- #1
TFM
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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