# 6.19: Untitled Page 142

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## Chapter 6

The elements of the column matrices are given explicitly by the following version of this theorem

*B * *Np*

R

*P * R *, *

*A * *N * 1 *, N * 2 *, ...., N *

(6‐82)

*A*

*AB*

*B*

*p*

*p*

*B * 1

in which the *global* net rate of production is related to the *local* net rate of production by Eq. 6‐30. A summary of the matrices presented in this chapter is given in Sec. 9.4 along with a discussion of the several matrices used in the study of chemical reaction kinetics.

6.5 Problems

Note that problems marked with the symbol will be difficult to solve without the use of computer software.

*Section* 6.1

6‐1. By “counting atoms” provide at least one version of a balanced chemical equation based on

*? * C H

+ ? O

? CO + ? C H O + ? H O + CO

2

6

2

2

4

2

2

that is different from the two examples given in the text.

*Section* 6.2

6‐2. Construct an atomic matrix for the following set of components: Sodium hydroxide ( Na OH ), methyl bromide ( CH Br ), methanol ( CH OH ), and 3

3

sodium bromide ( NaBr ).

6‐3. Construct an atomic matrix for a system containing the following molecular species: NH , O , NO , N , H O and NO .

3

2

2

2

2

6‐4. Begin with the statement that mass is neither created nor destroyed by chemical reaction

*A * *N*

*r*

0

*A*

*A * 1

and use it to derive Eq. 6‐20. Be careful to state any restrictions that might be necessary in order to complete the derivation.

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6‐5 The rank of a matrix is conveniently determined using the row reduced form of the matrix. Consider the atomic matrix given by Eq. 2 of Example 6.1 and use elementary row operations to develop the row reduced echelon form of that matrix.

6‐6. Using Eq. 6‐20, show how to obtain Eqs. 4 in Example 6.1.

6‐7. Use elementary row operations to express Eq. 5 of Example 6.1 in terms of the row reduced echelon form of the atomic matrix. Indicate how Eqs. 6 are obtained using the row reduced echelon form.

6‐8. First find the rank of the atomic matrix developed in Problem 6‐3. Next, choose N , H O , and NO as the pivot species and develop a solution for 2

2

2

*R*

, *R*

and *R*

.

NH

O

3

2

NO

6‐9. Express the atomic matrix in Eq. 12 of Example 6.2 in row reduced echelon form. Use that form to express R

and R

in terms of R

.

C2H6

C2H4

H2

6‐10. Represent Eqs. 13 of Example 6.2 in the form of the pivot theorem illustrated by Eq. 7 of Example 6.1.

6‐11. In this problem you are asked to consider the complete combustion of methanol, thus the molecular species under consideration are

CH OH *, * O *, * H O *, * CO

3

2

2

2

Develop the atomic matrix in row reduced echelon form, and use Axiom II with CO as the pivot species in order to determine the rates of production, *R*

,

2

CH3OH

*R*

, *R*

in terms of *R*

. Express your results in the form analogous to

O2

H2O

CO2

Eq. 7 of Example 6.1.

6‐12. Consider the complete combustion of methane to produce water and carbon dioxide. Construct the atomic matrix in row reduced echelon form, and show that Axiom II can be used to express the rates of production of methane, oxygen and water in terms of the single pivot species, carbon dioxide. Express your results in the form of the pivot theorem illustrated by Eq. 7 of Example 6.1.

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