Thermal Atmospheres

Why don't molecules of air rest on the surface of the earth ?
- Anonymous

The molecules pick up a little energy because they're surrounded by other things that have temperatures. That means that there's bits of thermal energy around traded back and forth randomly between all the little parts of things, like molecules.  There's a typical amount of thermal energy available for each different way a  part can rattle or spin. It's roughly kT, where k is Boltzmann's constant and T is the absolute temperature. At room temperature, that's about 4*10^-21 Joules. That's not a lot of energy, but the molecules of N₂ and O₂ are very light (low mass, m), so it's enough for them to bounce quite a height (h) the surface of the Earth despite the Earths gravitational field (g).  As they wander around different heights, with each averaging about kT of gravitational potential energy (mgh) over time, their typical height is several miles.

Mike W.

(published on 09/18/2013)

p(h) = (p0*e)^-(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!
- Hera (age 17)
Richmond, VA

I forget the name for that equation (maybe "Law of Atmospheres"? {yes, that's it}) , but it definitely follows from the Boltzmann law. The reason is this. Boltzmann (or least a modern version) shows that at some temperature T the probability of a quantum state being occupied is proportional to e^-E/kT Where E is the energy of the state, T is the absolute temperature, and k is Boltzmann's constant. Look at some quantum state for an N₂ molecule up at height h, vs. a similar state down at height 0. Everything about those states is the same, except the higher one has more energy by an amount mgh. So its probability is down by a factor e^-mgh/kT.

"e" is just a number, 2.718...., that happens to give convenient way to express exponentials. It's special because for small enough x:  e^x=1+x.  (e.g. e^0.01=1.01) If you use some other base for the exponents there's an inconvenient extra factor, e.g. 10^x=1+2.303x (e.g. 10^0.01=1.02303).

Mike W.

(published on 09/17/2013)