# 9.3: Mechanistic Matrix

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In this section we explore in more detail the reaction rates associated with chemical kinetic schemata of the type studied in the previous two sections. The *mechanistic matrix* ^{16} will be introduced as a convenient method of organizing information about reaction rates. This matrix is different than the *pivot matrix* discussed in Chapter 6, and we need to be very clear about the similarities and differences between these two matrices, both of which contain coefficients that are often referred to as *stoichiometric coefficients*. In some cases the mechanistic matrix is identical to the *stoichiometric matrix* and in some cases it consists of both a *stoichiometric matrix* and a *Bodenstein matrix*.

We begin by considering a system in which there are five species and three chemical kinetic schemata described by

Elementary chemical kinetic schema I:

\[A+B \stackrel{k_{I}}{\longrightarrow} C+D, \quad D \text{ is a by-product} \label{81a}\]

Elementary chemical kinetic schema II:

\[C+B \stackrel{k_{II}}{\longrightarrow} E, \quad E \text{ is the product} \label{81b}\]

Elementary chemical kinetic schema III:

\[C+D \stackrel{k_{III}}{\longrightarrow} A+B, \quad \text{ reverse of schema I} \label{81c}\]

In this example we *assume* that the *stoichiometric schemata* are identical in form to the *chemical kinetic schemata*, and we carefully follow the structure outlined in Sec. 9.1.1 in order to avoid algebraic errors. We begin with the first elementary step indicated by Equation \ref{81a}, and our analysis of this step leads to

Elementary chemical kinetic schema I:

\[A+B \stackrel{k_{I}}{\longrightarrow} C+D \label{82a}\]

Elementary stoichiometry I:

\[R_{A}^{I} = R_{B}^{I}, \quad R_{A}^{I} = -R_{C}^{I}, \quad R_{A}^{I} = -R_{D}^{I} \label{82b}\]

Elementary chemical reaction rate equation I:

\[R_{A}^{I} = -k_I c_A c_B \label{82c}\]

Elementary reference chemical reaction rate I:

\[r_I \equiv k_I c_A c_B \label{82d}\]

Here we should note that Equation \ref{82c} has the same form as Eq. \((9.1.19)\). In this case the term on the left hand side is an *elementary* rate of production while the term on the right hand side is referred to as an *elementary* chemical reaction rate. To illustrate the relation to Eq. \((9.1.19)\) we express Equation \ref{82c} as

\[\underbrace{\underbrace{R^I_{A}}_{\text{elementary rate of production}} = \underbrace{-k_I c_{A} c_B}_{\text{elementary chemical reaction rate}}}_{\text{elemntary chemical reaction rate equation}} \label{83}\]

In Equation \ref{82d} we have defined an elementary *reference* chemical reaction rate that is designated by \(r_I\), and we will choose a similar *reference quantity* for each chemical kinetic schema. The units of these reference quantities are \(moles/(volume \times time)\) and they will be designated by \(r_I\), \(r_{II}\) and \(r_{III}\). These reference chemical reaction rates will always be *positive*, and they must be distinguished from \(r_A\), \(r_B\), \(r_C\), etc. that were used in Chapter 4 (see Eq. \((4.1.6)\)) to represent the net *mass* rate of production of species \(A\), \(B\), \(C\), etc.

Moving on to the *second elementary step* indicated by Equation \ref{81b}, we create a set of results analogous to Eqs. \ref{82a}-\ref{82d} that are given by:

Elementary chemical kinetic schema II:

\[C+B \stackrel{k_{II}}{\longrightarrow} E \label{84a}\]

Elementary stoichiometry II:

\[R^{II}_C = R^{II}_B, \quad R^{II}_C = -R^{II}_E \label{84b}\]

Elementary chemical reaction rate equation II:

\[R^{II}_C = -k_{II}c_Bc_C \label{84c}\]

Elementary reference chemical reaction rate II:

\[r_{II} \equiv k_{II}c_Bc_C \label{84d}\]

Finally we examine the third elementary step indicated by Equation \ref{81c} in order to obtain

Elementary chemical kinetic schema III:

\[C+D \stackrel{k_{III}}{\longrightarrow} A+B \label{85a}\]

Elementary stoichiometry III:

\[R^{III}_C = R^{III}_D, \quad R^{III}_C = -R^{III}_A, \quad R^{III}_C = -R^{III}_B \label{85b}\]

Elementary chemical reaction rate equation III:

\[R^{III}_C = -k_{III}c_Cc_D \label{85c}\]

Elementary reference chemical reaction rate III:

\[r_{III} \equiv k_{III} c_C c_D \label{85d}\]

The net rate of production for each molecular species is given in terms of the elementary rates of production according to

Species A:

\[R_A = R^I_A + R^{II}_A + R^{III}_A \label{86a}\]

Species B:

\[R_B = R^I_B + R^{II}_B + R^{III}_B \label{86b}\]

Species C:

\[R_C = R^I_C + R^{II}_C + R^{III}_C \label{86c}\]

Species D:

\[R_D = R^I_D + R^{II}_D + R^{III}_D \label{86d}\]

Species E:

\[R_E = R^I_E + R^{II}_E + R^{III}_E \label{86e}\]

At this point we can use the elementary chemical reaction rates to express the net rates of production according to

Species A:

\[R_A = -r_I + 0 + r_{III} \label{87a}\]

Species B:

\[R_B = -r_I - r_{II} + r_{III} \label{87b}\]

Species C:

\[R_C = r_I - r_{II} - r_{III} \label{87c}\]

Species D:

\[R_D = r_I + 0 + r_{III} \label{87d}\]

Species E:

\[R_E = 0 + r_{II} + 0 \label{87e}\]

In matrix form these representations for the net rates of production are given by

\[\begin{bmatrix} R_{A} \\ R_{B} \\ R_{C} \\ R_{D} \\ R_{E} \end{bmatrix} = \begin{bmatrix} -1 & 0 & 1 \\ -1 & -1 & 1 \\ 1 & -1 & -1 \\ 1 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} r_{I} \\ r_{II} \\ r_{III} \end{bmatrix} \label{88}\]

Often it is convenient to express this result in the following compact form

\[\mathbf{R}_M = \mathbf{Mr} \label{89}\]

in which \(\mathbf{R}_M\) is the column matrix of all the net rates of production, \(\mathbf{M}\) is the mechanistic matrix^{17}, and \(\mathbf{r}\) is the column matrix of elementary chemical reaction rates. These quantities are defined explicitly by

\[\mathbf{R}_M = \begin{bmatrix} R_{A} \\ R_{B} \\ R_{C} \\ R_{D} \\ R_{E} \end{bmatrix}, \quad \mathbf{M} = \underbrace{\begin{bmatrix} -1 & 0 & 1 \\ -1 & -1 & 1 \\ 1 & -1 & -1 \\ 1 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix}}_{\text{mechanistic matrix}}, \quad \mathbf{r} = \begin{bmatrix} r_{I} \\ r_{II} \\ r_{III} \end{bmatrix} \label{90}\]

When reactive intermediates, or Bodenstein products, are present, the mechanistic matrix is decomposed into a *stoichiometric matrix* and a Bodenstein *matrix* and we give an example of this situation in the following paragraphs. Here it is crucial to understand that the column matrix on the left hand side of Equation \ref{88} consists of the *net molar rates of production* of all species including the reactive intermediates or Bodenstein products. It is equally important to understand that the column matrix on the right hand side of Equation \ref{88} consists of *chemical reaction rates* that are *not* net molar rates of production. Instead they are chemical reaction rates defined by Eqs. \ref{82d}, \ref{84d} and \ref{85d}. The definitions of these chemical reaction rates can be expressed explicitly as

\[\mathbf{r} = \begin{bmatrix} r_{I} \\ r_{II} \\ r_{III} \end{bmatrix} \equiv \begin{bmatrix} k_I c_A c_B \\ k_{II} c_B c_C \\ k_{III} c_C c_D \end{bmatrix} \label{91}\]

The matrix representations given by Equation \ref{88} and Equation \ref{91} can be used to extract the individual expressions for \(R_A\), \(R_B\), \(R_C\), \(R_D\), and \(R_E\) that are given by

Species A:

\[R_A = -k_I c_A c_B + k_{III} c_C c_D \label{92a}\]

Species B:

\[R_B = -k_I c_A c_B - k_{II} c_B c_C + k_{III} c_C c_D \label{92b}\]

Species C:

\[R_C = k_I c_A c_B - k_{II} c_B c_C - k_{III} c_C c_D \label{92c}\]

Species D:

\[R_D = k_I c_A c_B - k_{III} c_C c_D \label{92d}\]

Species E:

\[R_E = k_{II} c_B c_C \label{92e}\]

In addition to extracting these results directly from Equation \ref{88} and Equation \ref{91}, we can also obtain them from the schemata illustrated by Eqs. \ref{81a} - \ref{81c} in the same manner that was used in Sec. 9.1.1. In Chapter 6 we made use of the *pivot matrix* that maps the net rates of product ion of the *pivot species* onto the net rates of production of the *non-pivot species*. In this development we see that the *mechanistic matrix* maps the elementary chemical reaction rates onto *all* the net rates of production.

At this point we note that the row reduced echelon form of the mechanistic matrix is given by

\[\mathbf{M}^* = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \label{93}\]

This indicates that two of the net rates of production are linearly dependent on the other three. From Eqs. \ref{92a} - \ref{92e} we obtain

\[R_D = -R_A \label{94a}\]

\[R_C = -R_A - R_E \label{94b}\]

while the net rates of production for species \(A\), \(B\) and \(E\) are repeated here as

\[R_A = -k_I c_A c_B + k_{III} c_C c_D \label{94c}\]

\[R_B = -k_I c_A c_B - k_{II} c_B c_C + k_{III} c_C c_D \label{94d}\]

\[R_E = k_{II} c_B c_C \label{94e}\]

These net rates of production can be used with Axiom I to analyze chemical reactors such as the batch reactors studied in Section 8.2.

## Hydrogen Bromide Reaction

At this point we return to the hydrogen bromide reaction described briefly in Section 9.1 where Axiom II provided the result

Axiom II:

\[\begin{bmatrix} R_{\ce{H2}}\\R_{\ce{Br2}}\end{bmatrix} = \underbrace{\begin{bmatrix} -1/2 \\ -1/2 \end{bmatrix}}_{\text{pivot matrix}} \begin{bmatrix}R_{\ce{HBr}} \end{bmatrix} \label{95}\]

Use of local stoichiometry along with the chemical kinetic schema given by Eq. \((9.1.16)\) did not lead to a chemical reaction rate equation that was in agreement with the experimental result indicated by Eq. \((9.1.20)\). Clearly the molecular process suggested by Figure \(9.1.3\) is not an acceptable representation of the reaction kinetics and we need to explore the impact of *reactive intermediates* on the hydrogen bromide reaction. To do so, we propose the following chemical kinetic schemata:

Elementary chemical kinetic schema I:

\[\ce{Br2} \stackrel{k_{I}}{\longrightarrow} \ce{2Br} \label{96a}\]

Elementary chemical kinetic schema II:

\[\ce{Br} + \ce{H2} \stackrel{k_{II}}{\longrightarrow} \ce{HBr} + \ce{H} \label{96b}\]

Elementary chemical kinetic schema III:

\[\ce{H} + \ce{Br2} \stackrel{k_{III}}{\longrightarrow} \ce{HBr} + \ce{Br} \label{96c}\]

Elementary kinetic schema IV:

\[\ce{H} + \ce{HBr} \stackrel{k_{IV}}{\longrightarrow} \ce{H2} + \ce{Br} \label{96d}\]

Elementary kinetic schema V:

\[\ce{2Br} \stackrel{k_{V}}{\longrightarrow} \ce{Br2} \label{96e}\]

Here we note that Eqs. \ref{96a} - \ref{96e} are simply a more complex form of Eqs. \ref{81a} - \ref{81c}, thus we can follow the procedure outlined in the previous paragraphs assuming that the *stoichiometric schemata* are identical to the *chemical kinetic schemata*. We begin with the first elementary schema indicated by Equation \ref{96a} and our analysis of this schema leads to

#### SCHEMA I

Elementary chemical kinetic schema I:

\[\ce{Br2} \stackrel{k_{I}}{\longrightarrow} \ce{2Br} \label{97a}\]

Elementary stoichiometry I:

\[R^{I}_{\ce{Br2}} = -\frac{R^{I}_{\ce{Br}}}{2} \label{97b}\]

Elementary chemical reaction rate equation I:

\[R^{I}_{\ce{Br2}} = -k_I c_{\ce{Br2}} \label{97c}\]

Elementary reference chemical reaction rate I:

\[r_I \equiv k_I c_{\ce{Br2}}\label{97d}\]

The remaining schemata lead to an analogous set of equations given by

#### SCHEMA II

Elementary chemical kinetic schema II:

\[\ce{Br} + \ce{H2} \stackrel{k_{II}}{\longrightarrow} \ce{HBr} + \ce{H} \label{98a}\]

Elementary stoichiometry II:

\[R^{II}_{\ce{Br}} = R^{II}_{\ce{H2}}, \quad R^{II}_{\ce{Br}} = -R^{II}_{\ce{HBr}}, \quad R^{II}_{\ce{Br}} = -R^{II}_{\ce{H}} \label{98b}\]

Elementary chemical reaction rate equation II:

\[R^{II}_{\ce{Br}} = -k_{II} c_{\ce{Br}} c_{\ce{H2}} \label{98c}\]

Elementary reference chemical reaction rate II:

\[r_{II} \equiv k_{II} c_{\ce{Br}} c_{\ce{H2}} \label{98d}\]

#### SCHEMA III

Elementary chemical kinetic schema III:

\[\ce{H} + \ce{Br2} \stackrel{k_{III}}{\longrightarrow} \ce{HBr} + \ce{Br} \label{99a}\]

Elementary stoichiometry III:

\[R^{III}_{\ce{H}} = R^{III}_{\ce{Br2}}, \quad R^{III}_{\ce{H}} = -R^{III}_{\ce{HBr}}, \quad R^{III}_{\ce{H}} = -R^{III}_{\ce{Br}} \label{99b}\]

Elementary chemical reaction rate equation III:

\[R^{III}_{\ce{H}} = -k_{III} c_{\ce{H}} c_{\ce{Br2}} \label{99c}\]

Elementary reference chemical reaction rate III:

\[r_{III} \equiv k_{III} c_{\ce{H}} c_{\ce{Br2}} \label{99d}\]

#### SCHEMA IV

Elementary chemical kinetic schema IV:

\[\ce{H} + \ce{HBr} \stackrel{k_{IV}}{\longrightarrow} \ce{H2} + \ce{Br} \label{100a}\]

Elementary stoichiometry IV:

\[R^{IV}_{\ce{H}} = R^{IV}_{\ce{HBr}}, \quad R^{IV}_{\ce{H}} = -R^{IV}_{\ce{H2}}, \quad R^{IV}_{\ce{H}} = -R^{IV}_{\ce{Br}} \label{100b}\]

Elementary chemical reaction rate equation IV:

\[R^{IV}_{\ce{H}} = -k_{IV} c_{\ce{H}} c_{\ce{HBr}}\label{100c}\]

Elementary reference chemical reaction rate IV:

\[r_{IV} \equiv k_{IV} c_{\ce{H}} c_{\ce{HBr}} \label{100d}\]

#### SCHEMA V

Elementary chemical kinetic schema V:

\[\ce{2Br} \stackrel{k_{V}}{\longrightarrow} \ce{Br2} \label{101a}\]

Elementary stoichiometry V:

\[\frac{R^{V}_{\ce{Br}}}{2} = -R^{V}_{\ce{Br2}} \label{101b}\]

Elementary chemical reaction rate equation V:

\[R^{V}_{\ce{Br}} = -k_V c^{2}_{\ce{Br}} \label{101c}\]

Elementary reference chemical reaction rate V:

\[r_V \equiv k_V c^{2}_{\ce{Br}} \label{101d}\]

We begin our analysis of Eqs. \ref{96a} - \ref{96e} by listing the net molar rate of production of all five species in terms of the elementary rates of reaction according to

\[R_{\ce{Br2}} = R^I_{\ce{Br2}} + R^{II}_{\ce{Br2}} + R^{III}_{\ce{Br2}} + R^{IV}_{\ce{Br2}} + R^{V}_{\ce{Br2}} \label{102a}\]

\[R_{\ce{H2}} = R^I_{\ce{H2}} + R^{II}_{\ce{H2}} + R^{III}_{\ce{H2}} + R^{IV}_{\ce{H2}} + R^{V}_{\ce{H2}} \label{102b}\]

\[R_{\ce{HBr}} = R^I_{\ce{HBr}} + R^{II}_{\ce{HBr}} + R^{III}_{\ce{HBr}} + R^{IV}_{\ce{HBr}} + R^{V}_{\ce{HBr}} \label{102c}\]

\[R_{\ce{H}} = R^I_{\ce{H}} + R^{II}_{\ce{H}} + R^{III}_{\ce{H}} + R^{IV}_{\ce{H}} + R^{V}_{\ce{H}} \label{102d}\]

\[R_{\ce{Br}} = R^I_{\ce{Br}} + R^{II}_{\ce{Br}} + R^{III}_{\ce{Br}} + R^{IV}_{\ce{Br}} + R^{V}_{\ce{Br}} \label{102e}\]

At this point we can use the elementary chemical reaction rates to express the net rates of production according to

\[R_{\ce{Br2}} = -r_1 + 0 - r_{III} + 0 + \frac{1}{2} r_V \label{103a}\]

\[R_{\ce{H2}} = 0 - r_{II} + 0 + r_{IV} + 0 \label{103b}\]

\[R_{\ce{HBr}} = 0 + r_{II} + r_{III} - r_{IV} + 0 \label{103c}\]

\[R_{\ce{H}} = 0 + r_{II} - r_{III} - r_{IV} + 0 \label{103d}\]

\[R_{\ce{Br}} = 2r_I - r_{II} + r_{III} + r_{IV} - r_V \label{103e}\]

In matrix form these representations for the net rates of production are given by

\[\begin{bmatrix} R_{\ce{Br2}} \\ R_{\ce{H2}} \\ R_{\ce{HBr}} \\ R_{\ce{H}} \\ R_{\ce{Br}} \end{bmatrix} = \begin{bmatrix} -1 & 0 & -1 & 0 & 1/2 \\ 0 & -1 & 0 & 1 & 0 \\ 0 & 1 & 1 & -1 & 0 \\ 0 & 1 & -1 & -1 & 0 \\ 2 & -1 & 1 & 1 & -1 \end{bmatrix} \begin{bmatrix} r_{I} \\ r_{II} \\ r_{III} \\ r_{IV} \\ r_{V} \end{bmatrix} \label{104}\]

The compact form of this lengthy algebraic result can be expressed as

\[\mathbf{R}_M = \mathbf{Mr} \label{105}\]

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. These quantities are defined explicitly by

\[\mathbf{R}_M = \begin{bmatrix} R_{\ce{Br2}} \\ R_{\ce{H2}} \\ R_{\ce{HBr}} \\ R_{\ce{H}} \\ R_{\ce{Br}} \end{bmatrix}, \quad \mathbf{M} = \begin{bmatrix} -1 & 0 & -1 & 0 & 1/2 \\ 0 & -1 & 0 & 1 & 0 \\ 0 & 1 & 1 & -1 & 0 \\ 0 & 1 & -1 & -1 & 0 \\ 2 & -1 & 1 & 1 & -1 \end{bmatrix}, \quad \mathbf{r} = \begin{bmatrix} r_{I} \\ r_{II} \\ r_{III} \\ r_{IV} \\ r_{V} \end{bmatrix} \equiv \begin{bmatrix} k_I c_{\ce{Br2}} \\ k_{II} c_{\ce{Br2}} c_{\ce{H2}} \\ k_{III} c_{\ce{H}} c_{\ce{Br2}} \\ k_{IV} c_{\ce{H}} c_{\ce{HBr}} \\ k_{V} c^2_{\ce{Br}} \end{bmatrix} \label{106}\]

Here we note that the row reduced echelon form of the mechanistic matrix is given by

\[\mathbf{M}^* = \begin{bmatrix} 1 & 0 & 2 & 0 & -2 \\ 0 & 1 & 0 & -1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \label{107}\]

and this indicates that two of the net rates of production are linearly dependent on the other three. Some algebra associated with Eqs. \ref{103a} - \ref{103e} indicates that this dependence can be expressed in the form

\[2R_{\ce{H2}} + R_{\ce{H}} + R_{\ce{HBr}} = 0 \label{108a}\]

\[2R_{\ce{Br2}} + R_{\ce{Br}} - 2R_{\ce{H2}} - R_{\ce{H}} = 0 \label{108b}\]

Useful representations for the three independent net rates of production can be extracted from Equation \ref{104}; however, the analysis can be greatly simplified if we designate H and Br as *reactive intermediates* or *Bodenstein products* and then impose the condition of *local reaction equilibrium* expressed as

\[R_{\ce{H}} = 0, \quad R_{\ce{Br}} = 0 \label{109}\]

In order to make use of this simplification, it is convenient to represent Equation \ref{104} in terms of the chemical reaction rate expressions and then apply a \(row/row\) partition (see Sec. 6.2.6, Problem 6-22 and Appendix C1) to obtain

\[\begin{bmatrix} R_{\ce{Br2}} \\ R_{\ce{H2}} \\ R_{\ce{HBr}} \\ \hdashline R_{\ce{H}} \\ R_{\ce{Br}} \end{bmatrix} = \begin{bmatrix} -1 & 0 & -1 & 0 & 1/2 \\ 0 & -1 & 0 & 1 & 0 \\ 0 & 1 & 1 & -1 & 0 \\ \hdashline 0 & 1 & -1 & -1 & 0 \\ 2 & -1 & 1 & 1 & -1 \end{bmatrix} \begin{bmatrix} k_I c_{\ce{Br2}} \\ k_{II} c_{\ce{Br2}} c_{\ce{H2}} \\ k_{III} c_{\ce{H}} c_{\ce{Br2}} \\ k_{IV} c_{\ce{H}} c_{\ce{HBr}} \\ k_{V} c^2_{\ce{Br}} \end{bmatrix} \label{110}\]

Here the first partition takes the form

\[\begin{bmatrix} R_{\ce{Br2}} \\ R_{\ce{H2}} \\ R_{\ce{HBr}} \end{bmatrix} = \underbrace{\begin{bmatrix} -1 & 0 & -1 & 0 & 1/2 \\ 0 & -1 & 0 & 1 & 0 \\ 0 & 1 & 1 & -1 & 0 \end{bmatrix}}_{\text{stoichiometric matrix}} \begin{bmatrix} k_I c_{\ce{Br2}} \\ k_{II} c_{\ce{Br2}} c_{\ce{H2}} \\ k_{III} c_{\ce{H}} c_{\ce{Br2}} \\ k_{IV} c_{\ce{H}} c_{\ce{HBr}} \\ k_{V} c^2_{\ce{Br}} \end{bmatrix} \label{111}\]

in which the matrix of coefficients is the *stoichiometric matrix*. The second partition is given by

\[\begin{bmatrix}R_{\ce{H}} \\ R_{\ce{Br}} \end{bmatrix} = \underbrace{\begin{bmatrix} 0 & 1 & -1 & -1 & 0 \\ 2 & -1 & 1 & 1 & -1 \end{bmatrix}}_{\text{Bodenstein matrix}} \begin{bmatrix} k_I c_{\ce{Br2}} \\ k_{II} c_{\ce{Br2}} c_{\ce{H2}} \\ k_{III} c_{\ce{H}} c_{\ce{Br2}} \\ k_{IV} c_{\ce{H}} c_{\ce{HBr}} \\ k_{V} c^2_{\ce{Br}} \end{bmatrix} \label{112}\]

in which this matrix of coefficients is the *Bodenstein matrix* that maps the rates of reaction onto the net rates of production of the Bodenstein^{18} products. It is important to note that the stoichiometric matrix maps an array of chemical kinetic expressions onto the column matrix of the net rates of production of the three stable molecular species. This mapping process carried out by the *stoichiometric matrix* is quite different than the mapping process carried out by the pivot matrix that is illustrated by Equation \ref{95}.

If we impose the condition of *local reaction equilibrium* indicated by Equation \ref{109}, we obtain the following two constraints on the reaction rates

\[\begin{align} & R_{\ce{H}} = 0: && k_{II}c_{\ce{Br}}c_{\ce{H2}} -k_{III}c_{\ce{H}}c_{\ce{Br2}} - k_{IV}c_{\ce{H}}c_{\ce{HBr}} = 0 \end{align} \label{113a}\]

\[\begin{align} & R_{\ce{Br}} = 0: && 2k_Ic_{\ce{Br2}} - k_{II}c_{\ce{Br}}c_{\ce{H2}} + k_{III}c_{\ce{H}}c_{\ce{Br2}} + k_{IV}c_{\ce{H}}c_{\ce{HBr}} - k_V c^2_{\ce{Br}} = 0 \end{align} \label{113b}\]

These two results can be used to determine the concentrations of H and Br that take the form

\[c_{\ce{H}}=\frac{k_{II} c_{\ce{H2}} \sqrt{2 k_{1} / k_{V}} \sqrt{c_{\ce{Br2}}}}{\left(k_{III} c_{\ce{Br2}}+k_{IV} c_{\ce{HBr}}\right)}, \quad c_{\ce{Br}}=\sqrt{2 k_{I} / k_{V}} \sqrt{c_{\ce{Br2}}} \label{114}\]

On the basis of Eqs. \ref{108a} - \ref{108b} and Eqs. \ref{109} we see that there is only a single independent equation associated with Eqs. \ref{111} and we can use that equation to determine the net rate of production of hydrogen bromide as

\[R_{\ce{HBr}}=\frac{\left(2 k_{II} \sqrt{2 k_{I} / k_{V}}\right) c_{\ce{H2}} \sqrt{c_{\ce{Br2}}}}{1+\left(k_{IV} / k_{III}\right)\left(c_{\ce{HBr}} / c_{\ce{Br2}}\right)} \label{115}\]

A little thought will indicate that this result has exactly the same form as the experimentally determined reaction rate expression given by Eq. \((9.1.20)\).

In this section we have illustrated the use of the *mechanistic matrix* to provide a compact representation of chemical reaction rate equations. When reactive intermediates (Bodenstein products) are involved in the reaction process, and local reaction equilibrium is assumed, it is convenient to represent the *mechanistic matrix* in terms of the *stoichiometric matrix* and the *Bodenstein matrix* as illustrated by Eqs. \ref{110} through \ref{112}.