# 9.4: Two Matrices

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In this text we have made use of matrix methods to solve problems and to clarify concepts. Here we summarize our knowledge of the matrices associated with the conservation of atomic species and the matrices associated with the analysis of chemical reaction rate phenomena.

## Atomic matrix

The atomic matrix, \(\mathbf{A}\), was introduced in Sec. 6.2 in order to clearly identify the atoms and molecules involved in a particular process, and to provide a compact representation of Axiom II. The construction of the atomic matrix represents a key step in the analysis of chemical reactions since it identifies the molecular and atomic species that we *assume* are involved in the process under consideration. As an example, we consider the atomic matrix for the partial oxidation of ethane. The analysis begins with the following visual representation (see Example 6.4) of the molecules and atoms involved in this process.

\[ \text{ Molecular Species}\to \ce{C2H6} \quad \ce{O2} \quad \ce{H2O} \quad \ce{CO} \quad \ce{CO2} \quad \ce{C2H4O} \\ \begin{matrix} {carbon} \\ {hydrogen} \\ oxygen \end{matrix} \begin{bmatrix} { 2} & { 0} & {0} & {1} & {1} & {2} \\ { 6} & { 0} & {2} & { 0} & {0} & {4 }\\ {0} & { 2} & {1} & { 1 } & {2} & {1 } \end{bmatrix} \label{116}\]

In this case the atomic matrix is given by

Atomic matrix:

\[\mathbf{A} = \begin{bmatrix} { 2} & { 0} & {0} & {1} & {1} & {2} \\ { 6} & { 0} & {2} & { 0} & {0} & {4 }\\ {0} & { 2} & {1} & { 1 } & {2} & {1 } \end{bmatrix} \label{117}\]

and the column matrix of net molar rates of production takes the form

\[\mathbf{R} = \begin{bmatrix} R_{\ce{C2H6}} \\ R_{\ce{O2}} \\ R_{\ce{H2O}} \\ R_{\ce{CO}} \\ R_{\ce{CO2}} \\ R_{\ce{C2H4O}} \end{bmatrix} \label{118}\]

In terms of these two matrices Axiom II is given by

Axiom II:

\[\mathbf{A R} = 0 \label{119}\]

This represents a compact statement that atomic species are neither created nor destroyed by chemical reactions. The atomic matrix can always be expressed in *row reduced eschelon form* (see Sec. 6.2.5) and this allows us to express Equation \ref{119} as

Axiom II:

\[\mathbf{A}^* \mathbf{R} = 0 \label{120}\]

For the atomic matrix represented by Equation \ref{117} the row reduced echelon form is given by

\[\begin{bmatrix} 1 & 0 & 0 & 1/2 & 1/2 & 1 \\ 0 & 1 & 0 & 5/4 & 7/4 & 1 \\ 0 & 0 & 1 & -3/2 & -3/2 & -1 \end{bmatrix} \label{121}\]

The primary application of Axiom II takes the form of the **pivot theorem** that involves the **pivot matrix**.

## Pivot matrix

For the partial oxidation of ethane represented by Equation \ref{116}, we can use Eqs. \ref{118} and \ref{121} to conclude that Equation \ref{120} takes the form

\[\begin{bmatrix} 1 & 0 & 0 & 1/2 & 1/2 & 1 \\ 0 & 1 & 0 & 5/4 & 7/4 & 1 \\ 0 & 0 & 1 & -3/2 & -3/2 & -1 \end{bmatrix} \begin{bmatrix} R_{\ce{C2H6}} \\ R_{\ce{O2}} \\ R_{\ce{H2O}} \\ R_{\ce{CO}} \\ R_{\ce{CO2}} \\ R_{\ce{C2H4O}} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \label{122}\]

Referring to the developments presented in Sec. 6.2.5, we note that a column/row partition of this result can be expressed as

\[\begin{bmatrix} 1 & 0 & 0 & \vdots & 1/2 & 1/2 & 1 \\ 0 & 1 & 0 & \vdots & 5/4 & 7/4 & 1 \\ 0 & 0 & 1 & \vdots & -3/2 & -3/2 & -1 \end{bmatrix} \begin{bmatrix} R_{\ce{C2H6}} \\ R_{\ce{O2}} \\ R_{\ce{H2O}} \\ \hdashline R_{\ce{CO}} \\ R_{\ce{CO2}} \\ R_{\ce{C2H4O}} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \label{123}\]

Carrying out the matrix multiplication illustrated by this column/row partition leads to a special case of the **pivot theorem** given by

\[\begin{bmatrix} R_{\ce{C2H6}} \\ R_{\ce{O2}} \\ R_{\ce{CO}} \end{bmatrix} = \begin{bmatrix} -1/2 & -1/2 & -1 \\ -5/4 & -7/4 & -1 \\ 3/2 & 3/2 & 1 \end{bmatrix} = \begin{bmatrix} R_{\ce{CO}} \\ R_{\ce{CO2}} \\ R_{\ce{C2H4O}} \end{bmatrix} \label{124}\]

The general representation of the pivot theorem takes the form

Pivot Theorem:

\[\mathbf{R}_{NP} = \mathbf{PR}_P \label{125}\]

in which \(\mathbf{P}\) is the pivot matrix. The pivot theorem is ubiquitous in the application of the concept that atomic species are neither created nor destroyed by chemical reactions. In the *analysis* of chemical reactors presented in Chapter 7 the global form of Equation \ref{125} was applied repeatedly and we list that form here as

Global Pivot Theorem:

\[\pmb{\mathscr{R}}_{NP} = \mathbf{P}\pmb{\mathscr{R}}_P \label{126}\]

This important result can also be expressed as

\[\mathscr{R}_A = \sum^{B=N_p}_{B=1} P_{AB} \mathscr{R}_B, \quad A = N_p+1, N_p + 2, .....N \label{127}\]

in which \(P_{AB}\) represents the elements of the pivot matrix determined by Eq. \((6.4.1)\) through Eq. \((6.4.6)\). The *global* net rate of production that appears in Eqs. \ref{126} and \ref{127} is related to the *local* net rate of production by (see **Eq. 7.3**)

\[\mathscr{R}_A = \int_{\mathscr{V}} R_A dV, A = 1,2,...,N \label{128}\]

and it is important to remember that the units of \(R_A\) are moles per unit time *per unit volume* while the units of \(\mathscr{R}_A\) are moles per unit time.

## Mechanistic Matrix

In the *design* of chemical reactors, one needs to know how the *local* net rates of production are related to the concentration of the chemical species involved in the reaction. In the development of this relation, we encountered the **mechanistic matrix** that maps reference *chemical reaction rates* (see Eq. \((9.1.19)\)) onto all *net rates of production*. The general form is given by

\[\mathbf{R}_M = \mathbf{Mr} \label{129}\]

in which \(\mathbf{R}_M\) is the column matrix of all net rates of production, \(\mathbf{M}\) is the *mechanistic matrix*, and \(\mathbf{r}\) is the column matrix of elementary chemical reaction rates. An example of this result is illustrated by Eq. \((9.3.78)\) with the elementary chemical reaction rates respresented explicitly by Eq. \((9.1.80)\). This leads to

\[\underbrace{ \begin{bmatrix} R_{\ce{Br2}} \\ R_{\ce{H2}} \\ R_{\ce{HBr}} \\ R_{\ce{H}} \\ R_{\ce{Br}} \end{bmatrix}}_{\text{all species}} = \underbrace{\begin{bmatrix} -1 & 0 & -1 & 0 & 1/2 \\ 0 & -1 & 0 & 1 & 0 \\ 0 & 1 & 1 & -1 & 0 \\ 0 & 1 & -1 & -1 & 0 \\ 2 & -1 & 1 & 1 & -1 \end{bmatrix} }_{\text{mechanistic matrix}} = \underbrace{ \begin{bmatrix} k_I c_{\ce{Br2}} \\ k_{II} c_{\ce{Br2}} c_{\ce{H2}} \\ k_{III} c_{\ce{H}} c_{\ce{Br2}} \\ k_{IV} c_{\ce{H}} c_{\ce{HBr}} \\ k_{V} c^2_{\ce{Br}} \end{bmatrix} }_{\text{chemical reaction rates}} \label{130}\]

In many texts on chemical reactor design the mechanistic matrix is referred to as the stoichiometric matrix. However, when Bodenstein products^{19} are present, and they usually are, it is appropriate to partition the mechanistic matrix into a *stoichiometric matrix* and a *Bodenstein matrix* as indicated by Eqs. \((9.3.85)\) through \((9.3.87)\). The general partitioning of Equation \ref{129} can be expressed as

\[\mathbf{R}_M = \begin{bmatrix} \mathbf{R} \\ \mathbf{R}_B \end{bmatrix} = \mathbf{Mr} = \begin{bmatrix} \mathbf{S} \\ \mathbf{B} \end{bmatrix} \mathbf{r} \label{131}\]

and this leads to forms analogous to Eqs. \((9.3.86)\) and \((9.3.87)\). We list the first of these results as

Stoichiometric matrix:

\[\mathbf{R} = \mathbf{Sr} \label{132}\]

in which \(\mathbf{S}\) is the stoichiometric matrix composed of *stoichiometric coefficients.* The second of **Eqs. 16** is given by

Bodenstein matrix:

\[\mathbf{R}_B = \mathbf{B r} \label{133}\]

in which \(\mathbf{B}\) represents the Bodenstein matrix. In general, the Bodenstein products are subject to the approximation of local reaction equilibrium that is expressed as

Local reaction equilibrium:

\[\mathbf{R}_B = 0 \label{134}\]

and this allows one to extract additional constraints on the elementary chemical reaction rates. The result given by Equation \ref{132} represents a key aspect of reactor design that can be expressed in more detailed form by

\[R_A = \sum^{B=K}_{B=1} S_{AB} r_B, \quad A = 1,2, .....N \label{135}\]

Here \(S_{AB}\) represent the stoichiometric coefficients, \(r_B\) represents the elementary chemical reaction rates, and \(K\) represents the number of elementary reactions as indicated by Eq. \((9.1.32)\). In many texts on chemical reactor design the mechanistic matrix is referred to as the stoichiometric matrix. However, when Bodenstein products are present, and they usually are, it is appropriate to partition the *mechanistic matrix* into a *stoichiometric matrix* and a *Bodenstein matrix* as indicated by Eqs. \ref{131}.