Desorption in the Transition State Theory

The transition state theory of reaction rates [23] provides the link between macroscopic reaction rates and molecular properties of the reactants, such as translational, vibrational, and rotational degrees of freedom [27, 29, 30]. The desorption of a molecule M proceeds as follows  

Majs is the adsorbed molecule, the superscript referring to the transition state for desorption  

K is the equilibrium constant for the excitation of Mads into the transition state  

The factor kT/h represents the rate constant for the reaction over the activation barrier, in this case from the transition state to the gas phase. 

One degree of freedom of the adsorbed molecule serves as the reaction coordinate. For desorption, the reaction coordinate is the vibration of the molecule with respect to the substrate in the transition state this vibration is highly excited, and the chance that the adsorption bond breaks is given by the factor kT/h. All other degrees of freedom of the excited molecule are in equilibrium with those of the molecule in the ground state, and are accounted for by their partition functions. 

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The expression for the rate constant of desorption in the transition state theory is ... 

In the absence of transport limitations, the processes of adsorption, surface diffusion, surface reaction, and desorption can be treated via the transition state theory (Baetzold and Somorjai, 1976 Zhdanov et al, 1988). For example, the application of the TST to a single site adsorption process,... 

In this chapter, we discuss TPR and reduction theory in some detail, and show how TPR provides insight into the mechanism of reduction processes. Next, we present examples of TPO, TP sulfidation (TPS) and TPRS applied on supported catalysts. In the final section we describe how thermal desorption spectroscopy reveals adsorption energies of adsorbates from well-defined surfaces in vacuum. A short treatment of the transition state theory of reaction rates is included to provide the reader with a feeling for what a pre-exponential factor of desorption tells about a desorption mechanism. The chapter is completed with an example of TPRS applied in ultra-high vacuum (UHV), in order to illustrate how this method assists in unraveling complex reaction mechanisms. 

Equation (12) also contains a pre-exponential factor. In Section 3.8.4 we treated desorption kinetics in terms of transition state theory (Figure 3.14 summarizes the situations we may encounter). If the transition state of a desorbing molecule resembles the chemisorbed state, we expect pre-exponential factors on the order of ek T/h = 10 s . However, if the molecule is adsorbed in an immobilized state but desorbs via a mobile precursor, the pre-exponential factors may be two to three orders of magnitude higher than the standard value of 10 s . ... 

The classical approach for discussing adsorption states was through Lennard-Jones potential energy diagrams and for their desorption through the application of transition state theory. The essential assumption of this is that the reactants follow a potential energy surface where the products are separated from the reactants by a transition state. The concentration of the activated complex associated with the transition state is assumed to be in equilibrium... 

From the point of view of associative desorption, this reaction is an early barrier reaction. That is, the transition state resembles the reactants.46 Early barrier reactions are well known to channel large amounts of the reaction exoergicity into product vibration. For example, the famous chemical-laser reaction, F + H2 — HF(u) + H, is such a reaction producing a highly inverted HF vibrational distribution.47-50 Luntz and co-workers carried out classical trajectory calculation on the Born-Oppenheimer potential energy surface of Fig. 3(c) and found indeed that the properties of this early barrier reaction do include an inverted N2 vibrational distribution that peaks near v = 6 and extends to v = 11 (see Fig. 3(a)). In marked contrast to these theoretical predictions, the experimentally observed N2 vibrational distribution shown in Fig. 3(d) is skewed towards low values of v. The authors of Ref. 44 also employed the electronic friction theory of Tully and Head-Gordon35 in an attempt to model electronically nonadiabatic influences to the reaction. The results of these calculations are shown in... 

Temperature programmed desorption (TPD) or thermal desorption spectroscopy (TDS), as it is also called, can be used on technical catalysts, but is particularly useful in surface science, where one studies the desorption of gases from single crystals and polycrystalline foils into vacuum [2]. Figure 2.9 shows a set of desorption spectra of CO from two rhodium surfaces [14]. Because TDS offers interesting opportunities to interpret desorption in terms of reaction kinetic theories, such as the transition state formalism, we will discuss TDS in somewhat more detail than would be justified from the point of view of practical catalyst characterization alone. 

It is insti uctive to compare the values of pre-exponential factors for elementary step rate constants of simple surface reactions to those anticipated by transition state theory. Recall from Chapter 2 that the pre-exponential factor A is on the order ofkTjh= 10 s when the entropy change to form the transition state is negligible. Some pre-exponential factors for simple unimolecular desorption reactions are presented in Table 5.2.2. For the most part, the entries in the table are within a few orders of magnitude of 10 s . The very high values of the preexponential factor are likely attributed to large increases in the entropy upon formation of the transition state. Bimolecular surface reactions can be treated in the same way. However, one must explicitly account for the total number of surface... 

Reuter, Frenkel, and Scheffler have recently used DFT-based calculations to predict the CO turnover frequency on RuO2(110) as a function of 02 pressure, CO pressure, and temperature.31 This was an ambitious undertaking, and as we will see below, remarkably successful. Much of this work was motivated by the earlier success of ab initio thermodynamics, a topic that is reviewed more fully below in section 3.1. The goal of Reuter et al. s work was to derive a lattice model for adsorption, dissociation, surface diffusion, surface reaction, and desorption on defect-free Ru02(l 10) in which the rates of each elementary step were calculated from DFT via transition state theory (TST). As mentioned above, experimental evidence strongly indicates that surface defects do not play a dominant role in this system, so neglecting them entirely is a reasonable approach. The DFT calculations were performed using a GGA full-potential... 

The pre-exponential factor v can be correlated with the attempt frequency of the adsorbed molecules to overcome the adsorption potential. For atoms and small molecules this value is typically in the order of 1013 s. For molecules with a large number of atoms this perception is not appropriate. The pre-exponential factor actually takes into account the change of all translational and internal degrees of freedom during desorption. As a result of transition state theory (TST) considerations [3] the pre-exponential factor can be described as ... 

The only theoretical estimate of a prefactor for desorption is that of Pai and Doren [103], who used transition state theory in the harmonic approximation. Their predicted value for the prepairing mechanism was more than an order of magnitude below the measured value. Given the simplicity of the calculation and the uncertainty in experimental prefactors, it is premature to interpret the disagreement as evidence against the prepairing mechanism. 

As mentioned above, the adsorption kinetics for a kinetic-controlled mechanism is given by the balance of surfactant adsorption and desorption fluxes to and from the interface and for the Langmuir kinetics this balance has the form of Eq. (4.15). The rate constants kad and kdes are functions of the activation energies adsorption and desorption and can be specified on the basis of the molecular kinetic [9, 120] or transition state theory [121]. Eq. (4.15) was applied to adsorption kinetics data of surfactants at the water/air interface by many authors, for example in [24, 39, 83, 97, 122, 123, 124, 125, 126, 127]. In these works, it was shown that the values of kad and kdes are not constant hut depend on the surfactant bulk, the degree of adsorption layer saturation, or its lifetime. To obtain better correspondence with the experimental data, some authors had assumed that the adsorption and desorption activation energies depend on the degree of adsorption layer saturation. These rather complicated kinetic equation are more or less empirical, although they transforms into a valid adsorption isotherm at equilibrium... 

The reactions listed in Table 5.1 (adsorption, surface reaction, and desorption) are bimolecular (adsorption and surface reaction) or unimolecular (desorption). In transition state theory the rate of a gas-phase unimolecular reaction... 

The preexponential factors for each of the reaction steps can, in principle, be estimated using either gas kinetics or transition state theory. Desorption occurs when the adsorbate-adsorbent bond acquires the required activation energy for desorption in the form of vibrational energy. To a first approximation the vibrational frequency can be assumed to be approximately 10 s for the temperature at which desorption proceeds at a significant rate. The frequency of bond rupture is given by... 

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