Page 95 - Thermodynamics of Biochemical Reactions

90 Chapter 5 Matrices in Chemical and Biochemical Thermodynamics

for the underlying chemical reactions. As in the case of chemical reactions, the
apparent conservation matrix is related mathematically to the apparent
stoichiometric number matrix. Matrix notation is also useful in writing fundamen-
tal equations and Gibbs-Duhem equations and in calculating equilibrium compo-
sitions. There will be more applications of matrix operations in subsequent
chapters. More information on matrices is to be found in Smith and Missen
(1982) and in a textbook on linear algebra, such as Strang (1988).

5.1 CHEMICAL EQUATIONS AS MATRIX EQUATIONS

The conservation relationships in chemical reactions can be represented by
reaction equations or by conservation equations. When using reaction equations
in thermodynamics, it is important to remember that a reaction equation can be
multiplied by any positive or negative integer without changing the equilibrium
composition that will be calculated. Of course, the expression for the equilibrium
constant K of the reaction must be changed appropriately. When an equilibrium
calculation is made on a multireaction system, only an independent set of
reactions is used. An independent set of reaction equations is one in which no
equation in the set can be obtained by adding or subtracting other reactions in
the set. We will find that linear algebra provides a much more practical test of
independence. The number R of independent reactions in a set is unique, but the
particular reactions in the set are not. Any two reactions in a set of independent
reactions can be added, and this reaction can be used to replace one of the two
reactions without changing the equilibrium concentration that will be calculated.
These remarks apply in thermodynamics, but not in discussing rates of reactions.
The corresponding conservation equations are less familiar, but they contain
the same information as a set of independent chemical reactions. The conservation
equations for a system containing N, species are given by

N 5
Nijnj = nCi (5.1-1)
j= I
where n,, is the amount of component i, IV,~ is the number of units of component
i in species j, and nj is the amount of species j. For chemical reactions the
conservation equations are usually written in terms of amounts of elements and
electric charge, but they can also be written in terms of specified groups of atoms.
The things that are conserved are referred to as components. The amounts of
components in a closed system are not changed by chemical reactions. The
conservation equations for the components in a reaction system must be indepen-
dent; that is, no conservation equation in the set can be calculated by adding and
subtracting the other equations in the set. The number C of components for a
chemical reaction system is unique, but the components that are chosen are not.
Equation 5.1-1 for a reaction system can be written in matrix form as

An = n, (5.1-2)

where A is the conservation matrix made up of the N,, values, with a row for each
component and a column for each species. In equation 5.1-2, n is the column
matrix of amounts of species and n, is the column matrix of amounts of
components. The matrix product of the C x N, conservation matrix A and the
N, x 1 amount of species matrix II is equal to the C x 1 matrix tz, of amounts of
components. Equation 5.1-2 can be used to calculate amounts of components in
more complicated systems (see equation 5.1-27). The number N, of different
species in a system of chemical reactions is given by

N,=C+R (5.1-3)