Equation p(h): Explained and Derived From Boltzmann's Law

[yeveat]

p(h) = (p₀e)^-(mgh/kT)

where p0 is the atmosphere at sea level, m is the mass of an object at height h, g is the gravitational proportionality constant...

is there a specific name for this equation? is this derived from the Boltzmann’s distribution law? Also, I'm really confused about what e is... Thank you!

 

[johnqwertyful]

e is Euler's number, about 2.7. Search wiki for it.

It's the pressure at a height, h at a temperature T. k is Boltzmann's constant.

 

[johnqwertyful]

Also g is not the same thing as G. G is the gravitational proportionality constant, g is the acceleration due to gravity.

 

[Ken G]

Also,the p_o should not be inside the parentheses. Only the e is being raised to that power. It is not a Boltzmann factor, that would be statistical physics. This is a solution to a force equation, the hydrostatic equilibrium that looks like dp/dh = -rho * g, where rho = mp/kT from the ideal gas law and p is gas pressure.

 

[SteamKing]

It's the Law of Atmospheres:
http://www.cmmp.ucl.ac.uk/~als/teaching/1B28/Lecture notes/Lecture_8.pdf

See p. 4 of the attachment.

It predicts the number of gas molecules at a given height h, and consequently the pressure at that height in terms of the surface pressure Po.

N.B.: k is indeed Boltzmann's constant.

 

[D H]

And m is the mean molecular mass of air.