Statistical Mechanics - Physics at Oregon State University

3.3. DIFFERENTIAL RELATIONS AND GRAND POTENTIAL. 49
dS =
∂S
dU +
∂U V,N
∂S
∂S
dV +
dN (3.24)
∂V U,N ∂N U,V
Consider a reversible change in the state of the system in which we keep T
and V constant but change S,U, and N:
∂S
∂S
(∆S)T,V =
(∆U)T,V +
(∆N)T,V (3.25)
∂U V,N
∂N U,V
After dividing by (∆N)T,V and taking the limit of the variations to zero we find
∂S ∂S ∂U ∂S
=
+
(3.26)
∂N T,V ∂U V,N ∂N T,V ∂N U,V
∂S
−
∂N T,V
1
∂U ∂S
=
(3.27)
T ∂N T,V ∂N U,V
1 ∂F ∂S
=
(3.28)
T ∂N ∂N
from which we get, using the definition of the chemical potential,
∂S
= −
∂N
µ
T
T,V
U,V
U,V
(3.29)
The relation between the differential changes in dS and dU,dV , and dN is
now very simple and is usually written in the form
dU = T dS − pdV + µdN (3.30)
This is the standard formulation of the first law, as expected. It confirms the
point we learned in thermodynamics, that with S,V,N as independent variables
the partial derivatives of U give the remaining dependent variables. The set
S,V,N is therefore a natural set for U. Of course, if we for example use T,V,
and N as independent variables we can also calculate U(T,V,N) and the relation
between the differentials. In this case, however, the coefficients are not simple
functions of the dependent variables, but more complicated response functions:
dU = CV dT +
−p + T
∂S
∂U
dV +
dN (3.31)
∂V T,N ∂N T,V
The Helmholtz free energy F was introduced exactly for that reason, because it
has T,V,N as natural independent variables:
dF = −SdT − pdV + µdN (3.32)
One important class of processes are those which occur at constant T,V, and
N. A second group keeps T,V, and µ constant. This is certainly true in many