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!

Deriving Ideal Gas Law through partition function

  1. Jan 28, 2014 #1
    1. The problem statement, all variables and given/known data

    The pressure of a non-interacting, indistinguishable system of N particles can be derived from the canonical partition function

    [tex] P = k_BT\frac{∂lnQ}{∂V} [/tex]

    Verify that this equation reduces to the ideal gas law.

    3. The attempt at a solution

    I have a very poor background in quantum mechanics and probability and this is the first course I'm taking which relies on any of the two. The course I'm taking teaches thermodynamics through a blend of the classical point of view and the microscopic statistical point of view.

    From what I understand, for an indistinguishable system, the total partition function Q can be related to the individual molecular partition functions q by

    [tex] Q = \frac{q^N}{N!} [/tex]

    Using properties of logarithms,

    [tex] lnQ = Nlnq - lnN! [/tex]

    Since the partial with respect to V of lnN! is 0, I have so far:

    [tex] P = k_BTN\frac{∂lnq}{∂V} [/tex]

    But when I look at the molecular partition function q, I have absolutely no idea how I'd be able to relate it to volume to be able to differentiate. The definition of the molecular partition function q I have is:

    [tex] q = \sum{e^{-\frac{E_j}{k_BT}}} [/tex]

    The only thing that comes to my mind is that I know that the total energy is going to be the sum of translational, rotational, vibrational, and electronic energy levels in a molecule. It'd make sense to me for translational energy levels to have something to do with volume and pressure, but I'm not sure how to draw the connection or whether I'm even going down the right track. A lot of these concepts are completely new to me.

    Where else can I go with this, assuming this isn't completely off? I'd appreciate any help.
     
  2. jcsd
  3. Jan 29, 2014 #2

    Borek

    User Avatar

    Staff: Mentor

    Moved the problem to advanced physics, it is more densely populated by right kind of people, so probability of getting an answer should be higher.
     
  4. Jan 29, 2014 #3
    Hey Borek, since this never got any replies I started a new thread in the introductory physics subforum and got the help I needed,

    https://www.physicsforums.com/showthread.php?t=735495

    I have no idea how to lock threads on this site, so this one remained open. Sorry about that.
     
  5. Jan 29, 2014 #4

    Borek

    User Avatar

    Staff: Mentor

    Never start a new thread - report the old one and ask mentors to move the thread.

    Besides, you reposted just after 10 hours, that is worth a warning. You are lucky I have not noticed earlier.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Deriving Ideal Gas Law through partition function
  1. Ideal gas law qustion? (Replies: 2)

Loading...