Derivation of Maxwell's Velocity Distribution Function

A distribution function of the property, X, has as a definition that the property lies somewhere in the range X to X+DX.

For example, this is a normal distribution of the property, x, with a mean of 3 and one standard deviation of 0.2.

This is a statistical F distribution with 8 and 2 degrees of freedom

Distribution functions show the probability, P(X), of X in the range.

The probability holds whether the range is discrete, DX or continuous, dx.

We know that the probability of two independent events is the product of the probability of each event alone.

The velocity of a molecule is the vector sum of the n_{x}, n_{y}, n_{z} velocity components.

i.e. magnitude of the 3-d vector

Since the distribution function, F(n_{x},n_{y},n_{z}) depends only on velocities (not direction), it can be written as

The right side, due to the probability of independent events.

Because
, only an exponential can be a solution.
An appropriate trial solution with K and z as constants. The negative exponent is used to limit extreme velocities.

To establish the value of K and z, we have two observations.

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First)

The Probability of velocity has to be somewhere!

Trying out our GUESS solution for f(n_{x}),

<= From table of integrals

Thus

Second) Zeta, z, is obtained by finding the mean square speed of the molecules and substituting into a form of the ideal gas law, namely,

{EQU 3}

where n = moles, N_{0} = Avogrado, m=mass, and c^{2}="mean square speed".

*Aside: From Statistics, a general expression for computation of a mean is*

and substituting in the proposed solution for f(n_{x}):

<= Integral from Tables

Now Generalize to 3-d.

Since

Now substitute
for c^{2} in {EQU 1}.
another form of the Ideal Gas Law where k = boltzman const

Equating the two equations

&

Now plug z and K into the proposed solution

<= Proposed Solution

and simplify

I will now plug in some values and define the eqn for MathCad before continuing

One Dimensional Solution to the Velocity Distribution

By extension, to a 3d solution:

It would be nice to convert this expression over to n, molecular velocity, rather than the individual components. Note:

The volume element,
, may be written as
where dw is the infinitesimal solid angle and then integrate over the full sphere, i.e. 0 to 4p.