT/piM)^1/2. Hint: You will need an integral of the form ∫ (where a=0, and b=infinity) x^3*e^(-ax^2) dx = 1/2a^2.

## Homework Equations

The Maxwell speed distribution formula we are using is f=F(s)delta s where F(s)=4pi*(M/2piRT)^1/2*s^2*e^(-Ms^2/2RT)

## The Attempt at a Solution

I attempted to use the answer to the integral and say a= M/2RT, but that didn't work. I then thought a=M/2piRT, but I couldn't get it to be the correct answer. I don't think I need to do the integral just because it is already done for me, but I am stumped at how to relate the mean speed and this integral.

## Answers and Replies

Staff Emeritus
Homework Helper
The normalization constant of the distribution function isn't correct. It should be

$$F(s) = \sqrt{\frac{2}{\pi}\left(\frac{M}{RT}\right)^3} s^2 e^{-Ms^2/2RT}$$

Homework Helper
How do you define the mean speed for molecules ?

kashe
The mean speed is the sum off all the speeds of all the molecules divided by the number of molecules. I also know that the mean speed is related to the rms speed of the molecules by the equation mean speed= (8/3pi)^.5 times the rms speed.

Also I have been plugging in numbers and found the equation my book gave to me is equivalent to the normalization constant above.

Last edited:
Staff Emeritus
Homework Helper
What do you mean by f=F(s)Δs? Perhaps that's where the confusion is arising.

My point about the normalization constant comes from the fact that

$$\int_0^\infty 4\pi\sqrt{\frac{M}{2\pi RT}}s^2e^{-\frac{Ms^2}{2RT}}\,ds = \frac{2\pi R T}{M} \ne 1$$

kashe
The f(s) delta s. Is the fraction of molecules between two speeds. The delta s is just the change in speed you have, I.e. if you want the fraction between 300 and 310 m/s, you do the normalization constant with 300 and then multiply it by 10 for the fraction

Staff Emeritus
Homework Helper
kashe
Ok, I will attempt to integrate the constant you gave then, although I am not sure I know how to do it. I will work with it and post again and see if you can give some guiding help. Thanks.

kashe
Ok I am stuck. I tried to integrate the distribution constant but got stuck, and also wondered why I need to do it if in the hint on the problem they gave it to me. Although in the hint I am not sure where "x" went, because when I integrate the constant the "x" term should still be there. Any help on where to start? I tried many things but haven't got anywhere.

Staff Emeritus
Homework Helper