Page 96 - Thermodynamics of Biochemical Reactions

5.1 Chemical Equations as Matrix Equations 91

where R is the number of independent reactions. This equation can be interpreted
by pointing out that the number N, of unknown concentrations of species in an
equilibrium calculation is equal to the number of C components plus the number
R of independent reactions. Note that there is a conservation equation for each
component and an equilibrium constant expression for each reaction The species
in a chemical reaction system can be divided into C components and R
noncomponents. Various choices of components and non-components can be
made, but the numbers C and R are unique for a system of chemical reactions.
Consider a gaseous reaction system in which the only reaction is

CO + 3H2 = CH, + H,O (5.1-4)
When stoichiometric numbers are taken to be signed quantities, this chemical
equation can be written as

CO - 3H2 + CH, + HZO = 0 (5.1-5)
This may not look like a matrix equation, but it actually is. When we replace the
chemical formulas with column vectors that give the numbers of C, H, and 0
atoms in each species, equation 5.1-5 can be written as

0 i:r 13 [:I
-0 -3 2 + 4 + 2 = 0 (5.1-6)

This equation can be written as a matrix multiplication:

(5.1-7)

where the conservation matrix A is given by

CO H, CH, H,O
C 1 0 1 0
A= (5.1-8)
H 0 2 4 2
0 1 0 0 1
In the A matrix there is a column for each species and a row for each component.
Note that the components are taken to be atoms of C, H, and 0. (In equation
5.1-15 we will see that other choices of components can be made.) The
stoichiometric number matrix corresponding with equation 5.1 -5 is

rx5.1-4
CO -1
v = H, -3 (5.1-9)
CH, 1
H,O 1
In the v matrix there is a column for each reaction and a row for each species,
with the species in the same order as in the columns of the A matrix. The matrix
multiplication in equation 5.1-7 is represented in general by

AV = 0 (5.1 - 10)
Note that the matrix product of the C x N, conservation matrix and the N x R
stoichiometric number matrix is a C x R zero matrix.