Page 26 - Introduction to chemical reaction engineering and kinetics

8 Chapter 1: Introduction

where Anso, = the change in moles of SO, by reaction, and similarly for Ano, and AnSo3.
The coefficients in equations (C) and (D) form a matrix A in which each column represents
a species and each row an element:
( 1 09
1 0 1
A=223

The entries in A are the subscripts to the elements in the molecular formulas of the sub-
stances (in an arbitrary order). Each column is a vector of the subscripts for a substance,
and A is called a formula matrix.
In this case, A can be transformed by elementary row operations (multiply the second
row by 1/2 and subtract the first row from the result) to the unit-matrix or reduced row-
echelon form:

The form in (F) provides a solution for Anso and AnO in equations (C) and (D) in terms
of Anso,. This is
Anso = -AnsOs; and Ano, = -(1/2)Anso, ((-3

which may be written as

The numbers - 1, - 1/2, and 1 in (G’) are in proportion to the stoichiometric coefficients
in equation (B), which provides the same interpretation as in (G) or (G’). The last column
in (F) gives the values of the stoichiometric coefficients of SO, and 0, (on the left side)
in a chemical equation involving one mole of SO3 (on the right side):

+1so, + 10 = lS0, (W
2 2
or, in conventional form, on elimination of the fraction:

2s0, +o, = 2s0, U-U

SO, and O2 are said to be component species, and SO, is a noncomponent species. The
number of components C is the rank of the matrix A (in this case, 2):

rank (A) = C (1.4-6)

Usually, but not always, C is the same as the number of elements, M. In this sense, C is
the smallest number of chemical “building blocks” (ultimately the elements) required to
form a system of specified species.

More generally, a simple system is represented by

-$ viAi = 0 (l.4-7)
i = l