Relation of particle velocity to height

Molecules which are higher in the atmosphere are there ultimately because they had a higher kinetic energy to convert into gravitational potential energy on some time scale. (Never mind the fact of untold billions of collisions on their way to that height. On a very short time scale between collisions, some will rise incrementally higher than others from their starting points. It is that incremental height gain we consider here.)

If we have a population of particles which have energy sufficient to reach height z:

then a population with a little higher velocity can get a little higher

Relation of particle flux to velocity distribution

The number of particles traveling from z to z + dz per unit time may be related to the velocity distribution f(v). The number of particles per unit volume available for the trip is

For a given velocity vz, the number which would make it is given by multiplying the number per unit volume times the volume swept out

The general expression for the change in population (put in differential form) is:

So the flux is seen to be proportional to the upward velocity.

Relate the velocity distribution to the barometric formula

The difference in particle population with height from the barometric formula is

which must be driven by the available energy for the distribution. The differential change with height is

and it can be related to the net particle flux

For these small excursions in z, which are chosen small enough that no collisions occur, the height z attained can be related to the particle kinetic energy:

The constant C is used to lump together the scaling factors.

The form of the velocity distribution

The form of the velocity distribution in one dimension can be taken from the Boltzmann distribution in terms of particle kinetic energy. Here we are trying to argue the plausibility of that relationship is by specific correlation with the barometric formula, from which we have obtained:

Since the integral over all values of velocity must be equal to 1, the expression can be normalized to

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This is the final step in an exercise to show that one can arrive at the Boltzmann distribution by starting with two experimental observations: the equipartition of energy and the barometric formula. It is in no sense a derivation of the Boltzmann distribution, but one more exercise to show the self-consistency of a major relationship in nature upon which we depend heavily in the study of thermodynamics and kinetic theory.

The 3-Dimensional Velocity Distribution

The form of the velocity distribution in one dimension can be taken from the Boltzmann distribution in terms of particle kinetic energy. One way to argue the plausibility of that relationship is by specific correlation with the barometric formula, from which we have obtained: