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## Homework Statement

Calculate the integral

[tex]\left v\langle\right\rangle = \int^\infty_0 v f(v) dv[/tex].

The function

[tex]f(v)[/tex]

describing the actual distribution of molecular speeds is called the Maxwell-Boltzmann distribution,

[tex]f(v) = 4\pi (\frac{m}{2\pi kT})^3^/^2 v^2 e^{-mv^2/2kt}[/tex]

(Hint: Make the change of variable

[tex] v^2 = x [/tex]

and use the tabulated integral

[tex]\int^\infty _0 x^ne^\alpha^x dx = \frac{n}{\alpha^n^+^1}[/tex]

where

*n*is a positive integer and [tex] \alpha [/tex] is a positive constant.)

Express your answer in terms of the variables T, m, and appropriate constants.

## Homework Equations

## The Attempt at a Solution

I think I have got some way in, but I am not sure how to go from here:

firstly the:

[tex] 4\pi (\frac{m}{2\pi kT})^{3/2} [/tex]

is a constant, so can put in

*c*for now

[tex]f(v) = (c) v^2 e^{-mv^2/2kt}[/tex]

ANd can remove from the integration

[tex]v = c \int^\infty_0 v (v^2 e^{-mv^2/2kt}) dv[/tex]

then, replace [tex]v^2[/tex] with x:

[tex]v = c \int^\infty_0 v (x e^{-mx/2kt}) dv[/tex]

and change the integration,

[tex]x = v^2, thus \frac{dx}{dv} = 2x thus dv = \frac{dx}{2v}[/tex]

Which gives:

[tex]v = c \int^\infty_0 v (x e^{-mx/2kt}) \frac{dx}{2v}[/tex]

and

[tex]v = c \int^\infty_0 (x e^{-mx/2kt}) \frac{dx}{2}[/tex]

Taking out the half:

[tex]v = c/2 \int^\infty_0 x e^{-mx/2kt} dx[/tex]

rearraniging for the tabulated integral,

[tex]\alpha = -\frac{m}{2kt}[/tex]

So:

[tex]v = c/2 \int^\infty_0 x e^{-\alpha x} dx[/tex]

Which can be integrated using tabulated given in question:

[tex]c/2 \left[\frac{1}{\alpha^2}\right][/tex]

Putting back [tex]\alpha[/tex]

[tex]c/2 \left[\frac{1}{(- \frac{m}{2kt})^2}\right][/tex]

And:

[tex]c/2 \left[\frac{1}{(- \frac{m^2}{4k^2t^2})}\right][/tex]

Which I believe can go around to:

[tex]c/2 (\frac{4k^2 t^2}{m^2})[/tex]

Putting back the

*c*:

[tex](4\pi (\frac{m}{2\pi kT})^3^/^2)/2 (\frac{4k^2 t^2}{m^2})[/tex]

Which I think can be rarranged a bit more to give:

[tex](2\pi (\frac{m}{2\pi kT})^3^/^2) (\frac{4k^2 t^2}{m^2})[/tex]

But I am not quite sure where to go from here.

Any ideas? Does it look right?

TFM