6.16: Untitled Page 139

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)

Stoichiometry

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