Page 32 - Thermodynamics of Biochemical Reactions

26 Chapter 2 Structure of Thermodynamics

chemical reaction, F = N, - 1 + 2 = N, + 1, as shown by equation 2.4-2. The
independent intensive variables can be chosen to be 7: P, p,, . . . , /ihrS- I or 7; (pi I
or P, [pi].
Since F is the symbol for the number of independent intensive variables for
a system, it is also useful to have a symbol for the number of natural variables
for a system. To describe the extensive state of a system, we have to specify F
intensive variables and in addition an extensive variable for each phase. This
description of the extensive state therefore requires D variables, where D = F + p.
Note that D is the number of natural variables in the fundamental equation for a
system. For a one-phase system involving only PI/ work, D = N, + 2, as dis-
cussed after equation 2.2-12. The number F of independent intensive variables and
the number D of natural variables for a system are unique, but there are usually
multiple choices of these variables. The choice of independent intensive variables
F and natural variables D is arbitrary, but the natural variables must include as
many extensive variables as there are phases. For example, for the one-phase
system described by equation 2.2-8, the F = N, + 1 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, n,, n2,. . . , nN, or 7: P. xi, xz,. . . ,.Y,~,., and n (total amount in
the system).

2.5 LEGENDRE TRANSFORMS FOR THE DEFINITION
OF ADDITIONAL THERMODYNAMIC
POTENTIALS

The internal energy U has some remarkable properties and leads to many
equations between the thermodynamic properties of a system, but S, V; and in,)
are not convenient natural variables, except for an isolated system. As shown in
Section 2.2, Legendre transforms can be used to introduce other sets of N, + 2
natural variables. A Legendre transform is a change in natural variables that is
accomplished by defining a new thermodynamic potential by subtracting from the
internal energy (or other thermodynamic potential) one or more products of
conjugate variables. As mentioned after equation 2.2-2, examples of conjugate
pairs are T and S, P and r/; and p, and n,. More conjugate pairs are introduced
in Section 2.7. Callen (1985) emphasizes that no thermodynamic information is
lost in making a Legendre transform. For reviews on Legendre transforms, see
Alberty (1994d) and Alberty et al. (2001).
Legendre transforms are also used in mechanics to obtain more convenient
independent variables (Coldstein, 1980). The Lagrangian L is a function of
coordinates and velocities, but it is often more convenient to define the Hamil-
tonian H with a Legendre transform because the Hamiltonian is a function of
coordinates and momenta. Quantum mechanics is based on the Hamiltonian
rather than the Lagrangian.
In this section we will consider the Legendre transforms that define the
enthalpy H, Helmholtz energy A, and Gibbs energy G.

H=U+PV (2.5-1)
A=U-TS (2.5-2)

G= U+PV- TS (2.5-3)

If a system can be described by dU = TdS - PdLc: there are only four
thermodynamic potentials that can be defined in this way.
We will consider only the equations for the Gibbs energy and the enthalpy.
According to equation 2.5-3, the differential of G is given by

dG = dU + PdV+ VdP -- TdS - SdT (2.5-4)