equations for molar heat capacities

If I know the spectrum of something (say, diatomic oxygen molecule O2),
I can in principle compute its partition function. From that, I can
in principle compute the molar heat capacity. If I keep T as a variable
and ignore terms of certain orders, that should give me some explicit
approximate formulas for the molar heat capacities. On the other hand,
there are empirical formulas for the molar heat capacities, some of
which I mentioned in my original posting on this topic.

What I would like to know is whether the empirical formulas (that's all
the SHOMATE equations I was referred to seem to be) can be obtained,
at least approximately, by torturing the partition function until it gives
you the molar heat capacity? I've been trying to do this myself by hand and
am having a lot of problems with it, partly because of a lot of stupid mistakes
that it takes time to locate and and correct. It would be encouraging to know
that what I am trying to do is really possible before I get too bogged down
in this. Is it, or is there something about the empirical formulas that isn't
reflected in the spectroscopid approach?

If it is possible, I would think that someone would already have published
a paper illustrating the technique. If so, can someone please refer me to one
or more articles where this has been carried out in detail for some specific
molecules?

Just to clarify, I'm under the impression that the papers of Spencer et al,
which Denbigh's book cites and which I mentioned in my earlier posting,
really give empirical formulas, which I understand to mean based on
interpolating experimental measurements of the molar heat capacity.
So I am assuming that those papers really don't have the answer to
my question of deriving the equations from spectroscopic data.

Ignorantly,
Allan Adler
[Only registered users see links. ]

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The person who reads this list who can best answer your question is
probably Alan Harvey. At least one point is that the spectroscopic
values you see in books pretty much are for isolated molecules, and
neglect intermolecular interactions (low pressure limit). Another is
that they pretty much neglect anharmonicities, which at least for
polyatomics are not often measured (medium-low pressure limit).
Another is that Shomate equations are fitting forms, which can and
are used for condensed phases. So at least in those "limits" good luck.

O2 should be simple enough, at least for "room" temperatures, although
for 500 K and above you will have to include the low lying singlet levels.
A lot of the "priimary" data comes from the JANAF tables which used
statistical mechanics to extend calculations of enthalpy (and specific heat)
to different temperatures.

I would not be surprised if many of the Shomate eqs. in the webbook were
fit to the JANAF tables calculated from statistical mechanics.

josh halpern

Allan Adler wrote:

[Top-posting corrected for clarity]

Josh Halpern wrote:

I'm flattered -- don't recall having been invoked as an authority on this group
before. Maybe he knew I work for NIST (although anything I write here is not
in my official capacity). But I'm in a somewhat different area than the people
who do things like derive ideal-gas properties for the JANAF tables -- actually
those people are mostly gone due to budget cuts.

To the best of my knowledge, the procedure goes as follows. First, you make a
partition function (summation of all energy levels) the best you know how. And
how that is done depends on the molecule. For well-studied (and simple)
molecules, you can enumerate lots of levels essentially exactly, based on
spectroscopic data maybe supplemented by theoretical calculations. For other
molecules, you may have to make simplifying assumptions like harmonic
vibrations, separation of vibration and rotation, etc. Then you can derive
ideal-gas heat capacities from that. Those can then be fitted to a function
like the Shomate equation for practical use.

The original question was whether the Shomate equation is some approximation
from the rigorous partition function. I don't know the answer to that for
sure, since I don't know the origins of the Shomate equation, but I sort of
doubt it. The rigorous formulas have exponentials in them (see any stat mech
textbook in the section on the ideal diatomic gas). Which I suppose you could
expand in these polynomial terms if you want, but I suspect the form of the
equation is in some measure empirical.

You mentioned O2 -- for that you could certainly use the ideal diatomic gas
methods in the stat mech books at low and moderate temperatures. Higher-order
spectroscopic terms for rotation-vibration coupling and anharmonicity would get
you to somewhat higher temperatures. But the state-of-the-art numbers (like
those in the JANAF tables) would be obtained from a more rigorous summation of
energy levels. I tried to find where in the literature that was done for O2,
and it looks like a German dissertation which probably isn't an accessible
reference. But you can see a state-of-the-art approach for H2O (with
references that might also be of interest) in:
M. Vidler and J. Tennyson, "Accurate Partition Function and Thermodynamic Data
for Water," J. Chem. Phys., vol. 113, pp. 9766-9771 (2000).

--------------------------------------------------------------
Dr. Allan H. Harvey, Boulder, CO
"Any opinions expressed are mine, and should not be attributed to
my employer, my wife, or my cats."