Statistical Mechanics - Physics at Oregon State University

3.1. CHEMICAL POTENTIAL. 45
µ(T, V, N) =
∂F
∂N T,V
= ˆµ + Φ (3.7)
This is called the totalchemical potential. Two systems are in equilibrium
when the total chemical potential is the same. Ordinary, standard potentials
therefore give a direct contribution to the chemical potential. But even
without the presence of external potentials, the Helmholtz free energy will still
depend on the number of particles N and we will find a value for the chemical
potential. The value of the chemical potential without any external forces is
called the internal chemical potential. Hence the total chemical potential is the
sum of the internal chemical potential and the external potentials! Confusion
arises when one uses the term chemical potential without specifying total or
internal. Some textbooks assume that chemical potential means internal chemical
potential, others imply total chemical potential. Keep this in mind when
reading articles, books, and exam problems!! There is even more confusion,
since we often call the quantity U the internal energy. If that would mean the
energy without the effect of external potentials, the total energy would be the
sum of these two quantities. We could define
Utotal = Uinternal + NΦ (3.8)
Finternal = Uinternal − T S (3.9)
Ftotal = Finternal + NΦ (3.10)
µ = ˆµ + Φ (3.11)
in order to be consistent. But I prefer to call Utotal the internal energy and only
use µ and not ˆµ.
Origin of internal chemical potential.
The chemical potential is related to the partial derivative of the Helmholtz
free energy F = U − T S. Hence we have
∂U
∂S
µ =
− T
(3.12)
∂N T,V ∂N T,V
If the particles are independent, the internal energy is linear in N and the first
contribution to the chemical potential is independent of N (but does depend
on T and V). In real life, however, adding particles to a system changes the
interactions between the particles that are already present. The energy depends
on N in a non-linear manner. This is called correlation. But even if the particles
are independent, the entropy will be non-linear in N, since adding particles will
change the number of ways the particles already present can be distributed over