# 6.7: Untitled Page 130

- Page ID
- 18263

## Chapter 6

### Atomic

ecies

Sp

3:

*N R * *N R*

*N R * *N R * *N R * *N R*

0

(6‐35c)

31

1

32

2

33 3

34

4

35

5

36

6

This 3 6 system of equations always has the *trivial solution* *R * 0 , for *A*

*A * 1 *, * 2 *, .., * 6 , and the necessary and sufficient condition for a *non‐trivial solution* to exist is that the *rank* of the atomic matrix be less than the number of molecular species. For this special case of three atomic species and six molecular species, we express this condition as (Kolman, 1997)

Non‐trivial solution (Special):

r *rank * *N * 6

*JA *

(6‐36)

By *rank* we mean explicitly the *row rank* which represents the *number of linearly* *independent equations* contained in Eqs. 6‐35. It is possible that all three of Eqs. 6‐35 are independent and the rank associated with the atomic matrix is three, i.e., r *rank * 3 . On the other hand, it is possible that one of the three equations is a *linear combination* of the other two equations and the rank is two, i.e., r *rank * 2 . The general con it

d ion concerning the rank of e

th atomic matrix in

Eq. 6‐22 is given by

Non‐trivial solution (General):

r *rank * *N * *N*

*JA *

(6‐37)

hen

W

the rank is equal to *N * 1 , we have a special case of Eq. 6‐22 that leads to a

*single independent stoichiometric action*

*re*

. In that

special case, the *N * 1 net

rates of

roduction

p

can be specified in terms

of *R * and

Eq. 6‐22 can be

expressed as

*N*

*R*

*R *

(6‐38a)

*A*

*AN*

*N*

*R*

*R *

(6‐38b)

*B*

*BN*

*N*

*R*

*R *

(6‐38c)

*C*

*CN*

*N*

.

.

*R*

*R *

(6‐38 *n‐1*)

*N*1

*N*1 *N*

*N*

ere

H

,

, etc., are often referred to as *stoichiometric coefficients*; however, *AN*

*BN*

the authors prefer to identify these quantities as *elements of the* *pivot matrix* as dicated

in

in Example 6.1.

239

*6.2.4 Stoichiometric equations*

It is crucial to understand that Eqs. 6‐38 are based on the concept that atoms are neither created nor destroyed by chemical reactions. The bookkeeping associated with the conservation of atoms is known as stoichiometry, thus it is appropriate to refer to the equations given by Eqs. 6‐38 as stoichiometric equations. In addition, it is appropriate to identify Eqs. 6‐38 as a case in which there is a *single* independent net rate of production, and that this *single* independent net rate of production is identified as *R *

. In order to be clear about

*N*

stoichiometry and chemical kinetics, we place Eq. 6‐7 side‐by‐side with Eq. 6‐38a to obtain

2

*k c*

*A*

*R*

*, * chemical kinetics

*A*

1 *k c*

(6‐39a)

*A*

*R*

*R , * stoichiometry

(6‐39b)

*A*

*AN*

*N*

re

He we note that the symbol *R * in Eq. 6‐39a has *exactly the same physical* *A*

*significance * as *R * in Eq. 6‐39b. In both cases *R * is defined by Eq. 6‐17.

*A*

*A*

However, the description of the right hand side of these two representations of *R * is quite different. The right hand side of Eq. 6‐39a is a *chemical kinetic relation* *A*

while the right hand side of Eq. 6‐39b is a *stoichiometric relation*. The *chemical* *kinetic representation* depends on the complex processes that occur when molecules dissociate, active complexes are formed, and various molecular fragments coalesce to form products. The *stoichiometric representation* is based solely on the concept that atoms are conserved. The chemical kinetic representation may depend on temperature, pressure, and the presence of catalysts, while the stoichiometric representation remains *invariant* depending only on the conservation of atoms. In this chapter, and throughout most of the text, we will deal with stoichiometric relations based on Axiom II. In Chapter 9

we will examine chemical reaction rate equations, and in that treatment we will be very careful to identify *stoichiometric constraints* that are associated th

wi

*elementary stoichiometry*.

XAMPLE

E

6.1. Complete combustion of ethane

In this example we consider the complete combustion of ethane, thus the molecular species under consideration are identified as (see Figure 6‐1) C H *, * O *, * H O *, * CO

2

6

2

2

2