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Maxwell-Boltzmann Energy distribution

  1. Nov 21, 2016 #1
    1. The problem statement, all variables and given/known data
    find the average energy of a system with n energy states (0, 1E, 2E, 3E...nE)

    2. Relevant equations
    P(E) = e-BE/Z - where B=1/KbT and Z= ∑e(-BE)n
    <E>=∑(nE* (e-BE)n) /Z

    3. The attempt at a solution
    i feel like ive gone down the correct path - that is finding result of the sums.
    Z - if S=R0+R1+...+Rn i do S-RS and get S=R0-Rn+1/1-R ... so Z= e0-e-BE(n+1)/1-eBE
    now it gets tricky, i tried to evaluate the sum ∑(nE* (e-BE)n) in a similar way,
    so S2=0R0+1R1+2R2...
    and RS2=0R+1R2+2R3...
    so S2-RS2=S-1 (remember S=Z; "-1" cause R0 is missng)
    so S2=(1-e-BE(n+1)/(1-e-BE)2-1/(1-e-BE).

    substituting into average fomula and simplifying my answer is: (1/1-e-BE) - (1/1-eBE(n+1))
    is this correct?
  2. jcsd
  3. Nov 21, 2016 #2


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    You're missing a term on the right hand side of the last equation. (Don't forget to consider the last terms in S2 and RS2).
  4. Nov 23, 2016 #3
    oh right RS2 ends with nRn+1 so S2 ends with "-nRn+1" (the missing term)
    so my new answer is: (1/1-e-BE) + (-1-ne-BE(n+1)/1-e-BE(n+1))
  5. Nov 23, 2016 #4


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    OK, that looks right for S2 / Z. But that's not quite the answer for <E>.

    When you have expressions like 1/1-e-BE , you should include parentheses around the entire denominator: 1/(1-e-BE).
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