4.2 Fundamental Relations for Closed Simple Systems 159
Irreversible thermodynamicsornonequilibrium thermodynamicsis an extended
version of thermodynamics that deals with rates of entropy production and with rates
of processes and their driving forces. In irreversible thermodynamics, Eq. (4.2-3) is
assumed to be valid for nonequilibrium changes if the deviation from equilibrium is not
too large. This assumption is an additional hypothesis and does not follow from thermo-
dynamics. See the Additional Reading section for further information on irreversible
thermodynamics.
The equilibrium macroscopic state of a one-phase simple system is specified by
c+2 independent variables, wherecis the number of independent substances (com-
ponents) in the system. If the system is closed, the amounts of the substances are fixed
and only two variables can be varied independently. We takeUto be a function ofS
andVfor a simple closed system. An infinitesimal change inUthat corresponds to a
reversible process is given by the fundamental relation of differential calculus:dU(
∂U
∂S
)
V,ndS+(
∂U
∂V
)
S,ndV
(simple closed system;
reversible processes) (4.2-4)where the single subscriptnmeans that the amounts of all substances present are fixed.
Comparison of Eqs. (4.2-3) and (4.2-4) gives us two important relations:(
∂U
∂S)
V,nT (4.2-5)
(
∂U
∂V
)
S,n−P (4.2-6)
Maxwell Relations
From theEuler reciprocity relationshown in Eq. (B-13) of Appendix B, we can write(
∂^2 U
∂S∂V)
n(
∂^2 U
∂V ∂S
)
n(4.2-7)
A second derivative is the derivative of a first derivative,
(
∂^2 U
∂V ∂S)
n(
∂T
∂V
)
S,n(4.2-8)
(
∂^2 U
∂S∂V
)
n−
(
∂P
∂S
)
V,n(4.2-9)
Therefore(
∂T
∂V)
S,n−
(
∂P
∂S
)
V,n(a Maxwell relation) (4.2-10)