I am teaching thermo for the first time and using Keith Stowe "Intro.
to Stat. Mech. and Thermo." I have some questions about chemical
potential, that I hope Dan Schroeder and others on this list can
answer or help me get started on. I have been mulling over them for a
while.
In his discussion of the First Law, he writes the total internal energy as:
E = E_thermal + N*mu -(1)
where N is the number of molecules and mu is the chemical potential.
Here, the thermal energy is neither the internal energy nor heat, but
is the equipartition energy:
E_thermal = N*nkT/2 -(2)
where n is the number of degrees of freedom per molecule, assuming
all of the degrees of freedom are quadratic in their generalized
coordinates.
Substitute (2) into (1) and divide both sides by N to get the energy
per molecule as:
e = nkT/2 + mu -(3)
Let's apply Eq. (3) to a mixture of ice and water vapor in
equilibrium at some sublimation phase point. (Note: Stowe clearly
believes that this equation *is* applicable to a phase change - see
his page 75.)
Since we have an equilibrium mixture, we know that mu(ice) =
mu(vapor) and T(ice) = T(vapor). Let's take n(ice) = 9 from
Dulong-Petit and n(vapor) = 6 ignoring vibrational degrees of freedom
at these low sublimation temperatures; these values of n give good
agreement with the experimental heat capacities (cf. a previous email
of mine dated 1/27) and hence cannot be too far off the mark.
We thus conclude that:
e(ice) - e(vapor) = 3kT/2 in phase equilibrium
which is completely wrong. It doesn't even have the right sign, much
less agree with the (negative of the) latent heat of sublimation per
molecule which is what the left-hand side is supposed to give.
I have several more questions about chemical potential, but it seems
like the above is a crucial first step in trying to sort out what
Stowe is doing. Thanks, Carl
--
Dr. Carl E. Mungan, Assistant Professor http://uwf.edu/cmungan/
Dept. of Physics, University of West Florida, Pensacola, FL 32514-5751
office: 850-474-2645 (secretary -2267, FAX -3323) email: cmungan@uwf.edu