See attached file.
The function f(v) is the number of particles with velocities within a volume d^3v in velocity space. The constant A is a normalization constant. The formula is valid when the gas is in thermal equilibrium. Maxwell was the first to test this formula experimentally. He let a gas stream out of a container into a vacuum for a short time. The gas molecules then moved toward a very fast rotating disk. Molecules with different speeds would hit the disk at different positions. The disk is made out of a material that will cause the molecules hitting it to stick to the disk causing it to discolor. By investigating the disk one can then deduce the speed distribution.
a) The average of v_x is zero due to the symmetry of the speed distribution as a function of v_x.
b) The average of |v_x| can be computed as follows. We first normalize f(v) such that it gives the probability density for the speed:
f(v) = [m/(2 pi k T)]^(3/2) exp[-m v^2/(2 k T)] (1)
The integral of f(v)d^3 v is thus ...
A detailed solution is given.