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Partition Function

  1. Jan 19, 2014 #1
    1. The problem statement, all variables and given/known data

    For three systems A, B, and C it is approximately true that [itex]Z_{ABC}=Z_{A}Z_{B}Z_{C}[/itex]. Prove this and specify under what conditions this is expected to hold.

    2. Relevant equations

    Z is the partition function given by [itex]Z=∑e^{-ε/KT}[/itex]
    ε is energy, T is temperature and K is Boltzmann constant.

    3. The attempt at a solution

    let say that A is the translational, B is the vibrational and C is the rotational energy levels for diatomic molecule.

    To a good approximation the different forms of molecular energy are independent, so that we can write

    [itex]ε_{total}= ε_{A}+ε_{B}+ε_{C}[/itex]​

    Since [itex]Z=e^{-ε/KT}[/itex], the sum in the exponents becomes a product.

    [itex]Z_{total}=(∑e^{-ε/KT})_{A}(∑e^{-ε/KT})_{B}(∑e^{-ε/KT})_{C}[/itex]

    [itex]Z_{ABC}=Z_{A}Z_{B}Z_{C}[/itex] ​

    But what will be the conditions?
     
    Last edited: Jan 19, 2014
  2. jcsd
  3. Jan 19, 2014 #2

    lightgrav

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    You already mentioned one qualifier:

    When are the Energies in each different form NOT independent? (Think extremes, here!)
     
  4. Jan 19, 2014 #3

    haruspex

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    You are right that independence is important, but I don't think you've used that assumption in the right way.
    [itex]ε_{total}= ε_{A}+ε_{B}+ε_{C}[/itex] is true anyway. Don't you need the independence to get from
    ##Z_{total}=\Sigma_S e^{-ε_{tot}/KT}##
    to
    ##Z_{total}=\Sigma_A \Sigma_B \Sigma_C e^{-ε_{tot}/KT}##
    ?
    I.e. the microstates of the combined system are merely the combinations of the microstates of the separate systems.
     
  5. Jan 19, 2014 #4

    TSny

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    I might not be interpreting the word "independence" the same as way others are. I think of it as meaning non-interacting.

    Non-interaction of A, B, and C is important in being able to write εtotal = εA + εB + εC. An example where the energy cannot be written this way is a system with a potential energy of interaction U(A,B) between subsystems A and B.

    Also, there are systems for which the subsystems are strongly interacting but yet the sum over microstates of the total system can still be written as a multiple sum over the microstates of the subsystems. For example, consider a system of 3 interacting spins (A, B, and C) as in the 1D Ising model (See here, especially slide 5). (But the partition function of the total system does not factor into a product of individual partition functions due to the fact that the energy cannot be written as εtotal = εA + εB + εC .)

    Even for a system of three non-interacting particles A, B, and C (e.g., three non-interacting particles in a box), there is an important situation where the partition function does not factor as Z = ZA ZB ZC. Think about the case where the particles are indistinguishable. Note that Z doesn't factor even though εtotal = εA + εB + εC.
     
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