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Molecule Speed

This is for question #3 on the attached page...

You will need the following fact for Question #3.

integral_{-infinity}^{+infinity} e^(-x^2/2 sigma^2) dx / sqrt(2*pi*sigma^2) = 1

This is in Section 5.4: it just says that the function

e^(-x^2/2 sigma^2) / sqrt(2*pi*sigma^2)

is a density (it is the "famous" normal density with mean 0 and variance sigma^2).

By using symmetry about 0 and the above fact, you can finish Question #3 with one integration by parts.


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Solution for question #3 only.

integral (-inf to inf) [e^(-x^2/2 sigma^2) dx / sqrt(2*pi*sigma^2)] = 1
=> integral (-inf to inf) [e^(-x^2/2 sigma^2) dx] = sqrt(2*pi*sigma^2)
=>integral (0 to inf) [e^(-x^2/2 sigma^2) dx] = sqrt(2*pi*sigma^2)/2
=> integral (0 to inf) [e^(-c*x^2) dx] = sqrt(pi/c)/2

Because, probability function,
f(x) ...

Solution Summary

The solution discusses (Ross # 5.1 - Theoretical) The speed of a molecule in a uniform gas at equilibrium is a random variable whose probability density function is given by ....