Boltzmann distribution: isothermal atmosphere error?

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There is a well-known analysis of the distribution of particles by height in an isothermal atmosphere. It states that the probability of finding a particle at height ##h## is ##p(h) \propto e^{-\beta mgh}##, and then goes on to state that the number of particles at height ##h## is ##n(h) \propto e^{-\beta mgh}##.

Is this not strictly incorrect, and merely an approximation? Surely we need to count the number of available states at each ##h##, and this will be proportional to the available volume in a thin shell at height ##h## i.e. we should have ##n(h) \propto f(h) e^{-\beta mgh}## where ##f(h)## measures the available volume? (This is because, roughly speaking, we can fit more molecules into a thin shell as ##h## increases.)

I'm guessing that this isn't done as ##h## doesn't vary much compared to the Earth's radius?
 

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  • #2
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I'm guessing that this isn't done as h doesn't vary much compared to the Earth's radius?
It should be obvious form the fact that the potential used is proportional to h rather than 1/h!
 
  • #3
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It should be obvious form the fact that the potential used is proportional to h rather than 1/h!
Well, I take your point but I don't think that it's necessarily "obvious" - given that the way this kind of argument should be structured, it seems like poor pedagogy, to my mind, to make no mention of any kind of density-of-states derivation - this seems to abound in statistical physics treatments though.
 

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