Physics Friday 71: The Isothermal Atmosphere

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+dzAP(zA=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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