Assume that the atmosphere has a temperature (T = 235 K) that is independent of

Question

Assume that the atmosphere has a temperature (T = 235 K) 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.

Define h1/2(N2) as the altitude at which the partial pressure of nitrogen (28 g/mole) falls to 1/2 of its value at sea level. Define h1/2(H2) as the altitude at which the partial pressure of hydrogen (2 g/mole) falls to 1/2 of its value at sea level.

What is the difference between h1/2(H2) and h1/2(N2)?

h1/2(H2) - h1/2(N2) =

Explanation / Answer

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 = 28/6.02 x 10^23

so, 1.38 x 10^-23 x 235 x 0.693 x 6.02 x 10^23 = 28 x 9.8 x h

so, h= 4.93 m

similarly for hydrogen

1.38 x 10^-23 x 235 x 0.693 x 6.02 x 10^23 = 2 x 9.8 x h

so, h= 69.02 m

difference= 64.09 m

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