Consider an ideal gas in a vertical column with horizontal cross-sectional area *A*, under uniform gravity with gravitational accelleration *g*, at a uniform temperature *T* (the “isothermal atmosphere” model). What, then, is the probability of finding a particle at a given height *z* above the “floor” of our column?

The ideal gas law tells us that for an ideal gas of *N* particles, the themperature *T*, pressure *P*, and volume *V* obey the relation *PV*=*NkT*. Solving for the pressure, . We can note, though, that is the number density of molecules. Thus, we have . For our case, we have our properties varying with height *z*: .

Now, consider the volume of gas in a thin layer of height *dz* at an altitude *z*: The volume is *A dz*, so the number of molecules is *n*(*z*)*A dz*, and if the mass of a gas molecule is *m*, this gives the layer a mass of *mn*(*z*)*A dz*. This layer then has a weight of *mgn*(*z*)*A dz*. This force must be balanced by a pressure gradient:

*P*(*z*+*dz*)·*A*–*P*(*z*)·*A*=*A dP*=-*mgn*(*z*)*A dz*

and so we can cancel *A*, and we have differential equation

. Solving for *n*(*z*), we get , and plugging this in, we get:

. This is a simple seperable differential equation; our solution is

, and our pressure should decline exponentially with height. Further, , so the density declines exponentially as well.

Returning to our original problem, the probability of a particular molecule being found at a particular height should be proportional to the number density at that height. Thus, . However, *mgz* is the gravitational potential energy of a molecule at a height *z*. Thus, our exponential proportionality we just found is, in fact, simply the Boltzmann factor.

(I vaguely recall reading somewhere, though I don’t recall where, that reasoning similar to the above, along with observations of how the atmosphere changes with altitude, was what originally motivated the development of the Boltzmann factor).

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Tags: Boltzmann Factor, Friday Physics, Gravity, Ideal Gas, Ideal Gas Law, physics, Pressure

This entry was posted on May 8, 2009 at 12:01 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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