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!

Partition function of simple system

  1. Apr 29, 2014 #1
    1. The problem statement, all variables and given/known data

    2gwyweb.png

    A molecule has 4 states of energy -1, 0,0 and 1. Find its partition function and limit of energy as T → ∞.

    2. Relevant equations



    3. The attempt at a solution

    [tex]Z = \sum_r e^{-\beta E} = e^{-\beta} + 2 + e^{\beta}[/tex]

    [tex]U = -\frac{\partial ln(Z)}{\partial \beta} = \frac{e^{-\beta} - e^{\beta}}{e^{-\beta} + 2 + e^{\beta}}[/tex]

    As ##T→\infty##, ##exp(-\beta) \approx 1 - \beta## and ##exp(\beta) \approx 1 + \beta##.

    Thus,
    [tex]U \approx \frac{(1-\beta) - (1+\beta)}{2 + (1+\beta) + (1-\beta)} = -\frac{\beta}{2} = -\frac{1}{2kT}[/tex]

    The equipartition theorem should take over with Energy = 4 * (1/2)kT = 2kT = 2/β.
    But instead i'm getting -β/2.
     
  2. jcsd
  3. May 2, 2014 #2
  4. May 2, 2014 #3
    Ammm, okay...

    Your calculation is right, at least I got the same result for ##U##.

    my comment about the equipartition theorem: My experiences are that you really have to master thermodynamics to completely understand this theorem. Lots of results can be "guessed" if you truly understand the concept. I was never that good at it therefore I always had to do the long calculations.
    Ok, now to tell something that is actually useful:

    from http://chemwiki.ucdavis.edu/Physical_Chemistry/Statistical_Mechanics/Equipartition_Theorem (Degrees of freedom):
    "The law of equipartition of energy states that each quadratic term in the classical expression for the energy contributes ½kBT to the average energy."

    Let's take a molecule of ideal gas for example: One molecule has in fact ##6## degrees of freedom. ##3## of them precisely describe it's position and are called coordinates (x,y,z), the other ##3## are of course components of momentum (note that momentum is quadratic in energy ##E_k=\frac{p^2}{2m}##). Each component of momentum therefore contributes ##\frac{1}{2}kT##, so the average energy of molecule of ideal gas is ##\frac{3}{2}kT##.

    I guess all I am trying to say is that you have no quadratic degrees of freedom and therefore your calculation using equipartition theorem is wrong.

    ps: Keep in mind that I never mastered that theorem. I hope I didn't just make a fool out of myself.
     
  5. May 2, 2014 #4
    try expanding the exponential in different form
     
  6. May 2, 2014 #5
    Exp(-x) = 1/ exp(x) = 1/ (1+x)

    try in this form and show what you get, i hope this work
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Partition function of simple system
Loading...