# 10.6: Appendix F - Heterogeneous Reactions

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Our analysis of the stoichiometry of heterogeneous reactions is based on conservation of atomic species expressed as

Axiom II:

\[\sum_{A = 1}^{A = N}N_{JA} R_{A} = 0 \label{1}\]

We follow the classic continuum point of view^{1} and assume that this result is valid everywhere. That is to say that Axiom II is valid in homogeneous regions where quantities such as \(R_A\) change slowly and it is valid in interfacial regions where \(R_A\) changes rapidly. We follow the work of Wood *et al* ^{2} and consider the \(\gamma - \kappa\) interface illustrated in Figure \(\PageIndex{1}\).

The volume \(V\) encloses the \(\gamma - \kappa\) interface and extends into the homogeneous regions of both the \(\gamma\)-phase and the \(\kappa\)-phase. The total net rate of production of species \(A\) in the volume \(V\) is represented by

\[\int_V R_A dV \equiv \int_{V_{\gamma}} R_{A\gamma} dV + \int_{V_{\kappa}} R_{A \kappa} dV + \int_{A_{\gamma \kappa}} R_{A \mathbf{S}} dA \label{2}\]

Here the dividing surface that separates the \(\gamma\)-phase from the \(\kappa\)-phase is represented by \(A_{\gamma \kappa}\) and the heterogeneous rate of production of species \(A\) is identified by \(R_{A\mathbf{S}}\). This quantity is also referred to as the surface excess reaction rate^{3}. Multiplying Equation \ref{2} by the atomic species indicator and summing over all molecular species leads to

\[\int_{V} \sum_{A = 1}^{A = N}N_{JA} R_{A} dV = \int_{V_{\gamma}} \sum_{A = 1}^{A = N}N_{JA} R_{A_{\gamma}} dV \\ + \int_{V_{\kappa}} \sum_{A = 1}^{A = N}N_{JA} R_{A_{\kappa}} dV + \int_{A_{\gamma \kappa}} \sum_{A = 1}^{A = N}N_{JA} R_{A\mathbf{S}} dA , \quad J = 1,2,...,T \label{3}\]

From Equation \ref{1} we see that the left hand side of this result is zero and we have

\[0 = \int_{V_{\gamma}} \sum_{A = 1}^{A = N}N_{JA} R_{A\gamma} dV + \int_{V_{\kappa}} \sum_{A = 1}^{A = N}N_{JA} R_{A\kappa} dV \\ + \int_{A_{\gamma \kappa}} \sum_{A = 1}^{A = N}N_{JA} R_{A\mathbf{S}} dA , \quad J = 1,2,...,T \label{4}\]

At this point we require that the homogeneous net rates of production satisfy the two constraints given by

\[\sum_{A = 1}^{A = N}N_{JA} R_{A\gamma} = 0, \quad \sum_{A = 1}^{A = N}N_{JA} R_{A\kappa} = 0, \quad J = 1,2,...,T \label{5}\]

and this leads to the following form of Equation \ref{4}

\[\int_{A_{\gamma \kappa}} \sum_{A = 1}^{A = N}N_{JA} R_{A\mathbf{S}} dA = 0, \quad J = 1,2,...,T \label{6}\]

Catalytic surfaces consist of catalytic sites where reaction occurs and non-catalytic regions where no reaction occurs. Because the heterogeneous rate of production is highly non-uniform, it is appropriate to work in terms of the area average

\[ \langle R_{A\mathbf{S}} \rangle_{\gamma \kappa} = \frac{1}{A_{\gamma \kappa}} \int_{A_{\gamma \kappa}} R_{A\mathbf{S}} dA \label{7}\]

so that Equation \ref{6} takes the form

\[\sum_{A = 1}^{A = N}N_{JA} \langle R_{A\mathbf{S}} \rangle_{\gamma \kappa} = 0. \quad J = 1,2,...,T \label{8}\]

We summarize our results associated with Axiom II as

Axiom II (general)

\[\sum_{A = 1}^{A = N}N_{JA} R_A = 0. \quad J = 1,2,...,T \label{9}\]

Axiom II (homogeneous, \(\gamma\)-phase)

\[\sum_{A = 1}^{A = N}N_{JA} R_{A\gamma} = 0. \quad J = 1,2,...,T \label{10}\]

Axiom II (homogeneous, \(\kappa\)-phase)

\[\sum_{A = 1}^{A = N}N_{JA} R_{A\kappa} = 0. \quad J = 1,2,...,T \label{11}\]

Axiom II (heterogeneous, \(\gamma - \kappa\) interface

\[\sum_{A = 1}^{A = N}N_{JA} \langle R_{A\mathbf{S}} \rangle_{\gamma \kappa} = 0. \quad J = 1,2,...,T \label{12}\]

For a reactor in which only homogeneous reactions occur, we make use of Equation \ref{9} in the form

Axiom II:

\[\mathbf{AR} = \mathbf{0}, \quad \mathbf{R} = \begin{bmatrix} R_A \\ R_B \\ \mathbf{.} \\ \mathbf{.} \\ R_N \end{bmatrix} \label{13}\]

in which \(\mathbf{A}\) is the atomic matrix. For a catalytic reactor in which only heterogeneous reactions occur at the \(\gamma - \kappa\) interface, we make use of Equation \ref{12} in the form

Axiom II:

\[\mathbf{A} \langle \mathbf{R}_{\mathbf{S}} \rangle_{\gamma \kappa} = \mathbf{0}, \quad \langle \mathbf{R}_{\mathbf{S}} \rangle_{\gamma \kappa} = \begin{bmatrix} \langle R_{A\mathbf{S}} \rangle_{\gamma \kappa} \\ \langle R_{B\mathbf{S}} \rangle_{\gamma \kappa} \\ \mathbf{.} \\ \mathbf{.} \\ \langle R_{N\mathbf{S}} \rangle_{\gamma \kappa} \end{bmatrix} \label{14}\]

The pivot theorem associated homogeneous reactions is obtained from Equation \ref{13} and the analysis leads to Eq. \((6.4.3)\) which is repeated here as

Pivot Theorem (homogeneous reactions):

\[\mathbf{R}_{NP} = \mathbf{PR}_P \label{15}\]

The pivot theorem associated with heterogeneous reactions is obtained from Equation \ref{14} and is given here as

Pivot Theorem (heterogeneous reactions):

\[\left( \langle \mathbf{R}_{\mathbf{S}} \rangle_{\gamma \kappa} \right)_{NP} = \mathbf{P} \left(\langle \mathbf{R}_{\mathbf{S}} \rangle_{\gamma \kappa}\right)_P \label{16}\]

The fact that the axiom and the application take exactly the same form for both homogeneous and heterogeneous reactions has led many to ignore the difference between these two distinct forms of chemical reaction.

In general, measurement of the net rates of production are carried out at the macroscopic level, thus we generally obtain experimental information for the global net rate of production. For a homogeneous reaction, this takes the form

\[\mathscr{R}_A - \int_{\mathscr{V}} R_A dA, \quad A = 1,2,...,N \label{17}\]

while the global net rate of production for a heterogeneous reaction is given by

\[\mathscr{R}_A - \int_{\mathscr{A}_{\gamma \kappa}} \langle R_{A\mathbf{S}} \rangle_{\gamma \kappa} dA, \quad A = 1,2,...,N \label{18}\]

Here we note that the global net rate of production for both homogeneous reactions and heterogeneous reactions have exactly the same physical significance, thus it is not unreasonable to use the same symbol for both quantities. Given this simplification, the global version of the pivot theorem can be expressed as

Pivot Theorem (global form):

\[\pmb{\mathscr{R}}_{NP} = \mathbf{P} \pmb{\mathscr{R}}_P \label{19}\]

for both homogeneous and heterogeneous reactions.

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