The degree of rate control of catalyst-bound intermediates in catalytic reaction mechanisms: Relationship to site coverage

Highlights

• The Degree of Rate Control (DRC) for catalyst-bound intermediates is quantified..

• This DRC is proportional to its fractional population of sites times a constant -σ.

• σ = the sum over all transition states of X_i times n_i , where most X_i values = 0.

• X_i = the DRC of transition state i in the mechanism.

• n_i = the number of catalyst sites of the same type required in step i.

Abstract

The degree of rate control (DRC) quantifies how much the energy of each species in a reaction mechanism (e.g., catalyst-bound intermediates and transition states) affects the net reaction rate. It thus plays an important role in understanding catalyst activity and selectivity and in efforts to find better catalysts. We show here that under steady-state reaction conditions, the DRC for any catalyst-bound intermediate n (X_n) is proportional to its fractional population of catalyst sites (θ_n),

$X_n=-\sigma \times {\theta }_n$,

where the proportionality constant σ is given by

$\sigma =\sum _i{X}_i\times n_i$.

Here, X_i is the DRC of the transition state in step i, n_i is the number of catalyst sites of the same type required for the elementary step i, and the sum is over all transition states (or elementary steps) i. Since only a few transition states typically have non-negligible DRCs, this simple sum (or weighted average of elementary-step site requirements) includes only a few terms (and only one term when there is a single rate-determining step). We also show that the DRCs of reactants and catalyst depend upon the choice of zero-energy reference, but simplify to zero when their standard states are used as the zero-energy references.

Introduction

The generalized degree of rate control (DRC) is a mathematical approach used to identify rate-controlling species in multistep reaction mechanisms [1], [2]. It quantifies the extent by which a small decrease in the standard-state free energy of a species in the reaction pathway affects the reaction rate, which is of clear value in understanding how different catalysts might change the rate and selectivity. It has found many important applications in catalysis research [1], [3], [4], [5], [6], [7], including use for computational catalyst screening [6], [8]. The definition of the generalized DRC is [2]

$$X_i={\left(\frac{\partial \left(lnr\right)}{\partial \left(-G_i^0/\mathit{RT}\right)}\right)}_{G_{j\ne i}}$$

where r is the net steady-state reaction rate (per catalyst site) toward the product of interest (or for the consumption of one reactant) and G⁰_i is the standard-state Gibbs free energy of species i (i.e., a catalyst-bound intermediate or transition state). The derivative is taken holding constant the standard-state Gibbs free energies of all other species.

In the following treatment, we will define these G⁰_i free energies relative to the standard-state Gibbs free energy of the stoichiometrically combined reactants (plus the catalyst with no bound intermediate) as the zero-energy reference. (Maintaining element balance with every step in the mechanism simplifies picturing this zero-energy reference in a reaction energy diagram and analyzing DRCs.) However, we only do this as a matter of convenience, and one can also take other states as the zero-energy reference. Indeed, we show in the section on reference states that when quantum mechanical methods (such as density functional theory, DFT) are used to estimate the energies of species, it is often more convenient to choose the zero-energy reference differently, and even then the reference state typically varies between heterogeneous and homogeneous catalysis.

Since the term – G⁰_i/RT will be used frequently in the following sections, it will be replaced with a dimensionless variable (reduced free energy),

$$g_i=-G_i^0/\mathit{RT}$$

so that Eq. (1) can be rewritten as

$$X_i={\left(\frac{\partial \left(lnr\right)}{\partial g_i}\right)}_{g_{j\ne i}}$$

There are some similar measures of the sensitivity of the rate to the values of rate constants [5], [9]. Those apply to elementary steps, so that the contributions of the individual species in each step (e.g., the initial state intermediate, the transition state, and the final state) are coupled together. We focus here instead on the generalized DRC, because it relates directly to the free energies of individual species and has a more direct connection to the reaction energy diagram. There are only a few species with non-negligible DRCs, even in the most complex reaction mechanisms. Thus, the generalized DRC greatly simplifies analysis of the connection between reaction rate and species′ energetics. This benefit increases as the complexity of the mechanism grows [1]. Since it was proposed in 2009, the generalized DRC has shown its power in catalysis research. Wolcott et al. introduced a method for computational catalyst screening based on the generalized DRC [6]. Mao et al. [10] proved that the apparent activation energy of a multistep reaction equals the weighted average of the standard-state enthalpies of all the species in the reaction mechanism (relative to the reactants), each weighted by its generalized DRC, plus RT. Thus, knowing the values of the DRCs for the species in a reaction mechanism can be quite helpful. The DRC values quantify not only the extent to which different elementary steps control the rate, but also the extent to which the energy of each catalyst-bound intermediate and transition state affects the rate. They are very important in interpreting reaction kinetics, in understanding apparent activation energies, and in predicting better catalyst materials. Here, we show that the DRC for any catalyst-bound reaction intermediate n (X_n) is proportional to its fractional population of catalyst sites (θ_n),

$$X_n=-\sigma \times {\theta }_n$$

where the proportionality constant σ is a weighted average of elementary-step site requirements,

$$\sigma =\sum _i{X}_i\times n_i$$

The weighting factor X_i here is the DRC of the transition state in step i, n_i is the number of catalyst sites of the same type required for the elementary step i, and the sum is taken over all transition states (or elementary steps) i. This simple sum typically includes only a few terms. It has only one term when there is a single rate-determining step.

We defined n_i in Eq. (4) as the number of catalyst sites of the same type required for the elementary step i. This takes into consideration that a catalyst might have two different types of sites that bind different reactant gases but are both required in some rate-controlling step(s). An example of this type is treated in Reaction (3) below. This could occur in many real situations. For example, if a rate-controlling reaction occurred at the periphery of an oxide-supported metal nanoparticle, it might be a bond-forming reaction between one adsorbate bound to a site on the metal surface and another adsorbate bound to a nearby site on the oxide surface, or a bond-breaking reaction where the two product fragments bind to different types of sites (metal versus oxide sites).

We note that Eqs. (3), (4) relate the DRCs of adsorbates to the fraction of catalyst sites they occupy (θ_n) under the reaction conditions of interest. The fraction of free sites (θ_*) is 1 minus the sum of θ_n for all adsorbates. However, free sites themselves do not have a degree of rate control (since they are part of the zero-energy reference in Eq. (1)). This can be seen from the definition of DRC in Eq. (1), where the derivative is taken with respect to - G⁰_i/RT, where G⁰_i is the standard-state Gibbs free energy of species i (i.e., a catalyst-bound intermediate or transition state) relative to the standard-state Gibbs free energy of the stoichiometrically combined reactants (plus the adsorbate-free catalysts surface) as the zero-energy reference. Thus, it is impossible to determine the DRC for the free surface sites, since they are included in the reference-state energy, which cannot be changed in calculating DRCs. However, this is only true because we chose the reactants and catalyst (with no bound intermediates) as a convenient zero-energy reference state here. This changes when using other reference states, as shown below in the section on reference states.

Some attractive mathematical properties of DRCs have already been reported. The DRCs of transition states are usually positive (or zero), and the DRCs of intermediates are usually negative (or zero) [2]. For the DRCs of transition states, if there is a single rate-determining step (RDS), the transition state in the RDS has a DRC of 1, leaving the DRCs of other transition states near zero unless the reaction has branching. If several transition states show rate control, their DRCs sum to 1 [1], [9]. In a sequential fluid-phase reaction with no catalyst, the steady-state rate (for given steady-state reactant concentration(s)) is determined by the energy difference between the fluid-phase reactants and the transition state of the RDS [11], [12]. We can offer a further justification for this as follows. Within transition state theory (TST), one calculates the rate by assuming that the reactants of the RDS are in equilibrium with its transition state. If all the steps before this RDS are in equilibrium, then the initial step′s reactants are also in equilibrium with the transition state for the RDS. Thus, one need not even consider the intervening intermediates to calculate the steady-state rate for given steady-state reactant concentration(s). One can simply calculate the rate assuming there is a single step that converts initial reactants to the transition state for the RDS, applying TST as if it were a single elementary step, since they are in equilibrium. Thus, the intermediates do not exhibit significant rate control (their DRCs are close to zero).

In a catalyzed reaction, the story is quite different, because there is often a limited number of active reaction sites. By investigating several catalytic reactions involving adsorbed intermediates, Campbell and co-workers have discovered that the DRCs of the adsorbates are usually proportional to their fractional coverages [1], [2], with a proportionality constant almost always observed to be a positive integer that applies to all the surface-bound intermediates along the reaction pathway to a given product, which is often 2 in microkinetic models. Knowing how to predict this proportionality constant - σ between coverage and DRC of adsorbates using Eq. (4) should allow experimental estimation of DRCs of adsorbates, because their coverages can be measured by a variety of methods. Alternately, measuring the experimental proportionality between rate and adsorbate coverage would allow experimental determination of the value of σ, which would give experimental information about the number of sites required in the rate-controlling step(s) through Eq. (4).

In the following sections of this paper, we analyze many different classes of catalytic reaction mechanisms in both heterogeneous and homogeneous catalysis, elucidate the origin of this proportional relation, and show that the proportionality constant is given by the DRC-weighted average of elementary-step site requirements as given by Eq. (4). We first analyze six different reaction mechanisms where analytical rate expressions can be derived at steady state, and for several of these we also analyze multiple limiting cases where the rate laws simplify. All these rate laws give DRCs in agreement with Eqs. (3), (4). We next perform case studies of two more complex mechanisms where analytical rate expressions have not been determined, but where we have calculated the steady-state rates under various reaction conditions using a computer, based on the reaction energetics taken from the literature (which was calculated using DFT). In all these cases, the DRCs are in agreement with Eqs. (3), (4).