# 6.8: Untitled Page 131

- Page ID
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## Chapter 6

One form of the atomic matrix for this group of molecular species can be visualized as

Molecular Species C H

O

H O CO

2

6

2

2

2

*carbon*

2

0

0

1

(1)

*hydrogen*

6

0

2

0

*oxygen*

0

2

1

2

nd

a

for this particular arrangement the atomic matrix is given by

2 0 0 1

*N * 6 0 2 0

(2)

*JA *

0 2 1 2

A simple calculation (see Problem 6‐5) shows

t

tha the rank of the

matrix is

three

r *rank * *N * 3

(3)

*JA *

thus we have *three equations* and *four unknowns*. The three homogeneous equations that are analogous to Eqs. 6‐35 are given by

2 *R*

0 0 *R*

0

(4a)

C2H6

CO2

6 *R*

0 2 *R*

0

0

(4b)

C2H6

H2O

0 2 *R*

*R*

2 *R*

0

(4c)

O2

H2O

CO2

while the analogous matrix equation corresponding to Eq. 6‐22 takes the form

*R*

C2H6

2 0 0 1

0

*R*

O2

6 0 2 0

0

(5)

*R* H O

2

0 2 1 2

0

*R*

CO

2

It is possible to use intuition and the *picture* given by Eq. 6‐8 to express the net rates of production in the form

1

*R*

*R*

(6a)

C2H6

CO

2

2

7

*R*

*R*

(6b)

O2

C

4

O2

241

3

*R*

*R*

(6c)

H2O

CO

2

2

however, the use of Eqs. 4 to produce this result is more reliable. Finally, we note that these results for the net rates of production can be expressed in the form of the *pivot theorem* that is described in Sec. 6.4. In terms of the pivot theorem that can be extracted from Eq. 5 we have

*R*

C

2H6

1 2

*R*

*R*

(7)

O

7 4

2

CO

2

3 2

*R*

H

*column matrix*

2O

*of pivot species*

*pivot matrix*

*column matrix*

*of non‐pivot species*

This indicates that all the net rates of production are specified if we can determined the net rate of production for carbon dioxide, *R*

. Indeed,

CO2

all the rates of production can be determined if we know *any one*

* of the fou *

*r*

*tes*

*ra*

; however, we have chosen carbon dioxide as the *pivot* *species* in order to arrange Eqs. 4 and Eq. 5 in the forms given by Eqs. 6 and 7.

In the previous example we illustrated how Axiom II can be used to analyze the stoichiometry for the complete combustion of ethane. The process of complete combustion was described earlier by the single *stoichiometric schema* given by Eq. 6‐8 and the

n

coefficie ts that appeared in that schema

are evident in

Eqs. 6 and 7 of Example 6.1.

*6.2.5 Elementary row operations and column / row interchange operations* In working with sets of equations such as those represented by Eq. 6‐22, we will make use of elementary row operations and column/row operations in order to

ge

arran

the equations in a convenient form. Elementary row operations were described earlier in Sec. 4.9.1 and we list them here as they apply to the atomic

matrix:

I. Any row in the atomic matrix can be modified by

multiplying or dividing by a non‐zero scalar without

affecting the system of equations.

II. Any row in the atomic matrix can be added or

subtracted from another row without affecting the

system of equations.

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