Page 99 - Thermodynamics of Biochemical Reactions

94 Chapter 5 Matrices in Chemical and Biochemical Thermodynamics

This can be illustrated by starting with the transposed stoichiometric number
matrix 5.1-14, which is

CO H, CH, H,O CO,
vT = rx 5.1-4 -1 -3 1 1 0 (5.1-23)
rx 5.1-11 -1 1 0 -1 1

Row reduction of matrix 5.1-23 yields
CO H2 CH, H,O C02
v'=1 0 -z - -~ (5.1-24)
1
1
3
1
0 1 -- _- 4
1
1 -
4
2
The transpose of the conservation matrix can be obtained by using
(5.1-25)
= ( JCZj
This yields
CO H2 CH, H,O CO,
(5.1-26)

co2 4 : 0 0 1
_-
~ 3
This looks different from equation 5.1-8, but it yields 5.1-15 on row reduction,
which shows that the two matrices are equivalent.
Muthrrnutica is very useful for carrying out these matrix operations. The
operation for row reduction is RowReduce, and the operation for calculating a
basis for the null space is Nullspace. Row reduction is also used to determine
whether the equations in a set of conservation equations or reaction equations are
independent. Rows that are dependent come out as all zeros when this is done.
and they must be deleted because they do not provide any useful information.
We return to equation 5.1-1 for the system we have been discussing:

CO H, CH, H20 CO,
C 1 0 1 0 1 n(C0) nc(C)

4H2O)
n(CO2)
The product of a C x N, matrix and a N, x 1 matrix is a C x 1 matrix; note that
N, disappears as one of the dimensions of the resultant matrix. The amounts of
components in a reaction system are independent variables and consequently do
not change during a chemical reaction. The amounts of species are dependent
variables because their amounts do change during chemical reactions. Equation
5.1-27 shows that A is the transformation matrix that transforms amounts of
species to amounts of components. The order of the columns in the A matrix is
arbitrary, except that it is convenient to include all of the elements in the species
on the left so that the canonical form can be obtained by row reduction. When
the row-reduced form of A is used, the amounts of the components CO, H2, and
CH, can be calculated (see Problem 5.1).
As an example of a set of chemical reactions in aqueous solution that are of
biochemical interest, consider the hydrolysis of adenosine triphosphate to aden-
osine diphosphate and inorganic phosphate in the neighborhood of pH 7. The