Page 97 - Thermodynamics of Biochemical Reactions

92 Chapter 5 Matrices in Chemical and Biochemical Thermodynamics

Equation 5.1-10 applies to a multireaction system. For example, if in addition
to reaction 5.1-4, the following reaction occurs:

CO + H,O = CO, + H, (5.1-11)
We can add a column to the conservation matrix for CO, and a column to the
stoichiometric number matrix for this reaction to obtain

1-1 -l\

(5.1-2)

The new conservation matrix A is given by

CO H, CH, H,O CO,
c1 0 1 0 1
A= ( 5.1 - 13)
HO 2 4 2 0
0 1 0 0 1 2
The new stoichiometric number matrix v is given by
rx5.1-4 rx5.1-11
co -1 -1
H, -3
V= ( 5.1 - 14)
CH, 1
H,O 1 -
co, 0
Now we want to show that the conservation matrix can be written in terms
of other components. The easiest way to obtain an equivalent A matrix is to make
a row reduction (Gaussian elimination) to obtain the canonical form of the matrix
with an identity matrix in the left side. An identity matrix is a square matrix that
has ones on the diagonal with the other positions occupied by zeros. Subtracting
the first row of matrix 5.1-13 from the last row and using the difference with a
change in sign to replace the last row, subtracting the last row from the first, and
subtracting two times the third row from the second yields the canonical form of
the conservation matrix:

CO H, CH, H,O CO,
C O 1 0 0 1 2
A= (5.1 - 15)
H , O l O 3 2
CH, 0 0 1 -1 -1
The canonical form of a matrix is readily obtained using RowReduce in Muth-
matica. In equation 5.1-1 5 the conservation equations are for the conservation of
CO, H,, and CH, rather than for the atoms of C. H, and 0: in other words. the
components have been chosen to be CO, H,, and CH,. The last two columns
show how the noncomponents H,O and CO, are made up of the components.
They show that H,O is made up of CO + 3H, ~ CH,, and CO, is made up of
2CO + 2H, ~ CH,. If one of the conservation equations were redundant, it
would yield a row of zeros that would be dropped. Since there are three rows in
this A matrix that are not all zeros after row reduction, the A matrix has a rank
of 3, and so the number of components is given by

C = rank A (5.1 - 1 6)