Page 98 - Thermodynamics of Biochemical Reactions

5.1 Chemical Equations as Matrix Equations 93

In matrix 5.1-15 the three components are CO, H,, and CH,. However, if the
order of the columns were changed, other components would be chosen. Thus the
conservation matrix is not unique. A set of components must contain all the
elements that are not redundant. The rank of the stoichiometric number matrix
is equal to the number of independent reactions.
R = rankA (5.1 - 1 7)

Thus equation 5.1-3 (N, = C + R) can be written
N, = rank A + rank v (5.1 - 1 8)
Next we want to consider the fact that a stoichiometric number matrix can
be calculated from the conservation matrix, and vice versa. Since Av = 0, the A
matrix can be used to calculate a basis for the stoichiometric number matrix v.
The stoichiometric number matrix v is referred to as the null space of the A
matrix. When the conservation matrix has been row reduced it is in the form
A = [Z,,Zl, where Z, is an identity matrix with rank C. A basis for the null space
is given by

\.’ = ( YRZ) ( 5.1 - 19)

where Z is C x R and Z, is an identity matrix with rank R. It is necessary to say
that the null space calculated using equation 5.1-19, or using a computer, is a basis
for the null space because the stoichiometric number matrix for a system of
reactions can be written in many different ways, as mentioned before (equation
5.1-1). All of these forms of the v matrix satisfy equation 5.1-10.
Equation 5.1-19 shows that the stoichiometric number matrix corresponding
with conservation matrix 5.1-15 is

rx 5.1-14 rx 5.1-21
CO -1 -2
H, -3 -2
V= (5.1-20)
CH, 1 1
H,O 1 0
co, 0 1

Note that reaction 5.1-11 has been replaced by
2CO + 2H, = CH, + CO, (5.1-21)
which is stoichiometrically correct and independent. The correct equilibrium
composition can be calculated with either set of reactions. Stoichiometric number
matrices 5.1-14 and 5.2-20 have the same row-reduced form, and so they are
equivalent. This is an example of the fact thai the stoichiometric number matrix
for a system is not unique. It is important to realize that the equilibrium
composition that is calculated for a system of reactions is valid for all possible
reactions that can be obtained by adding and subtracting the reactions used in
the calculation of the equilibrium composition.
Equation 5.1-10 provides the means for calculating a basis for the
stoichiometric number matrix that corresponds with the conservation matrix.
Similarly the transposed stoichiometric number matrix provides the means for
calculating a basis for the transposed conservation matrix. This is done by using
the following equation, which is equivalent to equation 5.1-10:

vTAT = 0 (5.1 -22)
where “T” indicates the transpose. The transpose of a matrix is obtained by
exchanging rows and columns. Thus a set of independent reactions for a reaction
system can be used to calculate a set of conservation equations for the system.