Statistical Mechanics - Physics at Oregon State University

46 CHAPTER 3. VARIABLE NUMBER OF PARTICLES
available states. Hence even for independent particles the chemical potential
will depend on N.
Simple models of chemical potential effects often ignore any volume dependence.
We already know from thermodynamics that this cannot be sufficient,
because in that case the chemical potential and the temperature are not independent
variables. This can be easily demonstrated, since in this case we need
to have:
S = Nf( U
) (3.13)
N
for some function f(x). This gives
1
T = f ′ ( U
) (3.14)
N
or U = Ng(T ) and hence S = Nf(g(T )). In this case F = Nh(T ) is linear in N
and µ = h(T ). Even though such simple models are suspect, they can still be
useful in some discussions.
Basic formulation.
In summary we have:
The internal chemical potential is a thermodynamic variable which
is equivalent to a standard potential energy: the value of the difference
in internal chemical potential between two systems is equal to
the magnitude of the potential barrier needed to bring these two
systems into diffusive equilibrium.
In other words, two systems are in diffusive contact are in equilibrium when the
total chemical potentials are the same.
3.2 Examples of the use of the chemical potential.
Electricity is always useful.
As an example we consider a standard, old-fashioned battery. This simple
example is very useful to help us gain further understanding of what a chemical
potential is. The negative electrode is made of P b, the positive electrode of
P bO2 with a P b core. The electrolyte is diluted H2SO4, in other words mixed
with water . The chemical reaction at the negative electrode is
P b + SO 2−
4 → P bSO4 + 2e − + 0.8eV (3.15)