Page 36 - Thermodynamics of Biochemical Reactions

30 Chapter 2 Structure of Thermodynamics

Thus the Gibbs-Duhem equation represents one of the 2D thermodynamic
potentials that can be defined for a system, but this thermodynamic potential is
equal to zero. It should be emphasized that there is a single Gibbs-Duhem
equation for a one-phase system and that it can be derived from U, H, A, or G.
When work other than PV work is involved in a system (see Section 2.7),
Legendre transforms can be used to introduce intensive variables in addition to
T and P as natural variables in the fundamental equation for the system. Each
Legendre transform defines a new thermodynamic potential that needs a symbol
and a name. Since every system has 2O possible thermodynamic potentials, and
each needs a symbol and a name, nomenclature becomes a problem. Callen (1985)
showed how each possible thermodynamic potential can bc given an unambigu-
ous symbol. Callen nomenclature uses the symbol U[...], where ... is a list of the
intensive variables introduced as intensive variables in dcfining the particular
thermodynamic potential based on the internal energy. Thus the enthalpy H can
be represented by U[P], the Helmholtz energy A can be represented by U[T],
and the Gibbs energy G can be represented by U[T,P]. Since all possible
thermodynamic potentials can be represented in this way, this is a good method
to use when there is a possibility of confusion. However, in practice, it is
convenient to use symbols like U', H', A', and G' to represent transformed
properties that are similar to U, H, A, and G. When this is done, it is important
to specify which intensive properties have been introduced in defining these
primed properties.
The number of Maxwell equations for each of the possible thermodynamic
potentials is given by D(D - 1)/2, and the number of Maxwell equations for the
thermodynamic potentials for a system related by Legendre transforms is
[D(D - 1)/2]2D. Examples are given in the following section.

2.6 THERMODYNAMIC POTENTIALS FOR A
SINGLE-PHASE SYSTEMS WITH ONE SPECIES

The fundamental equation for U for a single-phase system with one species is
dU = TdS - PdV+ pdn (2.6-1)
Integration of this fundamental equation at constant values of the intensive
variables yields
U = TS + PV+ pn (2.6-2)

Since there are D = 3 natural variables, there are 23 - 1 = 7 possible Legendre
transforms. The Legendre transforms defining H, A, and G are given in equations
2.5-1 to 2.5-3, and the four remaining Legendre transforms are

U[p] = U - pn (2.6-3)
U[P,p] = U +PV- pn (2.6-4)
U[Tp] = U - TS - pn (2.6-5)

U[7;P,p] = u + PV- TS - p/1= 0 (2.6-6)
These four Legendre transforms introduce the chemical potential as a natural
variable. The last thermodynamic potential U[T, P,p] defined in equation 2.6-6 is
equal to zero because it is the complete Legendre transform for the system, and
this Legendre transform leads to the Gibbs-Duhem equation for the system.
Three of the eight thermodynamic potentials for a system with one species are
frequently used in statistical mechanics (McQuarrie, 2000), and there are generally
accepted symbols for the corresponding partition functions: U[q = A =
-RTln Q, where Q is the canonical ensemble partition function: