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 rotating molecule

  1. Nov 4, 2009 #1
    Hello

    I am working through a textbook here, struggling to follow a mathematical step. We are deriving the partition function Q due to pure rotation of a system containing molecules with quantum rotation energy levels:

    E = h2J(J+1) / 8pi2I

    Where J is the rotational quantum number, J = 0,1,2...

    I have substituted this into the partition function, where the degeneracy is (2J+1), and have made the approximation of turning the sum into an integral to obtain:

    Q(rot) = INT(J=0 to infinity) of (2J+1)exp[(-h2J(J+1))/(8pi2IkT)] dJ

    Suddenly they say we can eliminate the (2J+1) factor at the front of the integrand if we convert the integration variable from dJ to d(J2 + J). Nothing else changes:

    Q(rot) = INT(J=0 to infinity) of exp[(-h2(J2 + J))/(8pi2IkT)] d(J2 + J).

    They then integrate, taking the constant factor of the exponential down, evaluate whats left at the limits to get [1-0] and conclude the answer is this factor, which is:

    Q(rot) = (8pi2IkT)/(h2)

    I understand the last step, but can't see how they've done the substitution of the variable in the first place to remove the (2J+1) factor in the integral.

    Thanks for any help,

    Mike
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Partition function of rotating molecule
  1. Partition Functions (Replies: 1)

  2. Partition function (Replies: 2)

Loading...