Page 34 - Thermodynamics of Biochemical Reactions

28 Chapter 2 Structure of Thermodynamics

Substituting the integrated fundamental equation for U (equation 2.2-14) in
the Legendre transforms defining H, A, and G shows that

H = TS + Cp,n, (2.5 - 10)
A = -PV+ xprnr (2.5-1 1)
G = C&n, (2.5-12)

The interesting thing about these equations is that only the Gibbs energy G can
be calculated by adding contributions from individual species. These ther-
modynamic potentials can be determined as functions of other variables, but only
when they are determined as functions of natural variables can all of the ther-
modynamic properties be obtained by taking partial derivatives. Equations 2.5- 10
to 2.5-1 2 can also be obtained by integrating the corresponding fundamental
equations at constant values of the intensive variables.
The fundamental equation for the Gibbs energy (2.5-5) yields the following
Maxwell equations:

(2.5-1 3)

(2.5- 14)

(2.5-15)

(2.5- 16)

where S,(i) is the molar entropy of species i and V,(i) is its molar volume.
The Helmholtz energy is not very useful as a crterion for spontaneious change
and equilibrium in biochemistry because experiments are not done at constant
volume. However, the enthalpy is important in biochemistry because it is
connected with heat evolution and the change of the equilibrium constant with
temperature. The fundamental equation for the enthalpy is

N \
dH = TdS+PdV+ p,dn, (2.5- 17)
r=l
Since the enthalpy is defined by H = U + PI! its total differential is dH =
dU + PdV+ VdP. Substituting the equation dU = dq - PdK given earlier in
Section 2.2, yields dH = dq + VdP. At constant pressure the change in enthalpy
AH is equal to the heat q absorbed by the system in the process, which may be
irreversible. Thus thc change in enthalpy AH can be determined calorimetrically.
The change in enthalpy can also be determined using the Gibbs-Helmholtz
equation, which is introduced in the next paragraph, without using a calorimeter.
Equations 2.5-1 and 2.5-3 show that G = H - TS. Substituting the expression
for S from equation 2.5-7 yields

(2.5-1 8)

This is referred to as a Gibbs-Helmholtz equation, and it provides a convenient
way to calculate H if G can be determined as a function of 7: P, and [nrj. There
is a corresponding relation between the internal energy U and the Helmholtz
energy, which is defined by equation 2.5-2:

(2.5-1 9)
P In ;
This is also referred to as a Gibbs-Helmholtz equation.