# 9.14: Untitled Page 230

- Page ID
- 18363

## Chapter 9

If we impose the condition of *local reaction equilibrium* indicated by Eq. 9‐109, we obtain the following two constraints on the reaction rates *R * 0 :

*k c c*

*k c c*

*k c c*

0

(9‐113a)

H

II Br H2

III H B 2

r

IV H HBr

2 *k c*

*k c c*

*k c c*

I Br

II Br H

III H Br

*R*

0

2

2

2

:

(9‐113b)

Br

2

*k c c*

*k c*

0

IV H HBr

V Br

These two results can be used to determine the concentrations of the *Bodenstein* *products* that are given by

*k c*

2 *k k*

*c*

II H2

1

V

Br2

*c*

*, *

*c*

2 *k*

*k*

*c*

(9‐114)

H

Br

I

V

Br2

*k c*

*k c*

III Br

2

IV HBr

On the basis of Eqs. 9‐108 and Eqs. 9‐109 we see that there is only a single independent equation associated with Eqs. 9‐111 and we can use that equation to determine the net rate of production of hydrogen bromide as

2 *k * 2 *k k c c*

II

I

V H2

Br2

*R*

(9‐115)

HBr

1 ( *k*

*k*

)( *c*

*c*

)

IV

III

HBr

Br2

A little thought will indicate that this result has exactly the same form as the experimentally determined reaction rate expression given by Eq. 9‐19.

In this section we have illustrated the use of the *mechanistic matrix* to provide a compact representation of chemical reaction rate equations. When reactive intermediates (Bodenstein products) are involved in the reaction process, and local reaction equilibrium is assumed, it is convenient to represent the *mechanistic* *matrix* in terms of the *stoichiometric matrix* and the *Bodenstein matrix* as illustrated by Eqs. 9‐110 through 9‐112.

**9.4 Matrices **

In this text we have made use of matrix methods to solve problems and to clarify concepts. Here we summarize our knowledge of the matrices associated with the conservation of atomic species and the matrices associated with the analysis of chemical reaction rate phenomena.

429

Atomic matrix

The atomic matrix, A , was introduced in Sec. 6.2 in order to clearly identify the atoms and molecules involved in a particular process, and to provide a compact representation of Axiom II. The construction of the atomic matrix represents a key step in the analysis of chemical reactions since it identifies the molecular and atomic species that we *assume* are involved in the process under consideration. As an example, we consider the atomic matrix for the partial oxidation of ethane. The analysis begins with the following visual representation (see Example 6.4) of the molecules and atoms involved in this process.

Molecular Species C H

O

H O CO CO

C H O

2

6

2

2

2

2

4

*carbon*

2

0

0

1

1

2

(9‐116)

*hydrogen*

6

0

2

0

0

4

*oxygen*

0

2

1

1

2

1

In this case the atomic matrix is given by

2 0 0 1 1 2

Atomic matrix:

A

6 0 2 0 0 4

(9‐117)

0 2 1 1 2 1

and the column matrix of net molar rates of production takes the form

*R*

C2H6

*R*

O2

*R* H O

2

R

(9‐118)

*R*

CO

*R* CO

2

*R*

C

2H4O

In terms of these two matrices Axiom II is given by

Axiom II:

A R

0

(9‐119)

This represents a compact statement that atomic species are neither created nor destroyed by chemical reactions. The atomic matrix can always be expressed in *row reduced echelon form* (see Sec. 6.2.5) and this allows us to express Eq. 9‐119 as Axiom II:

A R

0

(9‐120)

430