Given:
Assume that the atmosphere has a temperature T that is independent of altitude. The pressure of the atmosphere is the sum of the partial pressures of the gases present (N2 , O2 , etc.). The partial pressure of each type of gas varies with altitude, h, according to the condition of constant chemical potential:
µ(h) = kT ln(n(h)/nQ ) + mgh = constant where n(h) is the gas density (particles/volume) at altitude h, and nQ is the quantum density (which is independent of h).
Find: At what altitude, h, does the partial pressure of nitrogen (molecular weight= 28) fall to 1/2 of its value at sea level? Give your answer in kilometers.
Solution:
µ(0) = kT ln(n(0)/nQ )
µ(h) = kT ln(n(h)/nQ ) + mgh
and you know µ(h) = µ(h!= 0) in equilibrium, so set them equal and isolate mgh on one side
kT ln(n(0)/nQ )  kT ln(n(h)/nQ ) = mgh
since ln(a)  ln(b) = ln(a/b), the nQs will cancel and that equation will become kT ln(n(0)/n(h)) = mgh.
The question is asking for when the partial pressure falls to n(h) = .5n(0), so the equation can be further simplified to kT ln(.5 ) = mgh, where m is the grams/particle of nitrogen (aka molweight * 1/avagadro's#).
Solve for h.
