Page 33 - Thermodynamics of Biochemical Reactions

2.5 Legendre Transforms for the Definition of Additional Thermodynamic Potentials 27

Substituting equation 2.2-8 yields the fundamental equation for the Gibbs energy
for a one-phase system without chemical reactions:

N,
dG = -SdT+ VdP + 2 ,uidni (2.5-5)
i= 1
This shows that the natural variables of G for a one-phase nonreaction system are
7; P, and (n,}. The number of natural variables is not changed by a Legendre
transform because conjugate variables are interchanged as natural variables. In
contrast with the natural variables for U, the natural variables for G are two
intensive properties and N, extensive properties. These are generally much more
convenient natural variables than S. V, and {n,). Thus thermodynamic potentials
can be defined to have the desired set of natural variables.
It is important to understand that the change in variables provided by using
a Legendre transform is quite different from the usual (much more frequent) type
of change in variables. For example, in Chapter 1 functions of [H'] were
converted to functions of pH by simply substituting [H'] = When a
Legendre transform of a thermodynamic potential is defined, the new variable
that is introduced is a partial derivative of that thermodynamic potential. For
example, when U is known as a function of S, V, and {n,}, the enthalpy is defined
by use of H = U + PV= U + V(dU/dV),, and the natural variables of H are
indicated by H(S, -(dU/dV),) or H(S, P). When the Gibbs energy is defined by
use of G = U + PV- TS = U + V(dU/?V), - S (dU/dS)., and the natural vari-
ables of G are indicated by G((dU/dS) v, -(?U/dV),) or G(7; P).
Since the natural variables of the Gibbs energy are 7; P, and {n,}, calculus
yields

Comparison with equation 2.5-5 indicates that

(2.5-7)

(2.5-8)

(2.5-9)

Thus, if G can be determined as a function of 7; P, and {ni}, all of the
thermodynamic properties of the system can be calculated. These N, + 2 equa-
tions (that is equations 2.5-7 to 2.5-9) are often referred to as equations of state.
Only N, + 1 equations of state are independent, and so if N, + 1 of them can be
determined experimentally, the remaining equation of state can be calculated.
The criterion of equilibrium for this one-phase system without chemical
reactions is dG d 0 at constant 7; P, and (n,}. In other words, the Gibbs energy
decreases in a spontaneous change in a system with constant T, P, and in,}. For
this system the F = N, + 1 independent intensive variables can be chosen to be
7; P, xl, x2,. . . , xN- and the D = N, + 2 natural variables can be chosen to be
7; P, xl, x2,. . ,xN-l and n (total amount in the system), or 7; P, n,, n2,. . . , n,,.
.
The derivation in equations like 2.5-4 to 2.5-9 can be repeated for H and A.
This shows that the natural variables for H are S, P, and (n,}, and for A are 7; V,
and {nL}. These thermodynamic potentials provide the following criteria for
spontaneous change and equilibrium: dH < 0 at constant S, P, and (n,); dA < 0
at constant 7; and {n,}.