# Partition function of rotating molecule

1. Nov 4, 2009

### mikeph

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