# 6.18: Untitled Page 141

- Page ID
- 18274

## Chapter 6

In this case we have *r * *rank * 3 and *N * 4 , and the net rates of production for methyl chloride, ethyl chloride and chlorine can be represented in terms of the net rate of production of hydrogen. The *column / row partition * of Eq. 6‐70 is illustrated by

(6‐71)

and this immediately leads to

1 0 0 *R* CH

4

0

3Cl

0 1 0 *R*

2 *R*

0

(6‐72)

C

2H5Cl

H

2

0 0 1

1

0

*R*

Cl

2

This can be solved for the column matrix of non‐pivot species according to

*R*

CH

4

3Cl

*R*

2

*R *

(6‐73)

C

2H5Cl

H

2

1

*R*

Cl

*column matrix*

2

*of pivot species*

*pivot matrix*

*column matrix*

*of non‐pivot species*

Here we have a *non‐trivial solution* in which the three rates of production of the *non‐pivot species* are specified in terms of the rate of production of the *pivot species*, *R*

. One should always keep in mind that the *null solution* is still possible for H2

this case, i.e., all four net rates of production may be zero depending on the conditions in our hypothetical reactor.

**6.4 Axioms and Theorems**

To summarize our studies of stoichiometry, we note that atomic species are neither created nor destroyed by chemical reactions. In terms of the *atomic matrix* and the *column matrix of net rates of production*, this concept can be expressed as Axiom II:

A R

0

(6‐74)

261

As indicated in the previous section, the *row reduced echelon form* of the atomic matrix can always be developed, thus we can express Eq. 6‐74 as

Row Reduced Echelon Form:

A R

0

(6‐75)

The product of the atomic matrix times the column matrix of net rates of production can be partitioned according to (see Sec. 6.2.6)

R

Column/Row Partition:

*NP*

I W

0

(6‐76)

R *P *

Here the *column partition* of A provides the *non‐pivot* *submatrix* I and the *pivot* *submatrix* W , while the *row partition* of R provides the *non‐pivot* *column submatrix* R

and the *pivot* *column submatrix* R . Carrying out the matrix multiplication *NP*

*P*

indicated by Eq. 6‐76 leads to

I R

W R

0

(6‐77)

*NP*

*P*

and operation of the unit matrix on R

provides the obvious result given by

*NP*

I R

R

(6‐78)

*NP*

*NP*

At this point we define the *pivot matrix* P according to Pivot Matrix:

P

W

(6‐79)

and we use this result, along with Eq. 6‐78, in Eq. 6‐77 to obtain the *pivot theorem* Pivot Theorem:

R

P R

(6‐80)

*NP*

*P*

These five concepts represent the foundations of stoichiometry, and they appear in various special forms throughout this chapter and in subsequent chapters.

When ionic species are involved, *conservation of charge* must be taken into account as indicated in Appendix E. Heterogeneous reactions can be analyzed using the framework presented in this chapter and the details are discussed in Appendix F.

Reactions involving *optical isomers* require some care that is illustrated in Problems 6‐32 and 6‐33.

In Chapter 7 we will make repeated use of the *global form* of the pivot theorem. To develop the global form we integrate Eq. 6‐80 over the volume *V * to obtain

Global Pivot Theorem:

R

P R

(6‐81)

*NP*

*P*

262