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I Boltzmann distribution: isothermal atmosphere error?

  1. Apr 18, 2016 #1
    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?
     
  2. jcsd
  3. Apr 19, 2016 #2

    ShayanJ

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    Gold Member

    It should be obvious form the fact that the potential used is proportional to h rather than 1/h!
     
  4. Apr 26, 2016 #3
    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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