Re: degrees of freedom of water

[Daniel Schroeder <DSCHROEDER@CC.WEBER.EDU>]

> Problem 9-23 in K. Stowe "Intro. to Stat. Mech. and Thermo." boils
> down to asking for the number of degrees of freedom per water
> molecule (call it n) using equipartition and the specific heat.
>
> I find n = 2*.018*c/8.314 where c is the specific heat at constant
> volume in J/kg/K.
>
> Thus I get n = 18 for water using c = 4186. The problem ends there,
> but my question is: What does this mean? In what way can a water
> molecule be said to have 18?
>
> Interestingly enough, if we try ice (c = 2000) we find n = 8.7 ~ 9 in
> accordance perhaps with Dulong-Petit; and if we try water vapor (c =
> 1520) we find n = 6.6 ~ 6 in accordance with our expectations of 3
> translations and 3 rotations (and maybe even a hint of the bending
> mode beginning to turn on?).
>
> Texts always seem to discuss solids and gases in connection with
> specific heat, but never liquids. Why *does* water have such a high
> heat capacity compared to ice or water vapor? Where does it store
> away all that extra energy?

Gee, I had forgotten that that problem was in Stowe.  That must
be where I first got it; I put a similar problem into my own book
(An Introduction to Thermal Physics, Problem 1.43) though I
worded it differently to try to make it clear that the
calculation doesn't really apply to water.  The equipartition
theorem applies only to "quadratic degrees of freedom", that is,
forms of energy whose formula is quadratic in some coordinate
or momentum.  This usually works well for gases and for solids
(at sufficiently high temperatures), but the forces between
molecules in a liquid do not fall into this category.

It would be interesting to try to model these intermolecular
forces in some approximate way, aiming at some modified version
of the equipartition theorem that would apply to a liquid.
The fact that the heat capacity of water is nearly independent
of temperature seems to imply that there is some underlying
simplicity.  I know that for a degree of freedom whose
energy is of the form q^n, for n>=1, you can derive a generalized
equipartition theorem which says that the average energy
is kT times 1/n.  In other words, the more strongly the
energy curves upward as a function of q, the smaller the
contribution of this degree of freedom to the average thermal
energy.  To model intermolecular forces, though, we want
an energy function that's actually concave-down.  This
suggests to me that the average energy should be kT times
some relatively large coefficient, but a derivation of
this coefficient eludes me.

Dan Schroeder
Weber State University
http://physics.weber.edu/schroeder/