While details of the Maxwell Boltzmann distribution are not really appropriate for high school students, the underlying principles are easily within their grasp.

The KMT is based upon the notion that both momentum and kinetic energy will be randomly distributed in a gas.

Energy Distribution

These arguments are modeled after those appearing in a seminal general chemistry text by George C.Pimentel and Richard D. Spratley. (ISBN 0-8162-6761-8).

Suppose you have 12 packets of energy, and you want to distribute these randomly among 4 particles labeled A, B, C, and D.

All 12 Packets into one particle:

Imagine writing out all of the possible combinations for every possible distribution of 12 packets among these 4 molecules.

Then, add up the number of times a molecule has 0 energy packets, then number of times it has one packet, etc.

Momentum Distribution

Momentum is given as mass times velocity. Velocity, a directed quantity, can be represented by a vector. Suppose we imagine all of the vectors for a given magnitude of momentum as emanating from a single point. If we do this, the tips of those vectors would form a sphere.

Suppose the surface of the sphere is divided up into some sort of uniform area units, and a vector goes to the center of each unit. What happens if the momentum is doubled? The length of the vector doubles. The surface of the sphere doubles.

Area₁ = 4πr²;

replace r with 2r;

Area2 = 4π(2r)² = 4 x 4πr² = 4 x Area_1.

In other words, the number of possible momenta goes up as the area of the sphere, which in turn is the square of the radius of the sphere. The function that describes this increase is a parabola.

The curve looks like this:

Energy and Momentum Distribution

When both energy and momentum are distributed together, the resulting probability is the product of the separate probabilities. In other words, the overall curve is the product a a parabolic curve (increasing) and an exponential curve (decaying). The result is the characteristic shape of the Maxwell Boltzmann distribution.

The figure above is one way of writing the Maxwell-Boltzmann distribution. Here u is speed. If we plot the probability of finding a gas particle with a speed in the range between that value u and a small increment to that value, du (like Du), then this is given by the function shown times du. Note the two features of this function, the parabolic feature and the exponential decay feature. These outcomes are summarized in the figure below.