Page 105 - Thermodynamics of Biochemical Reactions

100 Chapter 5 Matrices in Chemical and Biochemical Thermodynamics

The fundamental equation is

dG = - SdT+ VdP + p(fum2-)dn(fum2-) + p(ma12-)dn(ma12-)
+ ,u(Hfum- )dn(Hfum-) + p(Hmal-)dn(Hmal-) + ,u(H+)dn(H+)
+ P(H,O)dn(H,O) (5.4-8)
It is hard to divide the six chemical work terms into three terms for three chemical
reactions, but this can be done using equation 5.4-4 with the following
stoichiometric number matrix:
rx 5.4-5 rx 5.4-6 rx 5.4-7
fum2 -1 1 0
ma]' 1 0 1
V= (5.4-9)
Hfum - 0 -1 0
Hmal 0 0 -1
H+ 0 1 1
H2O 1 0 0
The row matrix of chemical potentials of species is p = {(p(fum2-), ,u(ma12-),
p(Hfum-), p(Hmal-), ,u(H+), p(H20)]}, and the column matrix of extents of
reaction is 5 = {{(,I, {(,), (r,)}. Therefore the last term in equation 5.4-4 is given
by

pvdc = (-,u(fum2-) - p(H,O) + ,u(ma12-))d(, + (p(fum2-) + p(H+)
- ,u(Hfum-))d(<, + (p(ma12-) + ,u(H+) - ,u(Hmal-))dt3 (5.4-10)
This shows how the fundamental equation for a system of chemical reactions can
be written in terms of the three extents of reaction (see Problem 5.5.)
It is important to be able to write the fundamental equation for a system of
chemical reactions in terms of components because components are involved in
the criterion for spontaneous change and equilibrium. We have seen earlier
(Section 2.3) that this is done by eliminating one chemical potential from the
fundamental equation with each independent equilibrium condition of the form
C vipi = 0 to obtain
dG = - SdT+ VdP + C ,uCidnCi (5.4-11)

i= 1
where pCi is the chemical potential of the species that corresponds with component
i. This equation can be written in terms of pc, which is the 1 x C matrix of
chemical potentials of components, and nc, which is the C x 1 column matrix of
amounts of components.

dG = - SdT+ VdP + p,dn, (5.4-1 2)
This one-phase system has C + 2 natural variables in agreement with D = F + 1,
where F = C - p + 2 is the number of intensive degrees of freedom. Thus
D = C - 1 + 2 + 1 = C + 2. Since the amounts of components are given by
n, = An (equation 5.1-2), the fundamental equation can also be written in the
form

dG = - SdT+ VdP + p,Adn (5.4-13)
where A is the transformation matrix that converts the matrix n into the matrix
n,. This form of the fundamental equation is useful for setting up the fundamental
equation for consideration of a reaction system described by a particular set of
components.