# 6.4: Untitled Page 127

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## Chapter 6

We refer to *N*

as the *atomic species* *indicator* and we identify the array of *JA*

coefficients associated with *N*

as the *atomic matrix* (Amundson, 1966, page 54) *. *

*JA*

To illustrate the structure of the atomic matrix, we consider the complete oxidation of ethane illustrated in Figure 6‐1. That process provides the basis for the following *visual* *representation* of the atomic matrix: Molecular Species C H

O

H O CO

2

6

2

2

2

*carbon*

2

0

0

1

(6‐15)

*hydrogen*

6

0

2

0

*oxygen*

0

2

1

2

This representation *connects* atoms with molecules in a convenient manner, and it is exactly what one uses to count atoms and balance chemical equations. There are two symbols that are useful for representing the *atomic matrix*. The first of these is given by

which has the obvious connection to Eq. 6‐14, while the

*JA*

*N*

second is given by A which has the obvious connection to the name of this matrix. In this text we will encounter both representations for the *atomic matrix* as indicated by

2 0 0 1

2 0 0 1

*N * 6 0 2 0 *, *

or

A

6 0 2 0

(6‐16)

*JA *

0 2 1 2

0 2 1 2

In order to use the atomic species indicator, *N*

, to construct an equation

*JA*

representing the concept that atoms are neither created nor destroyed by chemical reaction, we first recall the definition of *R *

*A*

*net * molar rate of production

*R*

per unit volume of species *A*

(6‐17)

*A*

owing to chemical reactions

which is consistent with the pictorial representation of *R*

given earlier in

CO2

Figure 4‐1. Next we form the product of the atomic species indicator with *R * to *A*

obtain

number of moles of *net* molar rate of production

*N R*

*J*‐type atoms per moleper unit volume of species *A* (6‐18) *JA*

*A*

of molecular species *A *owing to chemical reactions

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A little thought will indicate that the product of *N*

and *R * can be described as

*JA*

*A*

net molar rate of production per unit

*N R*

volume of *J*‐type atoms owing to the

(6‐19)

*JA*

*A*

molar rate of production of species *A*

and the axiomatic statement given by Eq. 6‐13 takes the form (Truesdell and Toupin, 1960, page 473)

*A * *N*

Axiom II:

*N*

*R*

0 *, *

*J * 1 *, * 2 *,...,T *

(6‐20)

*JA A*

*A * 1

This equation represents a precise mathematical statement that atomic species are neither created nor destroyed by chemical reactions, and it provides a set of *T * equations that *constrain* the *N * net rates of production, *R *, *A * 1 *, * 2 *,...,N *.

*A*

While Axiom II provides *T * equations, the equations are not necessarily independent. The number of independent equations is given by the *rank of the* *atomic matrix* and we will be careful to indicate that rank when specific processes are examined. If ions are involved in the reactions, one must impose the condition of *conservation of charge* as described in Appendix E. Some comments concerning *heterogeneous reactions* are given in Appendix F.

The net rate of production of species *A* indicated by *R * can also be expressed *A*

in terms of the *creation* and *consumption* of species *A* according to

molar rate of *creation* of

molar rate of *consumption* of

*R*

species *A* per unit volume species *A* per unit volume (6‐21) *A*

owing to chemical reactions owing to chemical reactions

Here we need to think carefully about the description of *R * given by Eq. 6‐17

*A*

where we have used the word *net* to represent the sum of the *creation* of species *A* and the *consumption* of species *A*. This means that Eqs. 6‐17 and 6‐21 are equivalent descriptions of *R * and the reader is free to chose which ever set of *A*

words is most appealing.

If we make use of the *atomic matrix* and the *column matrix of the net rates of* *production* we can express Axiom II as

234