# 9.15: Untitled Page 231

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## Chapter 9

For the atomic matrix represented by Eq. 9‐117 the *row reduced echelon form* is given by

1 0 0

1 2

1 2

1

A

0 1 0

5 4

7 4

1

(9‐121)

0 0 1 3 2 3 2

1

The primary application of Axiom II takes the form of the *pivot theorem* that involves the *pivot matrix*.

Pivot matrix

For the partial oxidation of ethane represented by Eq. 9‐116, we can use Eqs. 9‐118 and 9‐121 to represent Eq. 9‐120 in the form

*R*

C2H6

*R*

O2

1 0 0

1 2

1 2

1

0

*R* H O

2

0 1 0

5 4

7 4

1

0

(9‐122)

*R*

CO

0 0 1

3 2

3 2

1

0

*R* CO

2

*R*

C

2H4O

Referring to the developments presented in Sec. 6.2.5, we note that a column **/ **row partition of this result can be expressed as (9‐123)

Carrying out the matrix multiplication illustrated by this column **/ **row partition leads to a special case of the *pivot* *theorem* given by

*R*

C

1

*R*

2H6

1 2

1 2

CO

*R*

1

*R*

(9‐124)

O

5 4

7 4

2

CO2

3 2

3 2

1

*R*

*R*

CO

C

2H4O

431

The general representation of the pivot theorem takes the form Pivot Theorem:

R

P R

(9‐125)

*NP*

*P*

in which P is the pivot matrix. The pivot theorem is ubiquitous in the application of the concept that atomic species are neither created nor destroyed by chemical reactions. In the *analysis* of chemical reactors presented in Chapter 7

the global form of Eq. 9‐125 was applied repeatedly, and we can express the global form as

Global Pivot Theorem:

R

P R

(9‐126)

*NP*

*P*

Here

represents the column matrix of non‐pivot species *global* net rates of *NP*

R

production while

represents the column matrix of pivot species *global* net *P*

R

rates of production.

Mechanistic matrix

In the *design* of chemical reactors, one needs to know how the *local* net rates of production are related to the concentration of the chemical species involved in the reaction. In the development of this relation, we encountered the *mechanistic* *matrix* that maps reference *chemical reaction rates* (see Eq. 9‐18) onto *all* *net rates of* *production*. The general form is given by

R

M r

(9‐127)

M

in which R is the column matrix of all net rates of production, M is the M

*mechanistic matrix*, and r is the column matrix of elementary chemical reaction rates. For the hydrogen bromide reaction, Eq. 9‐127 provides the detailed representation given by

*k c*

*R*

I Br

Br

2

2

1

0 1

0

1 2

*k c c*

*R*

II Br H

H

0

1

0

1

0

2

2

0

1

1

1

0 *k*

*c c*

*R*

III H Br

(9‐128)

HBr

2

0

1 1 1

0

*R*

*k c c*

H

IV H HBr

2 1 1 1

1

2

*R*

Br

*k c*

V Br

*mechanistic matrix*

*all species*

*chemical *

*reaction rates*

In many texts on chemical reactor design the mechanistic matrix is referred to as the stoichiometric matrix. However, when Bodenstein products (Aris, 1965) are

432