# 6.18: Untitled Page 141

## 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)

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)

Stoichiometry

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