# 9.13: Untitled Page 229

- Page ID
- 18362

## Chapter 9

*R*

2

Br

1

0

1

0

1 2

*R*

H

2

0

1

0

1

0

R

*R*

*, *

M

M

HBr

0

1

1

1

0

*R*

0

1 1 1

0

H

2

1

1

1

1

*R*

Br

(9‐106)

*r *

*k c*

I B 2

r

I

*r *

*k c c*

II

II Br H2

r

*r*

*k c c*

III

III H B 2

r

*r*

*k*

*c c*

IV

IV H HBr

2

*r*

V

*k c*

V Br

Here we note that the row reduced echelon form of the mechanistic matrix is given by

1

0

2

0 2

0

1

0

1

0

M*****

0

0

1

0

0

(9‐107)

0

0

0

0

0

0

0

0

0

0

and this indicates that two of the net rates of production are linearly dependent on the other three. Some algebra associated with Eqs. 9‐103 indicates that this dependence can be expressed in the form

2 *R*

*R*

*R*

0

(9‐108a)

H2

H

HBr

2 *R*

*R*

2 *R*

*R*

0

(9‐108b)

2

Br

Br

H2

H

Useful representations for the three independent net rates of production can be extracted from Eq. 9‐104; however, the analysis can be greatly simplified if we designate H and Br as *reactive intermediates* or *Bodenstein products* and then impose the condition of *local reaction equilibrium* expressed as *R*

0 *, *

*R*

0

(9‐109)

H

Br

*Reaction Kinetics *

427

In order to make use of this simplification, it is convenient to represent Eq. 9‐104

in terms of the chemical reaction rate expressions and then apply a *row / row* partition (see Sec. 6.2.6, Problem 6‐22 and Appendix C1) to obtain (9‐110)

Here the first partition takes the form

*k c*

I Br2

*R*

*k c c*

II Br H

Br

1

0

1

0

1 2

2

2

*R*

0 1

0

1

0

*k c c*

III H Br

(9‐111)

H

2

2

0

1

1 1

0

*R*

*k c c*

HBr

IV H HBr

*stoichiometric matrix*

2

*k c*

V Br

in which the matrix of coefficients is the *stoichiometric matrix*. The second partition is given by

*k c*

I Br2

*k c c*

II Br H2

*R *

0

1 1

1

0

H

*k c c*

III H Br

(9‐112)

2

*R*

Br

2

1

1

1

1

*k c c *

IV H HBr

*Bodenstein matrix*

2

*k c*

V Br

in which this matrix of coefficients is the *Bodenstein matrix* that maps the rates of reaction onto the net rates of production of the *Bodenstein products* (Bodenstein and Lind, 1907). It is important to note that the stoichiometric matrix maps an array of chemical kinetic expressions onto the column matrix of the net rates of production of the three *stable molecular species*. This mapping process carried out by the *stoichiometric matrix* is quite different than the mapping process carried out by the *pivot matrix* that is illustrated by Eq. 9‐95.

428