Statistical Mechanics - Physics at Oregon State University

Chapter 3
Systems with a variable
number of particles.
3.1 Chemical potential.
More state variables?
The extensive system parameters we have studied are S,V,N, and U. Until
now we have allowed for variations in S (defining temperature) and V (defining
pressure). In many processes the number of particles is also allowed to change.
The corresponding intensive parameter is the chemical potential, µ. In this
section we will discuss the meaning of the chemical potential. In addition, the
system could have other extensive variables. Changes in these variables lead
to new intensive variables. The recipes in those cases are constructed in a
straightforward manner by copying the steps in the development of T,p, and µ.
Two systems are in diffusive contact if particles can flow back and forth
between them. Consider two systems with temperature T, and fixed volumes.
The number of particles in system 1(2) is N1(N2). The equilibrium configuration
is found by minimizing the Helmholtz free energy of the total system, since the
total system is at constant temperature and volume. If ∆N particles flow from
1 to 2, the change in free energy is
∆F =
∂F1
(−∆N) +
∂N T,V
We define the chemical potential by
µ(T, V, N) =
∂F2
(+∆N) (3.1)
∂N T,V
∂F
(T, V, N) (3.2)
∂N T,V
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