How is the kinetic energy related to the temperature?

Kinetic temperature

With a very large number of particles it makes sense to use statistical methods to study nature. When describing the molecular velocities in a gas, one expects the most probable distribution, since these are particle numbers in the order of magnitude of the Avogadro number. This most probable distribution (the Maxwell-Boltzmann distribution) assumes that the number of particles and the total energy are constant (conservation of energy). The adaptation of the probability distribution to these constraints is generally not a simple mathematical problem (see e.g. Richtmyer, et al.). A simpler way to get the solution is to look at the barometric altitude formula from atmospheric physics. The following derivation from Rohlf.

This derivation makes use of the fact that the mean kinetic energy of the molecules can be expressed in terms of the kinetic temperature. The conservation of energy in this case simply means that the kinetic energy is to be balanced with the potential (gravitational) energy, if one regards the atmosphere as an ideal gas.

From the expression for the kinetic temperature

this results in an experimentally verified expression for the kinetic energy of the molecules. The barometric altitude formula:

provides the description of an ideal gas system and can be used to derive a plausibility argument for Maxwell's velocity distribution. The following steps are necessary for this:

In one dimension of space this results in the expression:

If one includes all directions of the speed, the result is Maxwell's speed distribution:

It should be noted that a formula that depends on gravity was used for the velocity distribution, but "g" no longer appears in the end. The barometric formula was only used to link the conservation of particles and energy with the velocity distribution.