Analysis of reaction schemes using maximum rates of constituent steps

Significance

The design of active and selective catalysts is essential for a wide range of industrial applications and societal issues. A fundamental approach is to identify the key elementary steps and to elucidate the predicted reaction kinetics for potential reaction mechanisms. We present a methodology to analyze analytically the performance of catalytic reaction schemes by calculation of the maximum rates of the constituent steps. This proposed methodology can be used to identify the important transition states and adsorbed species, such that more detailed calculations can be carried out for these species, whereas more approximate methods can be used for the remaining species, thereby substantially reducing the computational time required to elucidate how catalyst performance is controlled by the fundamental surface chemistry.

Abstract

We show that the steady-state kinetics of a chemical reaction can be analyzed analytically in terms of proposed reaction schemes composed of series of steps with stoichiometric numbers equal to unity by calculating the maximum rates of the constituent steps, r_max,i, assuming that all of the remaining steps are quasi-equilibrated. Analytical expressions can be derived in terms of r_max,i to calculate degrees of rate control for each step to determine the extent to which each step controls the rate of the overall stoichiometric reaction. The values of r_max,i can be used to predict the rate of the overall stoichiometric reaction, making it possible to estimate the observed reaction kinetics. This approach can be used for catalytic reactions to identify transition states and adsorbed species that are important in controlling catalyst performance, such that detailed calculations using electronic structure calculations (e.g., density functional theory) can be carried out for these species, whereas more approximate methods (e.g., scaling relations) are used for the remaining species. This approach to assess the feasibility of proposed reaction schemes is exact for reaction schemes where the stoichiometric coefficients of the constituent steps are equal to unity and the most abundant adsorbed species are in quasi-equilibrium with the gas phase and can be used in an approximate manner to probe the performance of more general reaction schemes, followed by more detailed analyses using full microkinetic models to determine the surface coverages by adsorbed species and the degrees of rate control of the elementary steps.

Chemical reactions take place through sequences of elementary steps, and the dynamics of the overall stoichiometric reaction are typically controlled by key steps in this sequence of steps. For example, in the case of a catalytic reaction, the reaction kinetics are controlled by the energetics of the transition states for the rate-controlling steps and by the energetics of species that are abundant on the active sites (i.e., the Gibbs free energies of these transition states and adsorbed species relative to the reactants and products of the overall stoichiometric reaction). However, whereas the observed reaction kinetics are controlled by a limited number of transition states and adsorbed species, it is typically required to carry out detailed microkinetic analyses to identify the nature of these key transition states and adsorbed species. Thus, a general strategy to elucidate how the reaction kinetics are controlled by a proposed reaction mechanism is first to carry out density functional theory (DFT) calculations to determine the thermodynamic properties of all adsorbed species and transition states, and then to build a microkinetic model to determine the surface coverages by all adsorbed species and the forward and reverse rates of all elementary steps for a range of reaction conditions (1–5). Sensitivity analyses are then carried out using this microkinetic model to identify those transition states and adsorbed species that control the predicted reaction kinetics for reaction conditions of interest. Although this approach is effective to predict the performance of proposed reaction schemes, it requires information about all of the reaction steps, whereas only a fraction of these steps actually control catalyst performance. Moreover, this approach requires the numerical solution of coupled differential and/or algebraic equations, involving complexities associated with the solution of stiff differential equations and the formulation of initial guesses for solution of algebraic equations. Thus, it is desirable to develop an analytical methodology to identify the transition states and adsorbed species that are likely to be important, such that detailed DFT calculations can be carried out for these species, whereas more approximate methods (e.g., scaling relations) can be carried out for the remaining species. The analysis described here provides a simple approach to predict the reaction rate and the degrees of rate control for all of the elementary steps of a reaction mechanism by avoiding the need to solve systems of coupled differential equations and/or algebraic equations that constitute a full microkinetic model. This approach would be particularly useful to assess the feasibility of proposed reaction schemes to achieve desirable reaction kinetics for chemical processes of interest and would provide both conceptual as well as practical advantages in microkinetic modeling.

We present a methodology that can be used to identify the key intermediates and transition states that control the steady-state performance of a catalytic reaction consisting of a series of consecutive reaction steps passing through reactive intermediates. Reactions that may follow multiple pathways from reactants to products can be analyzed along each reaction pathway separately, allowing the dominant reaction pathway to be determined. Moreover, reactions leading to different products can be studied along each pathway separately, allowing the selectivity of the catalyst to be addressed.

Degree of Rate Control and Reversibility of Reaction Steps

A useful tool in the analysis of reaction schemes is the degree of rate control (X_RC) (6–9), defined as

$$X_{RC,i}={\left(\frac{\partial r}{\partial k_i}\right)\frac{k_i}{r}|}_{K_{eq,i},k_{j\ne i}}={\left(\frac{\partial \ln r}{\partial \ln k_i}\right)|}_{K_{eq,i},k_{j\ne i}},$$ [1]

where X_RC,i is the degree of rate control for step i, r is the rate of the overall stoichiometric reaction, k_i is the forward rate constant of step i, and the partial derivative is taken with the equilibrium constant for step i (K_eq,i) and all other rate constants (k_j) held constant. The sum of the values of X_RC,i is equal to unity for a reaction scheme leading to a single stoichiometric reaction (10). Thus, the reaction scheme has a rate determining step if X_RC,i = 1 for step i, whereas more generally, nonzero values of X_RC,i reflect the relative contributions of these steps in controlling the rate of the overall stoichiometric reaction. Campbell has shown more recently that the degree of rate control for step i can also be represented as the change in the rate of the overall stoichiometric reaction with respect to a change in the standard-state Gibbs free energy of the transition state for this step $G_i^{o†}$ (11):

$$X_{RC,i}={\left(\frac{\partial r}{-\partial G_i^{o†}}\right)\frac{RT}{r}|}_{G_{j\ne i}^o}={\left(\frac{\partial \ln r}{\partial \left(\frac{-G_i^{o†}}{RT}\right)}\right)|}_{G_{j\ne i}^o},$$ [2]

where R is the gas constant, T is the absolute temperature, and the partial derivative is taken with the Gibbs free energies of all other species $\left(G_j^o\right)$ held constant. If the values of X_RC,i are known, then it is possible to estimate how changes in the values of $G_i^{o†}$ for these steps would lead to changes in the rate of the overall stoichiometric reaction (12). Specifically, the above expression is integrated starting from a reference catalyst, having a rate r₀, transition states with Gibbs free energies equal to $G_{i,0}^{o†}$, and known values of X_RC,i, to a new catalyst with rate r and Gibbs free energies equal to $G_i^{o†}$:

$$\ln \left(\frac{r}{r_0}\right)={\displaystyle \sum _i{X}_{RC,i}\left(\frac{G_{i,0}^{o†}-G_i^{o†}}{RT}\right)}.$$ [3]

This expansion to estimate changes in the rate caused by changes in the Gibbs free energies of the controlling transition states assumes for simplicity that the degrees of rate control for the steps remain constant.

We now consider a four-step reaction scheme for the overall stoichiometric conversion of species A to B passing through reactive intermediates I₁, I₂, and I₃:

$$\begin{array}{l}1.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{A}\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{I}}_1\\ 2.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\text{I}}_1\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{I}}_2\\ 3.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\text{I}}_2\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{I}}_3\\ 4.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\text{I}}_3\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{B}\\ \text{overall}:\hspace{0.17em}\hspace{0.17em}\text{A}\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\mathrm{B}.\end{array}$$

We note the distinction between the case considered here of a reaction mechanism for an overall stoichiometric reaction passing through a series of reactive intermediates, versus a collection of reaction steps that produce intermediates at significant concentrations compared to the concentrations of the reactants (species A) or the final product (species B). For the case considered here, the concentrations of all reactive intermediates remain small compared with the concentrations of the reactants and products, such that the rate of consumption of the reactants is equal to the rate of production of the products throughout the course of the reaction, and the time derivatives of the intermediate concentrations are approximately zero (i.e., the steady-state approximation). Thus, we consider here the case where experimental results indicate that A is converted stoichiometrically to B, and the above reaction scheme can be treated as a reaction mechanism. Accordingly, the net rate of each reversible step is equal to the net rate of the overall reaction, such that the stoichiometric number of each step is equal to 1, where the stoichiometric number of a step is defined as the number of times that the step takes place for one turnover of the overall stoichiometric reaction (more generally, the net rate of step i is equal to σ_i times the net rate of the overall reaction, where σ_i is the stoichiometric number of step i). Hereafter, we discuss the reactions where the steady-state approximation is applicable and as such we address the overall stoichiometric reaction as the overall reaction. In the following analysis we consider the catalytic reactions in the limit of low coverage of the catalyst sites. In the later section we relax this constraint and account for the population of catalyst sites by adsorbed species.

We define the reversibility of step i (Z_i) as the rate of the reverse reaction divided by the rate of the forward reaction:

$$Z_1=\frac{k_{-1}{I}_1}{k_{1}A}=\frac{I_1}{K_{eq,1}A}$$ [4]

$$Z_2=\frac{k_{-2}{I}_2}{k_2{I}_1}=\frac{I_2}{K_{eq,2}{I}_1}$$ [5]

$$Z_3=\frac{k_{-3}{I}_3}{k_3{I}_2}=\frac{I_3}{K_{eq,3}{I}_2}$$ [6]

$$Z_4=\frac{k_{-4}B}{k_4{I}_3}=\frac{B}{K_{eq,4}{I}_3}$$ [7]

$$Z_1{Z}_2{Z}_3{Z}_4=\frac{I_1}{K_{eq,1}A}\frac{I_2}{K_{eq,2}{I}_1}\frac{I_3}{K_{eq,3}{I}_2}\frac{B}{K_{eq,4}{I}_3}=\frac{B}{K_{eq}A}=\beta ,$$ [8]

where k_i and k_−i are the forward and reverse rate constants for step i, K_eq,i is the equilibrium constant for step i, K_eq is the equilibrium constant for the overall reaction, β is the reversibility of the overall reaction, A and B are the activities of the reactant and product, and I_i are the activities of the intermediates. We note that Z_i can also be expressed as

$$Z_i=\exp \left(\frac{\mathrm{\Delta }{G}_i}{RT}\right),$$ [9]

where $\mathrm{\Delta }{G}_i$ is equal to the change in Gibbs free energy for step i (e.g., $\mathrm{\Delta }{G}_i$ = 0 as step i becomes equilibrated). It should be noted that $\mathrm{\Delta }{G}_i$ is the change in the Gibbs free energy at reaction condition and not the change in the standard state Gibbs free energy. Thus, the value of Z_i is equal to 0 for an irreversible step and approaches 1 for a quasi-equilibrated step. We can now write expressions for the net rate of each step in terms of the forward rate and the reversibility of the step, noting that the net rates of all of the elementary steps are equal to the net rate of the overall reaction, r :

$$r=r_1-r_{-1}=r_1\left(1-Z_1\right)=k_{1}A\left(1-Z_1\right)$$ [10]

$$r=r_2-r_{-2}=r_2\left(1-Z_2\right)=k_2{I}_1\left(1-Z_2\right)$$ [11]

$$r=r_3-r_{-3}=r_3\left(1-Z_3\right)=k_3{I}_2\left(1-Z_3\right)$$ [12]

$$r=r_4-r_{-4}=r_4\left(1-Z_4\right)=k_4{I}_3\left(1-Z_4\right).$$ [13]

The activities of the intermediates, I_i, can be expressed in terms of the reversibilities to give

$$r=k_{1}A\left(1-Z_1\right)$$ [14]

$$r=K_{eq,1}{k}_{2}A\hspace{0.17em}{Z}_1\left(1-Z_2\right)$$ [15]

$$r=K_{eq,1}{K}_{eq,2}{k}_{3}A\hspace{0.17em}{Z}_1{Z}_2\left(1-Z_3\right)$$ [16]

$$r=K_{eq,1}{K}_{eq,2}{K}_{eq,3}{k}_{4}A\hspace{0.17em}{Z}_1{Z}_2{Z}_3\left(1-\frac{\beta }{Z_1{Z}_2{Z}_3}\right),$$ [17]

where the value of Z₄ has been replaced by the reversibility of the overall reaction (β) and the reversibilities of steps 1, 2, and 3. We note that the net rates of these steps are controlled by four lumped parameters, C_i, defined as

$$r=C_{1}A\left(1-Z_1\right);C_1=k_1$$ [18]

$$r=C_{2}A\hspace{0.17em}{Z}_1\left(1-Z_2\right);C_2=K_{eq,1}{k}_2$$ [19]

$$r=C_{3}A\hspace{0.17em}{Z}_1{Z}_2\left(1-Z_3\right);C_3=K_{eq,1}{K}_{eq,2}{k}_3$$ [20]

$$r=C_{4}A\hspace{0.17em}{Z}_1{Z}_2{Z}_3\left(1-\frac{\beta }{Z_1{Z}_2{Z}_3}\right);C_4=K_{eq,1}{K}_{eq,2}{K}_{eq,3}{k}_4.$$ [21]

According to transition state theory, we write the rate constant for step i in terms of a frequency factor $\left({\nu }^{†}=k_{B}T/h\right)$ times the equilibrium constant for the formation of the transition state from the reactant of that step, where k_B and h are the Boltzmann and Planck constants, respectively:

$$k_i={\nu }^{†}{K}_{eq,i}^{†}.$$ [22]

It should be noted that in Eq. 22 both the enthalpic and the entropic changes are included in the definition of K_eq and as such the frequency factor is defined as ${\nu }^{†}=k_{B}T/h$. Using Eq. 22, the lumped parameters, C_i, are given by

$$C_1={\nu }^{†}{K}_{eq,1}^{†}$$ [23]

$$C_2={\nu }^{†}{K}_{eq,1}{K}_{eq,2}^{†}$$ [24]

$$C_3={\nu }^{†}{K}_{eq,1}{K}_{eq,2}{K}_{eq,3}^{†}$$ [25]

$$C_4={\nu }^{†}{K}_{eq,1}{K}_{eq,2}{K}_{eq,3}{K}_{eq,4}^{†}.$$ [26]

The physical significance of C_i can be related to the equilibrium constant for the formation of the transition state for step i from the reactant A $\left(K_{eq,A\to T{S}_i}^{†}\right)$:

$$C_1={\nu }^{†}{K}_{\mathit{eq},\mathit{A}\to {\mathit{TS}}_1}^{†}$$ [27]

$$C_2={\nu }^{†}{K}_{\mathit{eq},\mathit{A}\to {\mathit{TS}}_2}^{†}$$ [28]

$$C_3={\nu }^{†}{K}_{\mathit{eq},\mathit{A}\to {\mathit{TS}}_3}^{†}$$ [29]

$$C_4={\nu }^{†}{K}_{\mathit{eq},\mathit{A}\to {\mathit{TS}}_4}^{†}.$$ [30]

Thus, we can now express the net rates of the elementary steps in terms of lumped equilibrium constants from the reactant A to the transition states:

$$r={\nu }^{†}{K}_{\mathit{eq},\mathit{A}\to {\mathit{TS}}_1}^{†}A\left(1-Z_1\right)$$ [31]

$$r={\nu }^{†}{K}_{\mathit{eq},\mathit{A}\to {\mathit{TS}}_2}^{†}A{Z}_1\left(1-Z_2\right)$$ [32]

$$r={\nu }^{†}{K}_{\mathit{eq},\mathit{A}\to {\mathit{TS}}_3}^{†}A{Z}_1{Z}_2\left(1-Z_3\right)$$ [33]

$$r={\nu }^{†}{K}_{\mathit{eq},\mathit{A}\to {\mathit{TS}}_4}^{†}A{Z}_1{Z}_2{Z}_3\left(1-\frac{\beta }{Z_1{Z}_2{Z}_3}\right).$$ [34]

A simple calculation to assess whether an elementary step may contribute to a significant extent in a reaction scheme is to estimate the maximum rate of this step for the conditions of the catalytic reaction of interest. If the maximum estimated rate of this step is much slower than the observed rate of the catalytic reaction, then this step cannot play a significant role in the reaction scheme. As seen above, the maximum rate of step i corresponds to the case where the reversibilities of all other steps are equal to 1 and the reversibility of step i is equal to β, corresponding to the case where step i is rate-determining (X_RC,i = 1):

$$r_{1=rds}={\nu }^{†}{K}_{\mathit{eq},\mathit{A}\to {\mathit{TS}}_1}^{†}A\left(1-\beta \right)=r_{\max ,1}\left(1-\beta \right)$$ [35]

$$r_{2=rds}={\nu }^{†}{K}_{\mathit{eq},\mathit{A}\to {\mathit{TS}}_2}^{†}A\left(1-\beta \right)=r_{\max ,2}\left(1-\beta \right)$$ [36]

$$r_{3=rds}={\nu }^{†}{K}_{\mathit{eq},\mathit{A}\to {\mathit{TS}}_3}^{†}A\left(1-\beta \right)=r_{\max ,3}\left(1-\beta \right)$$ [37]

$$r_{4=rds}={\nu }^{†}{K}_{\mathit{eq},\mathit{A}\to {\mathit{TS}}_4}^{†}A\left(1-\beta \right)=r_{\max ,4}\left(1-\beta \right).$$ [38]

We note that we have defined r_max,i as the maximum rate for step i for the case where β is equal to zero. The maximum rate of an elementary reaction can also be obtained using Sabatier–Gibbs analysis (13, 14), in which the surface coverages by the reaction intermediates are first calculated assuming that the intermediates are in equilibrium with the reactants and/or the products of the overall reaction. The maximum rates of the elementary steps in the forward and/or reverse directions are then calculated from the maximum coverages calculated in this manner and the estimated activation barriers. This methodology is equivalent to the approach presented above in which equilibrium relations are written to form the transition states from the reactants and/or products of the overall reaction.

An initial assessment of the feasibility of a reaction scheme is to estimate the maximum rates of all of the steps and then verify that all of these values are comparable to or faster than the observed rate of the catalytic reaction under investigation.

The next phase of the analysis is to estimate the degrees of rate control for the various steps in the reaction scheme. This analysis is carried out by solving for the values of Z_i for the steps by noting that the net rates of all steps are equal to the net rate of the overall reaction, r:

$$r=r_{\max ,1}\left(1-Z_1\right)$$ [39]

$$r=r_{\max ,2}{Z}_1\left(1-Z_2\right)$$ [40]

$$r=r_{\max ,3}{Z}_1{Z}_2\left(1-Z_3\right)$$ [41]

$$r=r_{\max ,4}{Z}_1{Z}_2{Z}_3\left(1-\frac{\beta }{Z_1{Z}_2{Z}_3}\right).$$ [42]

The following expressions for the reversibilities are obtained:

$$Z_1=\frac{r_{\max ,1}{r}_{\max ,2}{r}_{\max ,3}+\beta r_{\max ,2}{r}_{\max ,3}{r}_{\max ,4}+r_{\max ,3}{r}_{\max ,4}{r}_{\max ,1}+r_{\max ,4}{r}_{\max ,1}{r}_{\max ,2}}{r_{\max ,1}{r}_{\max ,2}{r}_{\max ,3}+r_{\max ,2}{r}_{\max ,3}{r}_{\max ,4}+r_{\max ,3}{r}_{\max ,4}{r}_{\max ,1}+r_{\max ,4}{r}_{\max ,1}{r}_{\max ,2}}$$ [43]

$$Z_2=\frac{r_{\max ,1}{r}_{\max ,2}{r}_{\max ,3}+\beta r_{\max ,2}{r}_{\max ,3}{r}_{\max ,4}+\beta r_{\max ,3}{r}_{\max ,4}{r}_{\max ,1}+r_{\max ,4}{r}_{\max ,1}{r}_{\max ,2}}{r_{\max ,1}{r}_{\max ,2}{r}_{\max ,3}+\beta r_{\max ,2}{r}_{\max ,3}{r}_{\max ,4}+r_{\max ,3}{r}_{\max ,4}{r}_{\max ,1}+r_{\max ,4}{r}_{\max ,1}{r}_{\max ,2}}$$ [44]

$$Z_3=\frac{r_{\max ,1}{r}_{\max ,2}{r}_{\max ,3}+\beta r_{\max ,2}{r}_{\max ,3}{r}_{\max ,4}+\beta r_{\max ,3}{r}_{\max ,4}{r}_{\max ,1}+\beta r_{\max ,4}{r}_{\max ,1}{r}_{\max ,2}}{r_{\max ,1}{r}_{\max ,2}{r}_{\max ,3}+\beta r_{\max ,2}{r}_{\max ,3}{r}_{\max ,4}+\beta r_{\max ,3}{r}_{\max ,4}{r}_{\max ,1}+r_{\max ,4}{r}_{\max ,1}{r}_{\max ,2}}\hspace{0.17em}.$$ [45]

The degree of rate control for step i can be calculated from the reversibility of the step and the sensitivity, s_i, of the overall rate to changes in forward rate constant for the step:

$$s_i={\left(\frac{\partial r}{\partial k_i}\right)\frac{k_i}{r}|}_{k_{j\ne i}}$$ [46]

$$X_{RC,i}=s_i\left(1-Z_i\right),$$ [47]

where the partial derivative is taken with all other rate constants (k_j) held constant. An important distinction between the sensitivity (Eq. 46) and the degree of rate control (Eq. 1) is that in computing the degree of rate control, all of the equilibrium constants are held constant, whereas while computing the sensitivity of an elementary reaction, all of the rate constants are held constant except for the forward rate constant of the elementary reaction of interest. Eq. 47 shows that the degree of rate control approaches zero as the reversibility of the step approaches unity (10). In a sequence of steps for which the stoichiometric coefficients of the steps are equal to unity, the sensitivity for step i is related to the sensitivity of the preceding step i−1 (15):

$$s_i=Z_{i-1}{s}_{i-1}.$$ [48]

Accordingly, sensitivity of the rate with respect to changes in the forward rate constants is passed from one step to the following step for cases where the reversibility of the previous step approaches unity, whereas the rate is insensitive for steps following an irreversible step because the reversibility of the previous step approaches zero. The above relation for the degree of rate control X_RC,i in terms of the sensitivity s_i, and the relation for s_i in terms of s_i−1 can be written as

$$X_{RC,1}=s_1\left(1-Z_1\right)$$ [49]

$$X_{RC,2}=s_1{Z}_1\left(1-Z_2\right)$$ [50]

$$X_{RC,3}=s_1{Z}_1{Z}_2\left(1-Z_3\right)$$ [51]

$$X_{RC,4}=s_1{Z}_1{Z}_2{Z}_3\left(1-\frac{\beta }{Z_1{Z}_2{Z}_3}\right).$$ [52]

The values of Z_i can then be expressed in terms of r_max,i, and the value of s₁ is obtained by requiring that the sum of X_RC,i be equal to unity, leading to the following result:

$$X_{RC,1}=\frac{r_{\max ,2}{r}_{\max ,3}{r}_{\max ,4}}{r_{\max ,1}{r}_{\max ,2}{r}_{\max ,3}+r_{\max ,2}{r}_{\max ,3}{r}_{\max ,4}+r_{\max ,3}{r}_{\max ,4}{r}_{\max ,1}+r_{\max ,4}{r}_{\max ,1}{r}_{\max ,2}}$$ [53]

$$X_{RC,2}=\frac{r_{\max ,3}{r}_{\max ,4}{r}_{\max ,1}}{r_{\max ,1}{r}_{\max ,2}{r}_{\max ,3}+r_{\max ,2}{r}_{\max ,3}{r}_{\max ,4}+r_{\max ,3}{r}_{\max ,4}{r}_{\max ,1}+r_{\max ,4}{r}_{\max ,1}{r}_{\max ,2}}$$ [54]

$$X_{RC,3}=\frac{r_{\max ,4}{r}_{\max ,1}{r}_{\max ,2}}{r_{\max ,1}{r}_{\max ,2}{r}_{\max ,3}+r_{\max ,2}{r}_{\max ,3}{r}_{\max ,4}+r_{\max ,3}{r}_{\max ,4}{r}_{\max ,1}+r_{\max ,4}{r}_{\max ,1}{r}_{\max ,2}}$$ [55]

$$X_{RC,4}=\frac{r_{\max ,1}{r}_{\max ,2}{r}_{\max ,3}}{r_{\max ,1}{r}_{\max ,2}{r}_{\max ,3}+r_{\max ,2}{r}_{\max ,3}{r}_{\max ,4}+r_{\max ,3}{r}_{\max ,4}{r}_{\max ,1}+r_{\max ,4}{r}_{\max ,1}{r}_{\max ,2}}\hspace{0.17em}.$$ [56]

The net rate of the overall reaction is then calculated from the net rate of any step by substitution of the appropriate values of Z_i in terms of the maximum rates, r_max,i:

$$r=\frac{r_{\max ,1}{r}_{\max ,2}{r}_{\max ,3}{r}_{\max ,4}(1-\beta )}{r_{\max ,1}{r}_{\max ,2}{r}_{\max ,3}+r_{\max ,2}{r}_{\max ,3}{r}_{\max ,4}+r_{\max ,3}{r}_{\max ,4}{r}_{\max ,1}+r_{\max ,4}{r}_{\max ,1}{r}_{\max ,2}}\hspace{0.17em}.$$ [57]

The above relation for X_RC,i can be generalized for a sequence of n steps:

$$X_{RC,i}=\frac{{\displaystyle \underset{j=1,j\ne i}{\overset{n}{\Pi }}{r}_{\max ,\text{j}}}}{{\displaystyle \sum _{k=1}^n\left({\displaystyle \underset{j=1,j\ne k}{\overset{n}{\Pi }}{r}_{\max ,\text{j}}}\right)}}\hspace{0.17em},$$ [58]

the rate of the overall reaction is then given by:

$$r=\frac{\left({\displaystyle \underset{j=1}{\overset{n}{\Pi }}{r}_{\max ,\text{j}}}\right)\left(1-\beta \right)}{{\displaystyle \sum _{k=1}^n\left({\displaystyle \underset{j=1,j\ne k}{\overset{n}{\Pi }}{r}_{\max ,\text{j}}}\right)}}\hspace{0.17em},$$ [59]

the above relations show that it is possible to calculate the degree of rate control for each step (X_RC,i) and the rate of the overall reaction (r) by first calculating the maximum rate of each step (r_max,i). The expression for the rate of the overall reaction can also be written as

$$r=X_{RC,i}{r}_{\max ,i}\hspace{0.17em}\left(1-\beta \right),$$ [60]

indicating that the net rate of the overall reaction is equal to the product of the degree of rate control, the maximum rate of the step, and the reversibility of the overall stoichiometric reaction.

Reactions on Catalyst Surfaces

We note that the results derived above are valid for a heterogeneous, homogeneous, or biological catalytic reaction in the limit of low coverage of the catalyst sites (16). To account for population of catalyst sites by adsorbed species we now modify the reaction scheme to include adsorption and reaction on active sites, *:

$$\begin{array}{l}1.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{A}+*\rightleftharpoons \hspace{0.17em}{\text{I}}_1^{\ast }\\ 2.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\text{I}}_1^{\ast }\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{I}}_2^{\ast }\\ 3.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\text{I}}_2^{\ast }\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{I}}_3^{\ast }\\ 4.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\text{I}}_3^{\ast }\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{B}+*\\ \text{overall}:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{A}\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\mathrm{B}.\end{array}$$

As a first approximation, we assume that all adsorbed species are quasi-equilibrated with the gas phase, leading to the following relation for conservation of active sites:

$$\begin{array}{l}1={\theta }_{\ast }+{\theta }_{I_1}+{\theta }_{I_2}+{\theta }_{I_3}\\ 1={\theta }_{\ast }+K_{eq,1}{P}_A{\theta }_{\ast }+K_{eq,1}{K}_{eq,2}{P}_A{\theta }_{\ast }+K_{eq,1}{K}_{eq,2}{K}_{eq,3}{P}_A{\theta }_{\ast }\\ {\theta }_{\ast }=\frac{1}{1+K_{eq,1}{P}_A+K_{eq,1}{K}_{eq,2}{P}_A+K_{eq,1}{K}_{eq,2}{K}_{eq,3}{P}_A}\hspace{0.17em},\end{array}$$ [61]

where ${\theta }_{\ast }$ represents the fraction of the active sites that are free of adsorbed species, and we have assumed for simplicity that the reactant A and the product B are not strongly adsorbed on the catalyst. In general, assuming that the most abundant adsorbed species are in quasi-equilibrium with the gas or liquid phase, the value of ${\theta }_{\ast }$ can be estimated as

$${\theta }_{\ast }=\frac{1}{1+{\displaystyle \sum _{i=1}^n{K}_{ads,i}\left({\displaystyle \underset{j=1}{\overset{k}{\Pi }}{P}_j^{{\nu }_{ij}}}\right)}}\hspace{0.17em},$$ [62]

where the sum in the denominator is over the n abundant surface species, K_ads,i is the lumped equilibrium constant to form a specific adsorbed species from the reactants and/or products, and ν_ij are the stoichiometric coefficients in the lumped equilibrium to produce adsorbed species i from reactants and/or products j.

It is now possible to estimate the rate of the catalytic reaction by scaling the above expression for r, corresponding to low coverage of the catalyst sites, to take into account the fraction of the sites that is available for reaction, ${\theta }_{\ast }$. For example, if the catalytic reaction requires “m” adjacent sites, then r is scaled by ${\theta }_{\ast }^m$, leading to

$$r=\frac{\left({\displaystyle \underset{j=1}{\overset{n}{\Pi }}{r}_{\max ,\text{j}}}\right)\left(1-\beta \right)}{{\displaystyle \sum _{k=1}^n\left({\displaystyle \underset{j=1,j\ne k}{\overset{n}{\Pi }}{r}_{\max ,\text{j}}}\right)}{\left[1+{\displaystyle \sum _{i=1}^n{K}_{ads,i}\left({\displaystyle \underset{j=1}{\overset{k}{\Pi }}{P}_j^{{\nu }_{ij}}}\right)}\right]}^m}\hspace{0.17em}.$$ [63]

Analytical Strategy for Analysis of Reaction Schemes

We now suggest an analytical strategy to assess the feasibility of a proposed reaction scheme to describe the reaction kinetics for a catalytic reaction of interest, and then to use this reaction scheme to suggest directions for research to identify promising catalysts for the catalytic reaction. We note that the above approach is valid for heterogeneous, homogeneous, and biological catalysis, and it has been used for electrochemical processes (17–19). We illustrate this strategy below for the case of a heterogeneous catalytic reaction as illustrated in Fig. 1.

Fig. 1.

Schematic showing the strategy for analyzing a reaction scheme.

The first step is to estimate the thermodynamic properties (i.e., enthalpy of formation and absolute entropy) of the reactants, products, and reaction intermediates in the gas phase (or the liquid phase). The next step is to estimate values of binding energies for adsorption of all species on the surface, for example using experimental results from the literature, results from DFT calculations, or scaling relations (20–24). In addition, it is necessary to estimate values for the entropies of the adsorbed species, for example using results from experimental studies (25), correlations in the literature (26), results from DFT calculations of vibrational frequencies (27–34), combined with hindered translator and hindered rotor models for the three modes associated with motion parallel to the surface (35, 36), or by assuming models for the extent of mobility on the surface. It is now possible to estimate values for lumped equilibrium constants, $K_{ads,i}$, to form the adsorbed species from the reactants and/or products, followed by calculation of ${\theta }_{\ast }$. This analysis provides information about which species are expected to be abundant on the surface at various reaction conditions. Accordingly, the binding energies of these abundant species are significant in determining catalyst performance, and it would be desirable to obtain more precise values for these binding energies. In contrast, the binding energies of the remaining adsorbed species are kinetically insignificant, and more precise values are not required.

The next step in the analysis is to estimate values for the activation barriers for all steps in the reaction scheme. For example, the activation barrier for a step could be estimated from an experimental value for the reaction of interest or a similar reaction, estimated from results of DFT calculations (37), or estimated using Brønsted–Evans–Polanyi (BEP) scaling relations (38–40). These BEP relations provide estimates for the thermodynamic properties of transition states in terms of stable adsorbed species, thereby allowing the estimation of activation energies in terms of the binding energies used above to determine the surface coverages by adsorbed species. It is also required in this phase of the analysis to estimate values for changes in entropy associated with formation of the transition states from the adsorbed intermediates. It is typically sufficient for initial calculations to assume that these entropy changes are equal to zero for reactions taking place on the surface. It is now possible to estimate values for $K_{eq,A\to T{S}_i}^{†}$, corresponding the lumped equilibrium constants for the formation of the transition states from the reactants and/or products of the reaction, followed by calculation of the maximum rate for each step of the reaction scheme.

The values of $r_{\max ,\mathit{i}}$ estimated above can now be compared with the desired or measured rate of the overall stoichiometric reaction, r. If the values of $r_{\max ,\mathit{i}}$ for steps in the reaction scheme are much lower than the overall rate r, then the reaction conditions must be changed to increase the values of these maximum rates, or a new reaction scheme must be proposed. If the values of $r_{\max ,\mathit{i}}$ for all steps are sufficiently fast compared with the overall rate r, then it is possible to calculate values for the degrees of rate control, X_RC,i, to determine the extents to which each step controls the rate of the overall stoichiometric reaction. Accordingly, steps with finite degrees of rate control are significant in determining catalyst performance, and it would be desirable to obtain more precise values for the properties of the transition states for these steps. In contrast, the transition states of the remaining steps are kinetically insignificant, and more precise values for the properties of these transition states are not required.

Using the values of X_RC,i estimated above, it is now possible to identify the key transition states in the reaction scheme and then use the expansion of the rate in terms of changes in the Gibbs free energies of these key transition states to identify promising catalytic materials, as described by Campbell and coworkers (12). Optimization of catalyst performance is dictated by the Sabatier principle that the surface should stabilize strongly the key transition states identified from reaction steps with finite values of X_RC,i, whereas the surface should not bind adsorbed species too strongly, such that the values of $K_{ads,i}$ lead to values of ${\theta }_{\ast }$ that are approximately equal to 0.5 for the optimal catalyst (41).

In parallel with analysis of the reaction scheme in terms of the values of X_RC,i, the values of $r_{\max ,i}$ can also be used to predict the rate of the overall stoichiometric reaction. Thus, it is possible to estimate the reaction kinetics for the overall stoichiometric reaction, such as predicting the apparent activation energy by changing the reaction temperature, and predicting the apparent reaction orders by changing the partial pressures of the reactants and products. These predictions can then be compared with results from experimental measurements to determine whether the proposed reaction scheme is consistent with the experimental data. It is also possible to use the predictions of the overall rate to determine the reaction conditions for which the proposed reaction scheme would lead to favorable catalytic properties. More generally, it is possible to study how optimization of catalyst performance by changing the thermodynamic properties of the key transition states and adsorbed species is affected by the nature of the reaction conditions.

We note that the above analysis of proposed reaction schemes in terms of values of lumped equilibrium constants, $K_{eq,A\to T{S}_i}^{†}$, to form transition states from the reactants and/or products, coupled with calculation of ${\theta }_{\ast }$, is based on the assumptions that the stoichiometric coefficients of the reaction steps in the scheme are equal to unity, and the adsorbed species are in quasi-equilibrium with the gas phase. More generally, we show below that this approach can be used in an approximate manner as a preliminary method to probe the performance of more general reaction schemes, followed by more detailed analyses using full microkinetic models to determine the surface coverages by adsorbed species and the degrees of rate control of the elementary steps.

The expressions above are exact for the reversibilities (Eqs. 43–45), the degrees of rate control (Eq. 58), and the overall rate of the reaction (Eq. 59) obtained from the maximum rate analysis for a reaction mechanism with all stoichiometric coefficients equal to unity. More complex expressions are obtained for a more general reaction scheme where the stoichiometric coefficients are not all equal to unity. For example, consider a two-step reaction scheme wherein the elementary step 2 must occur twice for each catalytic cycle:

$$\begin{array}{l}1.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\text{A}}_2\hspace{0.17em}\rightleftharpoons \hspace{0.17em}2\text{A}\\ 2.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{A}+\text{B}\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{AB}\\ \text{overall}:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\text{A}}_2+2\text{B}\hspace{0.17em}\rightleftharpoons \hspace{0.17em}2\text{AB}.\end{array}$$

For this reaction mechanism it can be shown (SI Appendix) that the reversibilities, the degrees of rate control, and the overall reaction rate are

$$\begin{array}{c}{Z}_1=1+\left(\frac{\sqrt{\beta }}{2}\right)\xi +\frac{1}{8}{\xi }^2-\frac{1}{8}\xi {\left(16+8\sqrt{\beta }\xi +{\xi }^2\right)}^{1/2}{Z}_2={\left(\frac{\beta }{1+\left(\frac{\sqrt{\beta }}{2}\right)\xi +\frac{1}{8}{\xi }^2-\frac{1}{8}\xi {\left(16+8\sqrt{\beta }\xi +{\xi }^2\right)}^{1/2}}\right)}^{1/2}\\ \\ X_{RC,1}=\frac{\xi }{{\left(16+8\sqrt{\beta }\xi +{\xi }^2\right)}^{1/2}}\hspace{0.17em}{X}_{RC,2}=1-\frac{\xi }{{\left(16+8\sqrt{\beta }\xi +{\xi }^2\right)}^{1/2}}\\ \\ r=\frac{r_2}{8}\left[{\left(16+8\sqrt{\beta }\xi +{\xi }^2\right)}^{1/2}-4\sqrt{\beta }-\xi \right],\end{array}$$ [64]

where $\xi =\left(r_{\max ,2}/r_{\max ,1}\right)$. For the case when $\xi \approx 0$, i.e., $r_{\max ,2}<<r_{\max ,1}$, the overall rate of the reaction is

$$r=\frac{r_{\max ,2}}{2}\left(1-\sqrt{\beta }\right).$$ [65]

Far from equilibrium where $\beta <<1$, the maximum overall rate of reaction (written in term of the extent of the overall reaction) is r_max,2/2. Thus, maximum rate analysis can be used in an approximate manner for a more general reaction scheme if the maximum rates of the constituent elementary steps are defined as r_max,i/σ_i.

We now use four case studies to examine the applicability of our proposed approach to predict the overall rate and analyze the reaction kinetics of stoichiometric reactions based on the calculation of the maximum rates of the elementary steps of the reaction mechanism. In these studies, we compare the results of maximum rate analyses to results from full microkinetic analyses for the water–gas shift reaction on a Cu (111) surface. The DFT values for the binding energies of the surface intermediates and the activation energies of the elementary reactions were obtained from literature (42). The goal of these case studies is to compare the results of maximum rate analysis and the full microkinetic model. Accordingly, the DFT parameters used in these case studies were not optimized for predicting experimental reaction kinetics data. For example, it is known that the binding energy of carbon monoxide is a function of its surface coverage; also, at the experimental conditions the surface coverage of bidentate formate specie is high (42). However, these nuances of the mechanism were not included in these case studies. We present below a summary of the results for these case studies, and the details of these case studies are presented in SI Appendix.

Case Study I: Redox Mechanism for Water–Gas Shift

We first consider the redox mechanism for the water–gas shift reaction, involving the oxidation of CO by adsorbed oxygen atoms, O*. Atomic oxygen is obtained from H₂O by two successive H-atom abstraction steps (steps 3 and 4 in Table 1). The stoichiometric coefficients of all of the elementary steps in this example are equal to unity (Table 1). This reaction is analyzed for a copper catalyst at a temperature of 523 K and total pressure of 1 atm. The values of the equilibrium constants and forward rate constants for the elementary steps are listed in Table 2. The defining relations and values of the equilibrium constants for the lumped reactions that describe the formation of adsorbed intermediates from gas-phase reactants and products of the overall stoichiometric reaction are listed in Table 3. Table 4 lists the defining relations and values of equilibrium constants for lumped reactions that describe the formation of the transition states from the reactants and products of the overall stoichiometric reaction. Defining relations and values of the maximum rates are shown in Table 5.

Table 1.

Redox mechanism for water–gas shift (case I)

Step	Elementary reaction	Stoichiometric coefficients, σi
1	$\text{CO}\hspace{0.17em}+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{CO}*$	1
2	${\text{H}}_2\text{O}\hspace{0.17em}+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{H}}_2\text{O}*$	1
3	${\text{H}}_2\text{O}*+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{H}*\hspace{0.17em}+\hspace{0.17em}\text{OH}*$	1
4	$\text{OH}*+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{H}*\hspace{0.17em}+\hspace{0.17em}\text{O}*$	1
5	$\text{CO}*+\hspace{0.17em}\text{O}*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{CO}}_2*\hspace{0.17em}+*$	1
6	${\text{CO}}_2*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{CO}}_2\hspace{0.17em}+*$	1
7	$2\text{H}*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{H}}_2\hspace{0.17em}+\hspace{0.17em}2*$	1

Table 2.

Equilibrium and rate constants for elementary reactions (case I)

Step	Elementary reaction	$K_{eq,i}$	$k_{for,i}$, s−1
1	$\text{CO}\hspace{0.17em}+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{CO}*$	2.15 × 102	1.33 × 108
2	${\text{H}}_2\text{O}\hspace{0.17em}+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{H}}_2\text{O}*$	5.93 × 10−5	2.01 × 1011
3	${\text{H}}_2\text{O}*+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{H}*+\text{OH}*$	6.28 × 10−2	2.64 × 106
4	$\text{OH}*+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{H}*+\text{O}*$	1.18 × 10−5	5.24 × 101
5	$\text{CO}*+\hspace{0.17em}\text{O}*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{CO}}_2*+*$	1.03 × 103	2.05 × 105
6	${\text{CO}}_2*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{CO}}_2+*$	1.92 × 105	1.48 × 1012
7	$2\text{H}*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{H}}_2+\hspace{0.17em}2*$	4.50 × 101	5.32 × 102

Table 3.

Defining relations for surface coverages of adsorbed intermediates and values of equilibrium constants for adsorbed species (case I)

Adsorbed species, I*	Lumped reaction§	${\theta }_{I*}=K_{ads,i}\left({\displaystyle \prod _{j=1}^k{P}_j^{{\nu }_{ij}}}\right){\theta }_{\ast }$	$K_{ads,i}$	$K_{ads,i}\left({\displaystyle \prod _{j=1}^k{P}_j^{{\nu }_{ij}}}\right)$‡
CO*	$\text{CO}+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{CO}*$	$K_{ads,CO\ast }\hspace{0.17em}{P}_{CO}\hspace{0.17em}{\theta }_{\ast }$	2.15 × 102	1.51 × 101
H2O*	${\text{H}}_2\text{O}+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{H}}_2\text{O}*$	$K_{ads,H_{2}O\ast }\hspace{0.17em}{P}_{H_{2}O}\hspace{0.17em}{\theta }_{\ast }$	5.93 × 10−5	1.24 × 10−5
H*	$0.5\hspace{0.17em}{\text{H}}_2+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{H}*$	$K_{ads,H\ast }\hspace{0.17em}{P}_{H_2}^{0.5}\hspace{0.17em}\hspace{0.17em}{\theta }_{\ast }$	1.49 × 10−1	9.19 × 10−2
OH*	${\text{H}}_2\text{O}+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{OH}*+0.5\hspace{0.17em}{\text{H}}_2$	$K_{ads,OH\ast }{P}_{H_{2}O}{P}_{H_2}^{-0.5}\hspace{0.17em}{\theta }_{\ast }$	2.50 × 10−5	8.51 × 10−6
O*	${\text{H}}_2\text{O}+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{O}*+{\text{H}}_2$	$K_{ads,O\ast }\hspace{0.17em}{P}_{H_{2}O}{P}_{H_2}^{-1}\hspace{0.17em}{\theta }_{\ast }$	1.98 × 10−9	1.09 × 10−9
CO2*	${\text{CO}}_2+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{CO}}_2*$	$K_{ads,C{O}_2\ast }\hspace{0.17em}{P}_{C{O}_2}\hspace{0.17em}{\theta }_{\ast }$	5.21 × 10−6	4.43 × 10−7

^§Lumped reactions are written to form adsorbed intermediates from gas-phase reactants and products.

^‡$P_{CO},\hspace{0.17em}{P}_{H_{2}O},\hspace{0.17em}{P}_{H_2},\hspace{0.17em}\text{and}\hspace{0.17em}{P}_{C{O}_2}$ are 0.07 atm, 0.21 atm, 0.38 atm, and 0.085 atm, respectively.

Table 4.

Defining relations and values of equilibrium constants for the formation of transition states from gas-phase reactants and products of the overall stoichiometric reaction (case I)

Step	Reaction	$K_{\mathit{eq},\mathit{A}\to \mathit{TS},\mathit{i}}^{†}$	$K_{\mathit{eq},\mathit{A}\to \mathit{TS},\mathit{i}}^{†}$
1	$\mathrm{CO}+\ast \rightleftharpoons \mathrm{C}{\mathrm{O}}_{\ast }^{†}$	$K_{eq,1}^{†}$	1.22 × 10−5
2	${\mathrm{H}}_2\mathrm{O}+\ast \rightleftharpoons {\mathrm{H}}_2{\mathrm{O}}_{\ast }^{†}$	$K_{eq,2}^{†}$	1.84 × 10−2
3	${\mathrm{H}}_2\mathrm{O}+2\ast \rightleftharpoons {\mathrm{H}}_2{\mathrm{O}}_{\ast \ast }^{†}$	$K_{eq,2}{K}_{eq,3}^{†}$	1.44 × 10−11
4	${\mathrm{H}}_2\mathrm{O}+2\ast \rightleftharpoons \mathrm{O}{\mathrm{H}}_{\ast \ast }^{†}+0.5{\mathrm{H}}_2$	$K_{eq,2}{K}_{eq,3}{K}_{eq,7}^{0.5}{K}_{eq,4}^{†}$	1.20 × 10−16
5	$\mathrm{CO}+{\mathrm{H}}_2\mathrm{O}+2\ast \rightleftharpoons \mathrm{C}{\mathrm{O}}_{2\ast \ast }^{†}+{\mathrm{H}}_2$	$K_{eq,1}{K}_{eq,2}{K}_{eq,3}{K}_{eq,4}{K}_{eq,7}{K}_{eq,5}^{†}$	7.99 × 10−15
6	$\mathrm{C}{\mathrm{O}}_2+\ast \rightleftharpoons \mathrm{C}{\mathrm{O}}_{2\ast }^{†}$	$K_{eq,6}^{†}/K_{eq,6}$	7.08 × 10−7
7	${\mathrm{H}}_2+2\ast \rightleftharpoons {\mathrm{H}}_{2\ast \ast }^{†}$	$K_{eq,7}^{†}/K_{eq,7}$	1.08 × 10−12

Table 5.

Maximum rates for elementary steps of the mechanism (case I)

Step	$r_{\max ,i}$§‡, s−1	$r_{\max ,i}$, s−1
1	$r_{\max ,1}={\nu }^{†}\left(K_{\mathit{eq},\mathit{A}\to \mathit{TS},1}^{†}\right)\hspace{0.17em}{P}_{CO}$	9.28 × 106
2	$r_{\max ,2}={\nu }^{†}\left(K_{\mathit{eq},\mathit{A}\to \mathit{TS},2}^{†}\right)\hspace{0.17em}{P}_{H_{2}O}$	4.22 × 1010
3	$r_{\max ,3}={\nu }^{†}\left(K_{\mathit{eq},\mathit{A}\to \mathit{TS},3}^{†}\right)\hspace{0.17em}{P}_{H_{2}O}$	3.28 × 101
4	$r_{\max ,4}={\nu }^{†}\left(K_{\mathit{eq},\mathit{A}\to \mathit{TS},4}^{†}\right)\hspace{0.17em}{P}_{H_{2}O}\hspace{0.17em}{P}_{H_2}^{-0.5}$	4.46 × 10−4
5	$r_{\max ,5}={\nu }^{†}\left(K_{\mathit{eq},\mathit{A}\to \mathit{TS},5}^{†}\right)\hspace{0.17em}{P}_{CO}\hspace{0.17em}{P}_{H_{2}O}\hspace{0.17em}{P}_{H_2}^{-1}$	3.37 × 10−3
6	$r_{\max ,6}={\nu }^{†}\left(K_{\mathit{eq},\mathit{A}\to \mathit{TS},6}^{†}\right)\hspace{0.17em}{P}_{C{O}_2}$	6.55 × 105
7	$r_{\max ,7}={\nu }^{†}\left(K_{\mathit{eq},\mathit{A}\to \mathit{TS},7}^{†}\right)\hspace{0.17em}{P}_{H_2}$	4.49 × 100

^§$P_{CO},P_{H_{2}O},P_{H_2},\text{and}\hspace{0.17em}{P}_{C{O}_2}$ are 0.07 atm, 0.21 atm, 0.38 atm, and 0.085 atm, respectively.

^‡${\nu }^{†}=k_{B}T/h$.

The seven elementary steps and six reaction intermediates shown in Table 1 are included in a microkinetic model to determine the rate of the overall stoichiometric reaction and the degrees of rate control of the various steps. The following coupled differential equations are solved numerically to obtain the surface coverages at steady state, the rate of the overall stoichiometric reaction, the apparent activation energy, and the apparent reaction orders:

$$\begin{array}{l}\frac{\partial {\theta }_{CO}}{\partial t}=r_1-r_5=0\\ \frac{\partial {\theta }_{H_{2}O}}{\partial t}=r_2-r_3=0\\ \frac{\partial {\theta }_H}{\partial t}=r_3+r_4-2r_7=0\\ \frac{\partial {\theta }_{OH}}{\partial t}=r_3-r_4=0\\ \frac{\partial {\theta }_O}{\partial t}=r_4-r_5=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\\ \frac{\partial {\theta }_{C{O}_2}}{\partial t}=r_5-r_6=0.\end{array}$$

These differential equations are integrated over time, t, from a known starting point (e.g., a clean surface) to a surface operating at steady state where the derivatives are equal to zero. Table 6 shows a comparison of the overall rate of the reaction and the surface coverages for the mechanism of the water–gas shift reaction shown in Table 1. It can be seen that the results of the microkinetic model and the maximum rate analysis are in good agreement. The apparent activation energy and reaction orders obtained from the microkinetic model and the maximum rate analysis are presented in Table 7. The predicted degree of rate control (SI Appendix, Fig. S1), Arrhenius plot (SI Appendix, Fig. S2), and plots of rate versus partial pressures leading to the apparent reaction orders (SI Appendix, Fig. S3) show good agreement between the predictions of the microkinetic model and maximum rate analysis.

Table 6.

Comparison of overall rate and surface coverages of adsorbed intermediates calculated by maximum rate analysis and using the microkinetic model (case I)

Rate/coverage	Microkinetic model	Maximum rate analysis
Rate of reaction, s−1	1.68 × 10−6	1.51 × 10−6
CO*	9.32 × 10−1	9.32 × 10−1
H2O*	7.21 × 10−7	7.70 × 10−7
H*	5.46 × 10−3	5.69 × 10−3
OH*	6.14 × 10−7	5.26 × 10−7
O*	1.05 × 10−11	6.76 × 10−11
CO2*	2.10 × 10−8	2.74 × 10−8
Vacant sites	6.20 × 10−2	6.18 × 10−2

Table 7.

Predicted apparent activation energy and reaction orders from the microkinetic model and maximum rate analysis (case I)

Reaction kinetics	Microkinetic model	Maximum rate analysis
Apparent activation energy, kJ⋅mol−1	282	285
Reaction orders
H2O	1.07	1.00
CO	−1.84	−1.85
H2	−0.59	−0.55
CO2	−0.04	0.00

Case Study II: Redox Reaction Mechanism for Water–Gas Shift (OH Disproportionation)

We next consider a reaction mechanism where the stoichiometric coefficients of all of the elementary steps are not equal to unity. For this purpose we choose a redox mechanism for water–gas shift where the adsorbed atomic oxygen is obtained from the disproportionation of two adsorbed hydroxyl intermediates, as shown in Table 8. In this reaction scheme, the dissociation of adsorbed water must take place twice for each turnover of the reactants to products. The reaction mechanism is analyzed for a copper catalyst at a temperature of 523 K and a pressure of 1 atm. Table 9 shows a comparison of the overall rate of the reaction and the surface coverages for the mechanism of the water–gas shift reaction shown in Table 8. It can be seen that the results from the microkinetic model and the maximum rate analysis are in good agreement.

Table 8.

Redox reaction mechanism for water–gas shift (OH disproportionation, case II)

Step	Elementary reaction	Stoichiometric coefficients, σi
1	$\text{CO}+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{CO*}$	1
2	${\text{H}}_2\text{O}+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{H}}_2\text{O*}$	1
3	${\text{H}}_2\text{O*}+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{H*}+\text{OH*}$	2
4	$\text{OH*}+\text{OH*}\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{H}}_2\text{O*}+\text{O*}$	1
5	$\text{CO*}+\text{O*}\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{CO}}_2*+*$	1
6	${\text{CO}}_2*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{CO}}_2+*$	1
7	$2\text{H*}\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{H}}_2+2*$	1

Table 9.

Comparison of the rate and surface coverages of adsorbed intermediates predicted by maximum rate analysis and the microkinetic model (case II)

Rate/coverage	Microkinetic model	Maximum rate analysis
Rate of reaction, s−1	1.27 × 10−5	1.29 × 10−5
CO*	9.26 × 10−1	9.32 × 10−1
H2O*	7.69 × 10−7	7.70 × 10−7
H*	6.04 × 10−3	5.69 × 10−3
OH*	6.50 × 10−7	5.26 × 10−7
O*	6.87 × 10−11	6.76 × 10−11
CO2*	2.06 × 10−8	2.74 × 10−8
Vacant sites	6.81 × 10−2	6.18 × 10−2

The apparent activation energy and reaction orders obtained from the microkinetic model and the maximum rate analysis are presented in Table 10. The predicted degree of rate control (SI Appendix, Fig. S4), Arrhenius plot (SI Appendix, Fig. S5), and the plots of rate versus partial pressure leading to the apparent reaction orders (SI Appendix, Fig. S6) show good agreement between the predictions of the microkinetic model and maximum rate analysis.

Table 10.

Predicted apparent activation energy and reaction orders from the microkinetic model and maximum rate analysis

Reaction kinetics	Microkinetic model	Maximum rate analysis
Apparent activation energy, kJ⋅mol−1	213	221
Reaction orders
H2O	1.04	1.00
CO	−0.97	−0.92
H2	−1.01	−1.00
CO2	−0.05	0.00

Using the values of rate and equilibrium constants taken from the literature, the rate of the proposed reaction mechanism (Table 8) is predominantly controlled by the elementary reaction involving the formation of adsorbed carbon dioxide. However, to probe the applicability of our proposed methodology for a reaction mechanism where the rate is controlled by an elementary reaction having a stoichiometric number that is not equal to 1, we increase the activation energy of water dissociation step (step 3 in Table 8) to make it the rate-controlling step. The maximum rate of an elementary step is then obtained by the following equation:

$$r_{i,\max }=\frac{1}{{\sigma }_i}\left(\frac{\left({\displaystyle \underset{j=1}{\overset{n}{\Pi }}{r}_{\max ,\text{j}}}\right)\left(1-\beta \right)}{{\displaystyle \sum _{k=1}^n\left({\displaystyle \underset{j=1,j\ne k}{\overset{n}{\Pi }}{r}_{\max ,\text{j}}}\right){\left[1+{\displaystyle \sum _{i=1}^n{K}_{ads,i}\left({\displaystyle \underset{j=1}{\overset{k}{\Pi }}{P}_j^{{\nu }_{ij}}}\right)}\right]}^2}}\right).$$ [66]

The activation energy for the water dissociation step was varied from 1.15 eV (where the rate is predominantly controlled by the formation of adsorbed carbon dioxide) to 1.6 eV (where the rate is controlled by the dissociation of the adsorbed water). Fig. 2 shows that the maximum rate analysis approach is able to predict the degree of rate control over the entire range of the activation energy studied.

Fig. 2.

Comparison of predicted degrees of rate control using maximum rate analysis and from the full microkinetic model (case II). Blue, step 5, microkinetic model; pink, step 5, maximum rate analysis; red, step 3, microkinetic model; black, step 3, maximum rate analysis.

Case Study III: Carboxyl Reaction Mechanism for Water–Gas Shift

We now consider the “carboxyl” mechanism for the water–gas shift reaction (Table 11). The carboxyl intermediate formed in step 4 is in a cis configuration (cCOOH). In step 5, the cis-carboxyl species isomerizes into a trans-carboxyl species (tCOOH) where the H-atom is pointing away from the surface. The tCOOH then dissociates into adsorbed carbon dioxide and adsorbed hydrogen atom (step 6). The reaction mechanism is analyzed for a copper catalyst at a temperature of 523 K and a pressure of 1 atm. The stoichiometric coefficients of all of the elementary steps in this mechanism are equal to unity (Table 11). Table 12 shows a comparison of the overall rate of the reaction and the surface coverages for the carboxyl mechanism of the water–gas shift reaction shown in Table 11. It can be seen that the results from the microkinetic model and the maximum rate analysis are in good agreement.

Table 11.

Carboxyl reaction mechanism for water–gas shift (case III)

Step	Elementary reaction	Stoichiometric coefficients (σi)
1	$\text{CO}+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{CO*}$	1
2	${\text{H}}_2\text{O}+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{H}}_2\text{O*}$	1
3	${\text{H}}_2\text{O*}+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{H*}+\text{OH*}$	1
4	$\text{CO*}+\text{OH*}\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{cCOOH*}\hspace{0.17em}+*$	1
5	$\text{cCOOH*}\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\mathrm{t}\text{COOH*}$	1
6	$\text{tCOOH*}\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{CO}}_2*+\text{H*}$	1
7	${\text{CO}}_2*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\mathrm{C}{\mathrm{O}}_2\hspace{0.17em}+*$	1
8	$2\text{H*}\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\mathrm{H}}_2\hspace{0.17em}+\hspace{0.17em}2*$	1

Table 12.

Comparison of the rate and surface coverages of adsorbed intermediates from maximum rate analysis and using the microkinetic model (case III)

Rate/coverage	Microkinetic model	Maximum rate analysis
Rate of reaction, s−1	1.72 × 10−11	1.80 × 10−11
CO*	9.34 × 10−1	9.32 × 10−1
H2O*	7.12 × 10−7	7.70 × 10−7
H*	5.35 × 10−3	5.68 × 10−3
OH*	6.08 × 10−7	5.26 × 10−7
cCOOH*	3.07 × 10−13	3.09 × 10−13
tCOOH*	4.53 × 10−12	4.57 × 10−12
CO2*	2.03 × 10−8	2.74 × 10−8
Vacant sites	6.09 × 10−2	6.18 × 10−2

The apparent activation energy and reaction orders obtained from the microkinetic model and the maximum rate analysis are presented in Table 13. The predicted degree of rate control (SI Appendix, Fig. S8), Arrhenius plot (SI Appendix, Fig. S9), and plots of rate versus partial pressures leading to apparent reaction orders (SI Appendix, Fig. S10) show good agreement between the predictions of the microkinetic model and maximum rate analysis.

Table 13.

Predicted apparent activation energy and reaction orders from the microkinetic model and maximum rate analysis

Reaction kinetics	Microkinetic model	Maximum rate analysis
Apparent activation energy, kJ⋅mol−1	239	239
Reaction orders
H2O	1.03	1.00
CO	−0.91	−0.92
H2	−0.52	−0.50
CO2	−0.04	0.00

Case Study IV: Carboxyl Reaction Mechanism for Water–Gas Shift (Carboxyl + OH)

In this case study, we consider a “carboxyl” mechanism where the stoichiometric coefficients of all of the elementary steps are not equal to unity (Table 14). This mechanism for the water–gas shift reaction involves the formation of adsorbed carbon dioxide from the reaction of adsorbed carboxyl and hydroxyl intermediates (step 6 in Table 14). Accordingly, the dissociation of adsorbed water must take place twice for each turnover of the reactants to products. The reaction mechanism is analyzed for a copper catalyst at a temperature of 523 K and a pressure of 1 atm.

Table 14.

Carboxyl reaction mechanism for water–gas shift (carboxyl + OH, case IV)

Step	Elementary reaction	Stoichiometric coefficients, σi
1	$\text{CO}+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{CO*}$	1
2	${\text{H}}_2\text{O}+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\text{H}}_2\text{O*}$	1
3	${\text{H}}_2\text{O*}+*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{H*}+\text{OH*}$	2
4	$\text{CO*}+\text{OH*}\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\text{cCOOH*}+*$	1
5	$\text{cCOOH*}\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\mathrm{t}\text{COOH*}$	1
6	$\text{tCOOH*}+\mathrm{OH}\ast \rightleftharpoons \hspace{0.17em}{\text{CO}}_2*{+\hspace{0.17em}\text{H}}_2\text{O*}$	1
7	${\text{CO}}_2*\hspace{0.17em}\rightleftharpoons \hspace{0.17em}\mathrm{C}{\mathrm{O}}_2+*$	1
8	$2\text{H*}\hspace{0.17em}\rightleftharpoons \hspace{0.17em}{\mathrm{H}}_2+2*$	1

Table 15 shows a comparison of the overall rate of the reaction and the surface coverages for the carboxyl mechanism of the water–gas shift reaction shown in Table 14. It can be seen that results from the microkinetic model and the maximum rate analysis are in good agreement.

Table 15.

Comparison of the rate and surface coverages of adsorbed intermediates calculated using maximum rate analysis and the microkinetic model (case IV)

Rate/coverage	Microkinetic model	Maximum rate analysis
Rate of reaction, s−1	3.41 × 10−7	2.86 × 10−7
CO*	9.33 × 10−1	9.32 × 10−1
H2O*	7.14 × 10−7	7.70 × 10−7
H*	5.38 × 10−3	5.68 × 10−3
OH*	6.09 × 10−7	5.26 × 10−7
cCOOH*	3.06 × 10−13	3.09 × 10−13
tCOOH*	4.50 × 10−12	4.57 × 10−12
CO2*	2.04 × 10−8	2.74 × 10−8
Vacant sites	6.12 × 10−2	6.18 × 10−2

The apparent activation energy and reaction orders obtained from the microkinetic model and the maximum rate analysis are presented in Table 16. The predicted degree of rate control (SI Appendix, Fig. S11), Arrhenius plot (SI Appendix, Fig. S12), and the plots of rate versus partial pressure leading to the apparent reaction orders (SI Appendix, Fig. S13) show good agreement between the prediction of the microkinetic model and maximum rate analysis.

Table 16.

Predicted apparent activation energy and reaction orders from the microkinetic model and maximum rate analysis

Reaction kinetics	Microkinetic model	Maximum rate analysis
Apparent activation energy, kJ⋅mol−1	157	161
Reaction orders
H2O	2.03	1.94
CO	−0.91	−0.92
H2	−1.01	−0.96
CO2	−0.04	0.00

Using the values of the rate and equilibrium constants obtained from the literature, the rate of the proposed reaction mechanism (Table 14) is predominantly controlled by the elementary reaction involving the formation of adsorbed carbon dioxide. However, to study the applicability of our proposed methodology for a reaction mechanism where the rate is controlled by an elementary reaction having a stoichiometric number that is not equal to 1, we increase the activation energy of water dissociation step (step 3 in Table 14) to make it the rate-controlling step. The activation energy for the water dissociation step was varied from 1.0 eV (where the rate is predominantly controlled by the formation of adsorbed carbon dioxide) to 1.95 eV (where the rate is controlled by the dissociation of the adsorbed water). Fig. 3 shows that our proposed method is able to predict the degree of rate control for the entire range of the activation energy studied.

Fig. 3.

Comparison of predicted degrees of rate control using maximum rate analysis and from the full microkinetic model (case IV). Blue, step 6, microkinetic model; pink, step 6, maximum rate analysis; red, step 3, microkinetic model; black, step 3, maximum rate analysis.