# 6.16: Untitled Page 139

- Page ID
- 18272

## Chapter 6

*R*

C2H5OH

*R*

C H

1

1

1

0

0

2

2

4

0

*R* CH CHO

0

1

1 1

3

1

3

0

(6)

*R*

0

0

1

0 1

1

H O

2

0

*R* H2

*R* C4 H6

Here we note that our original choice of non‐pivot species, C H OH , 2

5

C H and H O , has been changed by the application of Eq. 5 that leads 2

4

2

to the non‐pivot species represented by C H OH , C H and CH CHO .

2

5

2

4

3

At this point we make use of some routine elementary row operations to obtain the desired *row reduced echelon form*

*R*

C2 H5OH

*R*

C2 H4

1

0

0

1

1

1

0

*R* CH

3 CHO

0

1

0 1

0

2

0

(7)

*R*

H

2O

0

0

1

0

1

1

0

*R*

H2

*R*

C

4 H6

Given

this

representation

of

Axiom

II

we

can

apply

a

*column / row* *partition * illustrated by

(8)

which immediately leads to

257

1

0

0

*R*

*R*

C

1

1

1

0

2H5OH

H

2O

0

1

0

*R*

1

0

2

*R*

0 (9)

C

2H4

H2

0

0

1

0

1

1

0

*R*

*R*

CH

3CHO

C

4H6

*non‐pivot*

*pivot*

*submatrix*

*submatrix*

Here the non‐pivot submatrix is the *unit matrix* that maps a column matrix onto itself as indicated by

1

0

0

*R*

*R*

C

2H5OH

C

2H5OH

0

1

0

*R*

*R*

(10)

C2H4

C2H4

0

0

1

*R*

*R*

CH

3CHO

CH

3CHO

*non‐pivot*

*submatrix*

Substitution of this result into Eq. 9 provides the following simple form

*R*

*R*

C

1

1

1

2H5OH

H

2O

*R*

1

0

2

*R*

(11)

C

2H4

H2

0 1

1

*R*

*R*

CH

3CHO

C

4H6

*pivot*

*submatrix*

From this we extract a representation for the column matrix of *non‐pivot* *species* in terms of the *pivot matrix* of stoichiometric coefficients and the column matrix of *pivot species*. This representation is given by

*R*

*R*

C

1

1

1

2H5OH

H

2O

*R*

1

0 2

*R*

(12)

C

2H4

H2

0

1 1

*R*

*R*

CH

3CHO

C

4H6

*pivot matrix*

*column matrix*

*column* *matrix*

*of non‐pivot species*

*of * *pivot* *species*

This is a special case of the *pivot theorem* in which we see that the net rates of production of the pivot species are mapped onto the net rates of production of the non‐pivot species by the pivot matrix. The matrix multiplication indicated in Eq. 12 can be carried out to obtain *R*

*R*

*R*

*R*

(13a)

C2H5OH

H2O

H2

C4H6

258