# 6.14: Untitled Page 137

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## Chapter 6

net rates of production. This case is illustrated in Eq. 6‐8 and discussed in detail in Example 6.2. When we have *multiple independent stoichiometric reactions*, such as the partial oxidation of carbon (Example 6.3) or the partial oxidation of ethane (Example 6.4), we need to measure more than one net rate of production in order to determine all the net rates of production.

*6.2.6 Matrix partitioning*

Axiom II provides an example of the multiplication of a *T * *N * matrix with a 1 *N * column matrix. Multiplication of matrices can also be represented in terms of *submatrices*, provided that one is careful to follow the rules of matrix multiplication. As an example, we consider the following matrix equation

*b *

1

*a*

*a*

*a*

*a*

*a * *b*

*c *

11

12

13

14

15 2

1

*a*

*a*

*a*

*a*

*a*

*b *

*c*

(6‐52)

21

22

23

24

25

3

2

*a*

*a*

*a*

*a*

*a * *b*

*c*

31

32

33

34

35 4

3

*b *

5

which conforms to the rule that the number of columns in the first matrix is equal to the number of rows in the second matrix. Equation 6‐52 represents the three individual equations given by

*a b * *a b * *a b * *a b * *a b*

*c *

(6‐53a)

11 1

12 2

13 3

14 4

15 5

1

*a b * *a b * *a b * *a b * *a b*

*c *

(6‐53b)

21 1

22 2

23 3

24 4

25 5

2

*a b * *a b * *a b * *a b * *a b*

*c *

(6‐53c)

31 1

32 2

33 3

34 4

35 5

3

which can also be expressed in compact form according to

AB

C

(6‐54)

Here the matrices A, B, and C are defined explicitly by

*b *

1

*a*

*a*

*a*

*a*

*a *

*b*

*c *

11

12

13

14

15

2

1

A

*a*

*a*

*a*

*a*

*a*

B

*b *

C

*c*

(6‐55)

21

22

23

24

25

3

2

*a*

*a*

*a*

*a*

*a *

*b*

*c*

31

32

33

34

35

4

3

*b *

5

In addition to the matrix multiplication that we have used up to this point, matrix multiplication can also be carried out in terms of *partitioned matrices*.

*Stoichiometry *

253

If we wish to obtain a *column partition* of the matrix A in Eq. 6‐52, we must also create a row partition of matrix B in order to conform to the rules of matrix multiplication that are discussed in Appendix C1. This *column / row* *partition * takes the form

(6‐56)

and the *submatrices* are identified explicitly according to

*a*

*a*

*a *

*b *

*a*

*a *

11

12

13

1

14

15

*b *

4

A

*a*

*a*

*a*

B

*b*

A

*a*

*a*

B

(6‐57)

11

21

22

23

1

2

12

24

25

2

*b*

5

*a*

*a*

*a *

*b *

*a*

*a *

31

32

33

3

34

35

Use of these representations in Eq. 6‐56 leads to

B

A

A

1

C

(6‐58)

11

12 B2

and matrix multiplication in terms of the submatrices provides A B

A B

C

(6‐59)

11 1

12

2

To verify that this result is identical to Eq. 6‐53, we use Eqs. 6‐57 and the third of Eqs. 6‐55 to obtain

*a*

*a*

*a * *b *

*a*

*a *

*c *

11

12

13

1

14

15

*b *

1

4

*a*

*a*

*a*

*b*

*a*

*a*

*c*

21

22

23

2

24

25

2

(6‐60)

*b*

5

*a*

*a*

*a * *b *

*a*

*a *

*c *

31

32

33 3

34

35

3

Carrying out the matrix multiplication on the left hand side of this result leads to

*a b*

*a b*

*a b *

*a b*

*a b *

*c *

11 1

12 2

13 3

14 4

15 5

1

*a b*

*a b*

*a b*

*a b*

*a b*

*c*

21 1

22 2

23 3

24 4

25 5

2

(6‐61)

*a b a b*

*a b *

*a b*

*a b *

*c *

31 1

32 2

33 3

34 4

35 5

3

At this point we add the two matrices on the left hand side of this result following the rules for matrix addition given in Sec.2.6 in order to obtain

254