2
The Potential Energy Diagram

The central theme in catalysis is the effect of the catalyst on the rate of a chemical reaction or on the product distribution, which is given by the relative rates of different reaction pathways. You can say that catalysis is all about what determines the chemical kinetics. A good catalyst is typically one that gives a high rate and a high selectivity toward the desired product. The reaction rate constant, k, for an elementary reaction is often written as an Arrhenius expression in terms of a prefactor, υ, and an activation energy, Ea:

where kB is the Boltzmann constant and T is the absolute temperature. Variations in the activation energy, when, for example, one catalyst or reactant is substituted with another or when a reaction proceeds through two different reaction mechanisms, are typically large (0.5–2 eV), while the thermal energy, kBT, is small (typically ranges from kBT = 0.0257 eV at T = 298 K to kBT = 0.1 eV at T = 1160 K). The rate constant is therefore very sensitive to the size of the activation energy. We will return with a more detailed discussion of how the Arrhenius expression comes about in Chapter 4. For now, it suffices to note that any discussion of reaction rates must start with a discussion of the origin of activation energies. For that reason, the starting point of this textbook is an understanding of the potential energy diagram (PED) for surface chemical transformations.

R versus kB

If the reaction and activation energies are expressed per amount of substance (in moles), then the Boltzmann constant kB (approximately 8.61·10−5 eV/K) should be substituted with the gas constant R (approximately 8.31 J/(mol·K)). Since we take an “atomic-scale” viewpoint throughout the book, expressing energies per atom, molecule, or elementary reaction, we shall be utilizing the Boltzmann constant.

van der Waals Interactions

The weak bonding due to induced dipole–induced dipole interactions go by several names. Sometimes, the interaction is referred to as London dispersion forces, and in other parts of the literature, they are called van der Waals forces. We shall here not dwell at the more detailed distinction.

2.1 Adsorption

We will start by considering the simplest possible potential energy diagrams: those that describe the elementary step of adsorption of a single atom or molecule on a surface. When an atom or a molecule (the adsorbate) approaches a surface, it will start interacting with the electronic states of the solid. At long distances, weak bonding called physisorption dominates. This is due to van der Waals forces, which are purely quantum mechanical in nature and which are relatively long ranged. They occur due to an attraction between mutually induced dipoles of the electron clouds surrounding the atom or molecule and in the surface. Closer to the surface, when the electron clouds of the adsorbate and surface atoms begin overlapping, chemical bonds may form. This stronger form of adsorption is called chemisorption. The strength of the interaction is measured by the change in potential energy of the system as a function of the distance, z, of the adsorbate above the surface:

In principle, the adsorption energy can be measured, and there are examples where this has been done, typically by inferring an adsorption energy from the measured rate of desorption (temperature-programmed desorption (TPD)). Another way to more directly measure adsorption energies is to measure the temperature increase of a surface as it becomes covered by adsorbates (calorimetry). There are, however, quite few systematic experimental data available. We have therefore chosen throughout this book to illustrate phenomena in terms of energies that are calculated through an approximate solution of the Schrödinger equation based on Kohn–Sham density functional theory (DFT). While DFT is not always in quantitative agreement with experiment, the values for adsorption energies on transition metal surfaces are typically within 0.1–0.2 eV of experiment in the cases where this has been tested. Trends from one catalyst to the next are usually described much better. The use of theoretical interaction energies allows us to always discuss surface reactions and catalysis in terms of the energetics, that is, at the most fundamental level. We will introduce experimental data where possible to illustrate important phenomena and to place the discussion on a firm experimental footing.

The adsorption energy is measured relative to the situation where the adsorbate is far away from the surface, that is, relative to the energy of the clean surface and the free adsorbate. This convention, which will be used throughout the book, means that negative adsorption energy signifies the formation of a chemical bond (the system being stabilized by formation of the bond). Two examples of potential energy diagrams for adsorption are shown in Figure 2.1.

Figure 2.1 Left: PED for the physisorption of Ar in the threefold position of the Cu(111) surface. The potential energy is shown as a function of the distance between the Cu surface and the adsorbate. The energy of the adsorbate at a distance of 6 Å is chosen as a reference. Due to the filled outermost electronic shell on the Ar atom, this species does not chemisorb to the surface at all, and the shallow physisorption minimum is clearly visible. Right: PED for the chemisorption of H on Cu(111) in ontop, bridge, and threefold position.

The potential energy diagrams (PEDs) (e.g., Fig. 2.1) contain significant information about an adsorption system. The minimum value of the PED defines the adsorption energy, since it gives the energy gained by adsorption. The location of the minimum defines the equilibrium distance of the adsorbate above the surface. An argon (Ar) atom has a closed outermost electron shell and therefore typically does not form chemical bonds to a metal surface. It only physisorbs with an energy of approximately −0.1 eV on a close-packed Cu surface. A hydrogen (H) atom, on the other hand, forms a strong chemisorption bond to the same surface. The potential energy also depends on the position of the adsorbate parallel to the surface. The most stable position of the H atom is found to be in a site where it has three Cu neighbors. The adsorption energy of −2.6 eV is comparable to the formation energy of an H2 molecule from H atoms of −2.5 eV per atom. It is thus energetically favorable for an H2 molecule to dissociate over a Cu(111), which is an important feature in understanding how and why a Cu surface can act as a catalyst, for instance, in the synthesis of methanol from CO and H2, 2H2 + CO → CH3OH.

Energy Units

Throughout this book, we will be using eV (the kinetic energy gained by an elementary charge accelerated through a potential of 1 V) as the energy unit. It is perhaps not a natural energy unit for chemical processes, since it is neither an SI unit (which would be joule) nor is it the typical atomic unit (hartree). It is, however, a very convenient unit to use when adopting an “atomic-scale” point of view of catalysis. Typical covalent bond strengths in molecules are on the order of 1–10 eV, for instance, and physisorption energies are on the order of 0.1–1 eV. Likewise, a reaction rate of 1 turnover per second corresponds to an activation energy of approximately 0.75 eV at room temperature. In terms of other energy units, 1 eV per atom or molecule = 96.49 kJ/mol, 23.06 kcal/mol, or 0.03675 hartree.

The PED for a molecule approaching a surface is considerably more complicated owing to the fact that it will depend not only on the position of the molecule relative to the surface but also on the intramolecular degrees of freedom as well as the rotational orientation of the molecule relative to the surface. Figure 2.2 shows the two-dimensional potential energy surface (PES), ΔE(z,R) for H2, dissociation over Cu(111) as a function of both the distance to the surface, z, and the H–H distance, R, for the molecule positioned parallel to the surface. This PES defines a surface chemical reaction. There are two minima, one where the molecule is far from the surface and one where the molecule has dissociated and the H atoms are well separated and bound to the surface. A considerable energy barrier separates the two minima. This can be observed experimentally by monitoring the probability for dissociation of H2 molecules scattering off a Cu surface as a function of the kinetic energy of the molecule (see Fig. 2.3). Only molecules with kinetic energies above or just below (due to quantum mechanical tunneling) the energy barrier will be able to dissociate.

Figure 2.2 Left: PES for H2 dissociation over Cu(111). The potential energy of the system is shown as a function of the Cu–H2 and H–H distance, respectively. H2 far from the Cu surface has been chosen as a reference. The lowest potential energy path for H2 splitting is marked with black crosses. Right: PED for H2 dissociation where the lowest potential energy (from the figure on the left) is plotted as a...