Boltzmann distribution: isothermal atmosphere error?

[strauser]

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?

 

[ShayanJ]

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!

 

[strauser]

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.