Transport and Equilibrium Reactions - Environmental Modeling - page 150

The following reaction example between hydrogen H , nitrotriacetic acid NTA
(a chelating agent in detergents influencing the metal ion activity in aqueous
systems), and cobalt (II) Co demonstrates the case:

HþNTA !

HNTA

CoNTA ! CoþNTA

CoNTAþH ! CoþNTA

There are five species involved in three reactions and the corresponding reaction
matrix is:

it is easy to check if the above given requirement is fulfilled. The
rank of the matrix is the maximum number of independent reactions. In the example
case MATLAB

In MATLAB

®

shows:

®

In fact, only two of the given reactions are independent. The rank is lower than 3
and thus not maximal. The chemical system is treated in more details by Fang et al.
( 2003 ). In order to obtain a maximum rank for the given example, one of the three
reactions has to be omitted. The resulting 2

5 matrix has maximum rank.
Without mentioning we assume a matrix of maximum rank in all following
theoretical derivations.

8.3 Speciation Calculations

The equation for a system of species in equilibrium is given by:

S T r eq ¼ 0 (8.15)

The problem with the system ( 8.15 ) is that it can not be computed directly,
because the exchange rates of the equilibrium reactions r eq are neither known,
nor are they given by an explicit expression. In order to reach an appropriate
formulation, the equations have to be added up in a way that eliminates the
unknown reaction rates. There are N s equations in the system ( 8.15 ), and there
are N r equilibrium conditions for the reactions. Usually N s >N r holds, i.e. the
system is overdetermined.
In order to complete the system ( 8.15 ), N s -N r additional conditions in form

of specified values for total concentrations can be given (Steefel and