Statistical Mechanics - Physics at Oregon State University

32 CHAPTER 2. THE CANONICAL ENSEMBLE
dU = T dS − pdV (2.35)
If other extensive parameters are allowed to change, we have to add more
terms to the right hand side. The change in energy U, which is also called
the internal energy, is equal to the work done on the system −pdV plus the
flow of heat to the system if the system is in contact with some reservoir, T dS.
When the entropy of the system does not change, the complete change in internal
energy can be used to do work. Therefore, the internal energy is a measure of
the amount of work a system can do on the outside world in processes in which
the entropy of the system does not change. Most processes are of a different
nature, however, and in those cases the internal energy is not useful.
An important group of processes is described by changes in the volume at
constant temperature. In order to describe these processes we introduce the
Helmholtz free energy F:
which has the differential
F = U − T S (2.36)
dF = −SdT − pdV (2.37)
Therefore, in a process at constant temperature the work done on the system
is equal to the change in the Helmholtz free energy. The Helmholtz free energy
measures the amount of work a system can do on the outside world in processes
at constant temperature. Since both the entropy and temperature are positive
quantities, the Helmholtz free energy is always less than the internal energy.
This is true in general. The amount of energy which can be retrieved from a
system is usually smaller than the internal energy. Part of the internal energy
is stored as heat and is hard to recover.
2.5 Changes in variables.
The variables we have introduced so far are S,T,U,V, and p. These variables are
not independent, and if we specify two the other three are determined. We can
choose, however, which two we want to specify. The cases we have considered
either use (U,V) as independent variables and calculate all other quantities from
S(U,V) or use (T,V) as independent variables and calculate all other quantities
through the partition function Z(T, V ) and U(T,V) as a derivative of log(Z).
This implies restrictions on functions like S(U,V). For example, if we know
S(U,V) the temperature follows from the partial derivative
∂S
∂U . If we specify
V
the temperature, the energy U is found by solving
1
T =
∂S
∂U
V
(U, V ) (2.38)
Since we know that this equation has a unique solution, there are no two
values of U for which S(U,V) has the same slope! Hence T is a monotonous