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Boltzmann Factor Type Question

  1. Feb 25, 2013 #1
    1. The problem statement, all variables and given/known data

    A system in thermal equilibrium at temperature T consists of N particles that have two
    energy states separated by an energy Δε.

    If the number of particles in the two states is N1 and N2, show that:

    [itex]N{1}[/itex] = N([itex]\frac{1}{1+exp(-Δε/k{B}T}[/itex])) and [itex]N{2}[/itex] = N([itex]\frac{exp(-Δε/k{B}T}{1+exp(-Δε/k{B}T}[/itex]))

    2. Relevant equations
    [itex]\frac{N{1}}{N{2}}[/itex] = [itex]\frac{exp(-E{1}/k{B}T}{exp(-E{2}/k{B}T}[/itex]

    Δε=E1 - E2

    3. The attempt at a solution

    Really struggling to see where to get started with this the lectures and the lecture notes we have are not helping.
  2. jcsd
  3. Feb 25, 2013 #2


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    From the way the problem is worded, I think Δε should be Δε = E2 - E1

    See if you can show [itex]\frac{N{2}}{N{1}}= {exp(-Δε/k_{B}T)}[/itex]

    Also, what must the sum N1+N2 equal?
    Last edited: Feb 25, 2013
  4. Feb 25, 2013 #3
    Well N1 + N2 must equal N?

    So now that I have it in terms of Δε I can rearrange for expressions for N1 and N2 then add them for N?
  5. Feb 25, 2013 #4


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    You have two equations for N1 and N2. So you should be able to solve for N1 and N2 separately.
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