Re: chemical potential

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Carl Mungan's question about chemical potential:

> 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)

This is news to me.  I don't have a copy of Stowe's book, but this
equation seems crazy on the face of it.  Maybe his meaning of his
internal energy E is some weird non-standard expression that he defines
for his own purposes.

> 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)
> ....
> 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.

You have a nonsense result because either the equation (1) is a
nonsense equation, or Stowe means something different by it than
you are interpreting it to mean.

The way I understand the 1st order phase transition is like this: the
stable phase in any region of (T,p) space is which ever one has the
lowest chemical potential.  In one region of (T,p) space the chemical
potential [mu]_v for the vapor phase is numerically lower than the
chemical potential [mu]_i for the solid ice phase.  In this region the
vapor phase is the thermodynamically stable phase.  In another region of
(T,p) space the value of [mu]_v is greater than [mu]_i.  In this other
region the ice phase is the stable phase.  (For now we assume we are
away in (T,p) space from any region where the liquid water phase may have
the lowest chemical potential.)  These two chemical potentials (i.e.
[mu]_v & [mu]_i) have different functional forms but there is a boundary
line in (T,p) space (which separates the two regions specified above)
where they have the same numerical value.  On this boundary line (i.e.
the sublimation curve) both phases are stable and can mutually coexist in
equilibrium.

For each phase the chemical potential is related to the internal energy,
the pressure, the density, the entropy, and the temperature according to
the equation: [mu] = e + p*v - s*T.  Here e is the internal thermal
energy per particle (composed of a sum of two parts--one being the
internal kinetic energy and the other being the internal potential
energy), p is the pressure, v is the volume per particle (inverse number
density), s is the entropy per particle, and T is the absolute
temperature.  At all points in thermodynamic (T,p) space we have that the
equilibrium value of [mu] is continuous *including* across the
coexistence curve, but the equilibrium value of [mu] does switch between
which functional form [mu]_v or [mu]_i is the correct (i.e. lowest)
equilibrium one used as the system point in (T,p) space crosses that
curve at the phase transition.  In general the ice phase has a different
value of e and v and s at a given (T,p) point than the vapor phase has.
On the coexistence curve where [mu]_v = [mu}_i we have:
(e_v) + p*(v_v) - (s_v)*T = (e_i) + p*(v_i) - (s_i)*T .  Let
De == (e_v) - (e_i), Dv == (v_v) - (v_i), and Ds == (s_v) - (s_i)
on the coixestence curve.  Then we have: T*Ds = De + p*Dv.  Here T*Ds is
the latent heat per particle of the transition, Dv is the discontinuity
in the specific volume per particle for the transition, and De is the
discontinuity in the per particle internal energy for the transition.
This De discontinuity is entirely due to a jump discontinuity in the
value of the internal *potential* energy per particle at the transition.
The internal *kinetic* energy per particle is completely continuous
across the coexistence curve.  The value of p*Dv represents the work per
particle done on the environment by volume expansion as the transition
from ice to vapor takes place.  Thus the latent heat of sublimation is
the sum of the increase in the internal potential energy and the external
expansion work done against the environment as the vapor forms and pushes
the system's boundary walls outward.  As this phase transition occurs the
internal kinetic energy, the chemical potential, the temperature and the
pressure are all held constant.

> 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

I have no idea what Stowe is doing in the mentioned section of his book.
I think someone with a copy of the book would need to address that issue.

David Bowman
David_Bowman@georgetowncollege.edu