1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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_{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


    User Avatar
    Homework Helper

    You already mentioned one qualifier:

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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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}##
    ##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


    User Avatar
    Homework Helper
    Gold Member
    2017 Award

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted