[CHM 330] Helpful Advice on Calculating Partition Functions
Andrew J. Pounds
pounds_aj at mercer.edu
Fri Mar 11 08:32:28 EST 2022
As we noted in class the partition function Q is built by summing over
the energy levels. As the index for the sum increases these values get
smaller and smaller. As an example, in the problem dealing with the
harmonic oscillator, the vibrational partition function is
Q=∑i=0Ne−iΔE/kbTQ=\sum_{i=0}^N e^{-i\Delta E/k_bT}because the argument
of the exponential (other than i) is fixed, we can calculate it before
taking the sum. Let's imagine that value is 0.75. Our new equation is
then just
Q=∑i=0Ne−i0.75Q=\sum_{i=0}^N e^{-i 0.75}
Rather than explicitly calculating all of those terms individually and
summing them up, you can use that $125 piece of tech most of you are
toting around - your TI-8X calculator.
There are many YouTube videos on how to do summations on your
calculators. If you can't find one let me know and I can recommend
one. For any problems I give you the maximum number of terms should not
exceed 100 -- which takes my HP calculator less than a second to solve
(I don't have a TI-8X available for testing). When I plug in the
summation from above I get 1.8952551344.
You don't have to learn how to do this -- but it could definitely save
you some time when doing problems that require you to calculate the
fraction of molecules in a given vibrational level.
--
*/Andrew J. Pounds, Ph.D./*
/Professor of Chemistry and Computer Science/
/Director of the Computational Science Program/
/Mercer University, Macon, GA 31207 (478) 301-5627/
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