Molecular Heterogeneous Catalysis, Wiley (2006), 352729662X

The Ag2+ is regenerated by anodic oxidation.

The most important issue that has to be addressed in the design of these cycles is that of selectivity. The catalyst that regenerates the catalytic reagent should not induce undesired reactions with reactant molecules or products.

2.2 Physical Chemistry of Intrinsic Reaction Rates

2.2.1 Introduction

According to the Sabatier principle introduced in Section 2.1.1.1, the rate of a catalytic reaction is a maximum when the interaction between reactant and catalyst is at an optimum value. The key to the Sabatier principle then involves understanding the free energies of adsorption as a function of catalyst structure and composition. In the 1960 ’s, Tanaka and Tamaru[11] noted that trends in the interaction energies of di erent molecules with varying catalyst surfaces are often very similar. Theoretical surface studies provide a justification as to why. As we will discuss in detail in Chapter 3, the strength of the surface-chemical bond directly depends on the electronic structure of the metal surface. In the 1980 ’s it had been proposed that electronic properties such as the local density of

states at the Fermi level or d-electron occupation are parameters that critically control catalysis. More recent studies by Hammer and Nørksov[12] show that it is possible to tie

the reactivity to the interaction with the relative positioning of the d-valence band center with respect to the Fermi level at the surface. The Hammer–Nørskov model has rapidly become the standard probe of surface reactivity and is being used to design surfaces with specific reactivity. As we will illustrate in the next chapter, these changes in surface electronic structure can be related to the degree of coordinative unsaturation of the surface atoms. In Section 2.2.2 we discussed the point that an optimum catalyst has maximized the balance between the rates of intermediate forming reactions and the rates of reactions by which the product molecules leave the surface.

While the relationship between the electronic properties and the reaction enthalpy is important in understanding energetics, the more important thermodynamic feature to focus on is the free energy. Indeed, in Chapter 4 the maximum for the rate of a zeolitecatalyzed reaction is not found for the zeolite with the smallest pore size (maximum adsorption enthalpy) but for medium-sized micropores where adsorbates have a higher entropy, and as a consequence, their concentration is a maximum. The gain in entropy often balances the loss in adsorption enthalpy.

In the subsections that follow we will focus on the factors that maximize the rate constant for elementary surface reaction steps. Again we will stress the need explicitly to include entropic contributions. According to transition-state reaction rate theory[13a] , the

where ∆G‡ refers to the free energy di erence between the transition state and the initial reactant state and Γ is the transmission factor. Expression (2.5) is a maximum when ∆G‡ is a minimum, which implies a maximum stabilization of the transition state. A condition for the transition-state reaction rate expression to be valid is that the vibrational and rotational modes in the transition state are in equilibrium with the corresponding modes of the reactant ground state. This implies that energy transfer between these modes is

28 Chapter 2

fast compared with the reaction rate. This condition is usually satisfied for intramolecular reactions. Reactions that occur on surfaces between adsorbed reaction intermediates can also be considered to belong to this category of reactions. When molecules collide in the gas phase or collide with a surface, their contact time can be short compared with the energy transfer time. In such a case, energy transfer can become rate limiting. The transition-state theory rate expression then predicts an upper bound to the reaction rate. For a proper treatment, a molecular dynamics approach should be used. This situation is unusual but can occur for dissociation reactions of gas-phase molecules impinging upon a surface. The residence time will be short when the interaction potential is small. This can be the case for small coordinatively saturated molecules such as methane [13b,15].

2.2.2 The Transition-State Theory Definition of the Reaction Rate Constant; Loose and Tight Transition States

In order to appreciate the use of transition-state rate expressions, it is important to be reminded of the di erent time scales of the processes that underpin the chemistry we wish to describe. The electronic processes that define the potential-energy surface on which atoms move have characteristic times that are of the order of femtoseconds, 10−15 sec, whereas the vibrational motion of the atoms is on the order of picoseconds, 10−12 sec. The overall time scale for bond activation and formation processes that control catalysis vary between 10−4 and 102 sec. This implies that on the time scale of the elementary reaction in a catalytic process, many vibrational motions occur. If energy transfer is e cient, then the assumption that all vibrational modes except the reaction coordinate of the chemical reaction are equilibrated is satisfied. Kramers[14] defined this condition as Eb > 5kT . Under this condition the transition state reaction-rate expression applies:

where Eb − E0 refers to the barrier height of the reaction as computed from electronic structure calculations. Except when quantum-mechanical tunneling or reaction dynamics become important, the transmission factor Γ can be assumed to be one. This is a reasonable assumption for most surface reactions which have activation barriers that satisfy the Kramers condition (see Ref. 15 for more details). Q‡ and Q0 are the partition functions of transition state and initial state, respectively.

The pre-exponential factor, νe , and the activation barrier, Eact, are the kinetic parameters that are necessary to describe a reaction, deduced from the Arrhenius reaction-rate expression in the following way:

30 Chapter 2

Figure 2.7. Schematic illustration of reaction coordinate and saddle point definition of the transition state[15].

electrons into its antibonding π orbital and initiate dissociation. After dissociation, the atomic C and O intermediates are each strongly adsorbed to the metal surface. The di erence in the projection of the O coordinates from the C coordinates onto the surface can be taken as the measure of the actual reaction coordinate. The reaction coordinate is zero before reaction and after reaction it equals the C and O distance between the adsorbed C and O atoms.

When the C–O bond length increases from its initial length (in its adsorbed state), the potential energy of the system increases until a maximum is reached. Further increases in the reaction coordinate lower the potential energy until it reaches the equilibrium value of the fully dissociated system. The barrier for the transition state is defined as the saddle point on the potential energy surface. The derivatives of the energy with respect to all degrees of freedom at this point are zero, and at a minimum for all modes, except that which corresponds to the reaction coordinate for which the potential energy is at a maximum.

Γ is defined as the probability that once the system reaches the top of the transitionstate barrier, it actually passes over it. In transition-state reaction rate theory Γ is usually taken to be equal to one. This is often a reasonable approximation. However, when the reaction system is viscous, as in a solution, or has di usional restrictions, Γ can be substantially less. Molecular dynamics simulations for the motion of the system near the transition state can be carried out in order to determine Γ.

By way of example, we show how the transition-state rate expression can be used to determine the rates for both surface desorption and the dissociation of CO at low surface coverage on the terrace sites of the transition-metal substrate. This also allows

for an illustration of the concepts of tight and loose transition states and their respective definitions[15,16].

Transition states which contain a high degree of entropy are typically called loose, whereas those with a low degree of entropy are considered tight. The entropy of a CO molecule adsorbed to a metal surface is not negligible, but is quite small since the main

contribution to the entropy is the liberating motion of the oxygen atom versus the surface normal. The entropy of the surface atoms is essentially zero. The strong surface bonds give rise to fairly high vibrational frequencies and, hence, the vibrational partition functions are nearly equal to one.

A diatomic molecule such as CO will only dissociate with a kinetically acceptable barrier if the C–O bond is significantly weakened in the transition state. This implies a strong electronic interaction between metal-surface electrons and the lowest unoccupied states of CO. In the transition state, the molecular axis of CO is nearly parallel to the metal surface such that the overlap between molecular π and π orbitals of CO and transition-metal surface orbitals is maximized. This weakens the molecular bond due to back-donation of surface electrons into the anti-bonding π states on CO. In the transition state, the C and O atoms have nearly the same distance as those in the separared product state, and thus their interaction with the surface is quite large. The entropy of this transition state is very similar to the final state and hence essentially equal to zero. This defines the transition state of the dissociating molecule on the transition-metal surface as a tight transition state.

The transition state for a molecule desorbing from the surface is quite di erent to that of the reacting molecule. The desorbing molecule in its transition state is nearly free from the surface. The molecule can therefore be considered to freely rotate and to move freely parallel to the surface. Hence the activation entropy for desorption is quite large. The ratio of the rate of CO bond cleavage, rdiss , over the rate of desorption is

The values for the activation energies, Eb, include the zero point vibrational-frequency corrections. The transition state for desorption can be characterized as a loose transition state. The entropy of this state is quite high, which is the result of the weak interaction of the desorbing molecule with the catalyst.

On most of the group VIII metals the heats of adsorption are comparable with their heats of dissociation. This implies that the adsorbed molecule will desorb with a rate which is 104 times faster than that for dissociation owing to the di erence in the activation entropy. The reaction path that proceeds through a transition state of maximum entropy is the preferred path. For catalysis, the implication is that dissociation of adsorbed molecules will occur at those pressures and temperatures where equilibrium between the gas phase and surface maintains a significant surface concentration of the dissociated molecules. The temperature has to be chosen such that the rate constant for dissociation is in the proper time regime. The observation that for a transition-metal surface reaction, the transition entropy is low, but that for a reaction step between the surface and gas phase is large, is quite general.

2.2.3 The Brønsted–Evans–Polanyi Reaction-Rate Expression Relations

In order to establish equations which describe the reaction rate from first principles, theories based on non-equilibrium statistical mechanics have to be used[15] . However, useful empirical relations, which have some theoretically justification, exist. They linearly relate the activation energies for a reaction with some property of the reaction. Hammett,

32 Chapter 2

for example, showed a direct correlation between the rate of reaction for a family of di erent substituted benzoic acid species with a measure of the electronic properties of the substituent (the Hammett substituent parameter in physical organic chemistry)[17a] . Hammett, therefore, established a linear relationship between the logarithm of the rate and the substituent parameter. The correlations were extended over a full range of different types of reactions whereby changes in the reaction were described by changes in the slope of the line. A similar relationship for catalysis was establish by Brønsted, who showed that the rate constant of an acid-catalyzed reaction was correlated with the catalyst’s equilibrium acid dissociation constant, KA[17b]. Evans–Polanyi[17c] showed that the molecular reaction rates could be directly correlated with the overall reaction enthalpy. The Evans–Polanyi and Brønsted relationships are quite similar as the Brønsted relationship is simply a subset of the more general Evans–Polanyi relationship. Some have combined the two in what is named the BEP relationship, attribut to the three authors Brønsted, Evans, and Polyani. The BEP relationship is perhaps the most widely used linear relationship in all of catalysis. The BEP relationship is valid when one compares related elementary reaction steps that proceed through nearly the same intermediate structures and have similar reaction coordinates.

For the simplified parabolic energy curves of reactant and product states shown in Fig. 2.8, if one shifts the reaction energies but holds the coordinates of the final and reactant states fixed one can derive the following expression[15] :

E‡, the activation energy, and ∆ER the reaction energy are defined in Fig. 2.8. ∆E0‡ is the activation energy of the system when the overall reaction energy ∆ER = 0. Expressions (2.16) are valid as long as |EER‡ | 1.

0

In order to discuss the physical meaning of α, we need to introduce the concept of early and late transition states. In the previous section we discussed in detail the transition state for CO dissociation over transition-metal surfaces and described the reaction as an example of a late transition state. The transition state is late along the reaction coordinate since the transition-state structure is close to the final dissociated state. Transition states which are early along the reaction coordinate are called early transition states and thus resemble the initial reaction states (see Chapters 4 and 7 for the definition of the pretransition state). The activation energies for the protonic zeolite reactions correlate with deprotonation energies (see Fig. 2.9) and are examples of intermediate transition states that also vary with the energies of the initial states[18,19]. When:

α0.5 (α ≈ 0), the transition state is early; ∆S ≈ 0

α0.5 (α ≈ 1), the transition state is late;

and

∆S < 0 for surface reactions (tight transition states); ∆S > 0 for desorption (loose transition state),

Figure 2.8. Parabolic energy curves of reactant and product states that di er in reaction energies. The transition states are defined as the cross-section of the parabolas.

the general expression for the activation energy becomes:

Because of microscopic reversibility, the microscopic balance of the elementary reaction steps gives a relation between the forward (denoted with +) and backward reaction (denoted with -):

As a result, we have the following three important relations for the activation free energies of an elementary reaction step:

An interesting consequence of these relationships is that they help us to understand the trends in reactivity for selected reactions.

Let us discuss this again for the case of the CO dissociation reaction. In the next chapter on the reactivity of transition-metal surfaces, we will help to justify why the bond energy for surface atoms decreases from left to right across a given row of the periodic table. In addition, we demonstrate that this change in the adsorption energy is much less for adsorbed molecules than it is for adsorbed atoms. The adsorption energy of a molecule is much smaller than that of an adsorbed atom. The reaction energy ∆ER for the dissociation of an adsorbed CO molecule can change from exothermic to endothermic. It decreases when the transition-metal to which adsorption occurs changes within a row from the left to the right in the periodic table. The corresponding activation energy for dissociation increases (α ≈ 1) from left to right across a row of transition metal elements in the periodic table. The rate of recombination, which is the microscopic reverse reaction,

34 Chapter 2

however, will have a nearly constant activation energy (α ≈ 0) and be rather independent of transition metal.

The other important elementary step for CO is its desorption from the surface. The activation energy for desorption is equal to the adsorption energy and, hence, the trend for desorption will be opposite of that for dissociation. The desorption of a molecule becomes easier as one moves across a row of the periodic table from left to the right where the binding energy of the molecule to the surface is weakest. This also has an interesting consequence for trends in selectivity for di erent catalytic systems. For instance, the reaction products for the hydrogenation of CO are methanol, methane and/or hydrocarbons (i.e. Fischer–Tropsch type of selectivity). The selectivity of these reactions is governed by the reactivity of CO and its fragments, since hydrogen will readily dissociate on most metals without an appreciable activation energy. In order to produce methanol, the intramolecular CO bond should not be broken and, hence, the favorable transition metals are those

that are overall endothermic (∆ERdiss > 0) for CO activation such as Cu and Pd. Once CO dissociates the selectivity for methanation versus hydrocarbon growth is determined

by the rate of CH4 formation and desorption versus the rate of carbon-carbon coupling. Here there is competition in the requirements for the metal substrate.

Figure 2.9. The activation energies of ethane activation as a function of deprotonation energies as computed for di erent clusters modeling the zeolite proton (see also Chapter 4). ∆Eact and ∆Hdepr are in kcal/mol.

The interaction between the metal and the reactant has to be strong enough to dissociate CO, but weak enough so that CH4 is readily formed and desorbs from the surface. Nickel is the preferred metal for methanation. Fe, Co and Ru, on the other hand, are the preferred Fischer–Tropsch catalysts mainly because they e ciently dissociate CO and suppress

methane formation because of the stronger metal-carbon bond. The formation of C–C bonds is expected to be almost independent of the metal (α ≈ 0). CO dissociation has the lowest barrier on the metal with highest reactivity (α ≈ 1). Metals such as Fe, Co, and Ru are preferred for the Fischer–Tropsch reaction. For ethylene hydrogenation the site requirement varies as a function of coverage. At low coverage ethylene adsorbs to two surface-metal atoms and at high coverage it reacts from a position where it is only bound to a single metal atom.

To apply the BEP relation, one has to be careful to select properly the elementary reaction step to which one wishes the relation to apply. In zeolite-catalyzed reactions an important elementary reaction step is the protonation of a hydrocarbon, to give as the reaction intermediate, a protonated carbenium ion. The proton activates the molecule to rearrange and after reaction the proton is back-donated to the solid. Often intermediate cationic molecular fragments can stabilize themselves by forming covalent bonds with the zeolite framework oxygen atoms. This will be explained in much more detail in Chapter 4. Figure 2.9 shows that for several such reactions there is a linear relationship between the deprotonation energy of a particular cluster representing the zeolite proton and computed activation energies. The proportionality constant between the activation energy and the deprotonation energy was calculated to be 1/3. This implies that the intermediate carbocation is stabilized by its interaction with the negatively charged zeolite wall. It will be clear in Chapter 4 that the intermediate is a transition state and that the product state has a strong bond with the zeolite oxygen atom. This interaction varies also with the O–H proton energy and explains the low proportionality constant.

2.3 The Reactive Surface-Adsorbate Complex and the Influence of the Reaction Environment

2.3.1 Introduction

Catalysis is controlled by the rate at which the active surface complex turns over in the catalytic reaction cycle. This involves making and breaking of bonds between an adsorbate and the surface site to which it is bound as well as within the adsorbate. For a bimolecular reaction, this involves the bonds between two adsorbates and between the adsorbates and their respective surface sites. The active surface site for the reaction of small molecules typically contains anywhere between one atom such as a metal atom redox site in a homogeneous organometallic complex or in a metal-loaded zeolite such as the Fe2+ cation in ZSM-5 (Chapter 4, page 193) on up to 7-10 atoms for a special metal surface site such as the special C5 site for Fe involved in the activation of N2 in the synthesis of ammonia from nitrogen and hydrogen[20] (Chapter 7, page 333). The active sites and their environments for these examples are shown in Fig. 2.10. The conversion of larger molecules in enzymes, on the other hand, may involve multiple activating contacts between the adsorbate and the cavity surface. The type, number and the strength of bonds in the active adsorbate surface complex are all important in controlling the overall rate at which the catalytic cycle turns over (Chapter 7, page 319). The principles of the adsorbate surface chemical bonds and their transformations are followed in detail in Chapters 3, 4, 5, and 6 for metals, zeolites, metal oxides and sulfides, and enzymes, respectively.

As was presented in the Introduction to this chapter, two di erent views on the relationship between the active site and the catalytic cycle currently exist. The first follows the view taken by Langmuir suggesting that all of the sites on the surface are same and