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Partition function of rotating molecule

  1. Nov 4, 2009 #1

    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,

  2. jcsd
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