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Help me convert Boltzmann distribution/partition function into Geometric series

  1. Dec 14, 2007 #1
    1. The problem statement, all variables and given/known data

    3. The following calculaltion shows how the ratio of e to kT affects the
    populations of different energy levels. kT is sometimes called the thermal
    energy; if it is small relative to e, a particle will not be able to access higher
    energy states.
    Consider a harmonic oscillator with spacing νe/kT between energy levels, where
    ν can take on the values 0, 1, 2, etc.Calculate the fraction of particles in the ν = 3
    energy level for e/kT = 4, 1, and 0.2.

    Note: The definition of a harmonic oscillator is that the energy levels are equally spaced so the spacing between energy levels is identical.

    2. Relevant equations

    The only equations you need for this is the Boltzmann distribution and partition function,

    Boltzmann Distribution:
    ni/N = e^(-eiB)/z

    z = Partition function = summation of e^(-eiB)

    B= 1/kT

    k = Boltzmann constant = 1.381e-23 J/K

    (Sorry for the poor notation but you can look these up online)

    3. The attempt at a solution

    I know how to do this problem and how to set it up, but the problem here is I don't know how to express the expression as an infinite series.

    For example, let's say e/kT = 1:

    ni/N = e^(-3 * 1)/(1 + e^(-1) + e^(-2) + e^(-3).....)

    My professor says the equation can be expressed in the following geometric series:

    1/(1 - e^(-e/kT))

    I'm pretty sure that doesn't work because it doesn't result in sensible answers. Does anyone here know how to correctly express it in terms of a series? I realize my explaination is rather hazy but please consult this pdf (problem #3) for more information...you will see if you use his series you will not get the answers he has listed

    http://ded.chm.jhu.edu/~pchem/AnswerKeys/Resources/HW Wk13 Ans.pdf

  2. jcsd
  3. Dec 14, 2007 #2

    Shooting Star

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    Homework Helper

    Do you know the sum of an infinite GP? (Don't put constants equal to 1; sometimes it becomes difficult afterwards to see exactly where to put it back.)
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