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Maxwell Distribution
This model illustrates the concept of thermal motion of gas molecules (also known as molecular chaos). Gas molecules undergo multiple collisions with each other. Each collision leads to a change in magnitude and direction of molecular velocities. As a result, a statistical velocity distribution can be established in a gas vessel with large number of molecules. This velocity distribution is a function of absolute temperature. The directions of molecular movements after the collision are unpredictable, while the velocity magnitudes obey a certain law, known as the Maxwell's distribution law.
If we measure the velocities of a large number N of gas molecules and concentrate on a small velocity interval from v to v + Δv, a certain number ΔN of molecules is bound to have a velocity in the chosen interval Δv.

It is convenient to plot the quantity (ΔN/Δv) versus v on a graph. The large number N in this function is represented by a smooth curve ψ(v) with its maximum at v = vm = ((2kT)/m)0.5, which is referred to as the most probable velocity. Here, m is the mass of the molecule and k = 1.38∙10-23 J/K is the Boltzmann constant.
An important parameter of Maxwell's distribution is the so-called mean square velocity
vrms = < v2 >0.5, where < v2 > is the mean value of velocity squared. It has been proven that

vrms = ((3kT)/m)0.5 = ((3RT)/μ)0.5, where μ is the molar mass.
As it follows from this equation, the mean kinetic energy of transitional motion of gas molecules is
< E > = (m< v2 >)/2 = 3/2kT

Maxwell's distribution is one of the most important statistical relationships in molecular physics.
Observe the graph of Maxwell distribution at a given temperature on your computer screen. The root-mean-square vrms and the most probable molecular velocities vp are calculated for you. Change the temperature of gas T, and observe the displacement of the maximum on the graph. All molecules with velocities that belong to the selected interval are painted green. You can modify the average velocity of molecules by changing the interval on the speed axis. Note that the number of green molecules is at its maximum when the selected velocity interval is situated near the maximum of the distribution curve. The velocity of these molecules is the same as the most probable velocity vp.

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