Quasi-equilibrium surface coverage

Eqs. (1,4,5) show that to determine the equilibrium properties of an adsorbate and also the adsorption-desorption and dissociation kinetics under quasi-equilibrium conditions we need to calculate the chemical potential as a function of coverage and temperature. We illustrate this by considering a single-component adsorbate. The case of dissociative equilibrium with both atoms and molecules present on the surface has recently been given elsewhere [11]. 

Thermodynamic control (Figure 1, right) is based on adsorption of substances until quasi-equilibrium stage. In this case, the surface ratio of the adsorbed species is defined by the ratio of products of their concentration and binding constant. This deposition is much less influenced by poorly controllable fluctuations of external conditions and provides much better reproducibility. The total coverage can be almost 100%. Because of these reasons, the thermodynamic control is advantageous for preparation of mixed nanostructured monolayers for electrochemical applications including a formation of spreader-bar structures for their application as molecular templates for synthesis of nanoparticles. 

Surface Coverage and Reaction Rate. If precursor complex formation is fast relative to electron transfer and product release, it can be treated as a quasi-equilibrium step ... 

Liquid phase hydrogenation catalyzed by Pd/C is a heterogeneous reaction occurring at the interface between the solid catalyst and the liquid. In our one-pot process, the hydrogenation was initiated after aldehyde A and the Schiff s base reached equilibrium conditions (A B). There are three catalytic reactions A => D, B => C, and C => E, that occur simultaneously on the catalyst surface. Selectivity and catalytic activity are influenced by the ability to transfer reactants to the active sites and the optimum hydrogen-to-reactant surface coverage. The Langmuir-Hinshelwood kinetic approach is coupled with the quasi-equilibrium and the two-step cycle concepts to model the reaction scheme (1,2,3). Both A and B are adsorbed initially on the surface of the catalyst. Expressions for the elementary surface reactions may be written as follows ... 

By using the quasi-equilibrium equations of steps 1, 2, 4 and 5, it is possible to eliminate the unknown 0 parameters in the rate equation (6.15).The surface coverage of S( )2 can be eliminated by using step 2 ... 

In alkaline solutions, the mechanism is apparently the same as that found in acid solutions at high values of the overpotential, namely the first charge-transfer step is at quasi-equilibrium, with the ion-atom recombination step following as the rate-determining step at high surface coverage. This scheme is also confirmed by the high value of the isotope separation factor observed in this system. 

The quasi-equilibrium approximation relies on the assumption that there is a single rate-determining step, the forward and reverse rate constants of which are at least 100 times smaller than those of all other reaction steps in the kinetic scheme. It is then assumed that all steps other than the rds are always at equilibrium and hence the forward and reverse reaction rates of each non-rds step may be equated. This gives simple potential relations describing the varying activity of reaction intermediates in terms of the stable solution species (of known and potential-independent activity) that are the initial reactants or final products of the reaction. The variation of the activities of reaction intermediates is, however, restricted by either the hypothetical solubility limit of these species or, in the case of surface-confined reactions and adsorbed intermediates, the availability of surface sites. In both these cases, saturation or complete coverage conditions would result in a loss of the expected... 

Thus, the mechanism of catalytic processes near and far from the equilibrium of the reaction can differ. In general, linear models are valid only within a narrow range of (boundary) conditions near equilibrium. The rate constants, as functions of the concentration of the reactants and temperature, found near the equilibrium may be unsuitable for the description of the reaction far from equilibrium. The coverage of adsorbed species substantially affects the properties of a catalytic surface. The multiplicity of steady states, their stability, the ordering of adsorbed species, and catalyst surface reconstruction under the influence of adsorbed species also depend on the surface coverage. Non-linear phenomena at the atomic-molecular level strongly affect the rate and selectivity of a heterogeneous catalytic reaction. For the two-step sequence (eq.7.87) when step 1 is considered to be reversible and step 2 is in quasi-equilibria, it can be demonstrated for ideal surfaces that... 

Quasi-equilibrium assumption for all steps, but the second one gives expressions for surface coverage of dinitrogen, hydrogen and ammonia... 

The quasi-equilibrium hypothesis is applied on the adsorption steps. The surface coverage of hydrogen, formaldehyde and the aldol from the adsorption quasi-equilibria are inserted into the rate equations (10.108), after which the following expressions are obtained for the hydrogenation rates rf. 

Assuming that the dimerization is rate-determining, two electrons are transferred per molecule of HMMH and z = 2. It is also reasonable to suppose that the first two reactions are at quasi-equilibrium. According to reaction 3, the current density is proportional to the square of the functional surface coverage by MHads-... 

According to the latter authors the rationalization of the Elovich equation is simple. It is based on the application of the absolute rate theory (ART) to adsorption on an energetically heterogeneous surface with a rectangular distribution of activation energies for adsorption. An implicit assumption is then made that the adsorption process is at quasi-equilibrium, that the process proceeds in a stepwise fashion, and that the activation energy for adsorption increases linearly with surface coverage. 

Let us consider the situation when adsorption runs at quasi-equilibrium conditions. This means that at a surface coverage 0, all the surface correlation functions... 

As a matter of fact, most of the experiments on adsorption kinetics were done at lower coverages. In the literature we often find statements such as. "After the first very fast kinetics, a slower kinetics is observed, which follows the Elovich equation. "The first fast kinetics, at very low surface coverages, is probably governed by purely kinetic (collision) factors, the effectiveness of the transfer of the energy excess to the solid phase as well as possible changes in the structure of the adsorbed phase. The last effect is due to the saturation of chemical bonds, which may be most essential for the first molecules adsorbed on a strongly "unsaturated," "empty" solid surface. After a certain first portion of molecules are adsorbed, the rate of adsorption falls to the limits where it runs as at quasi-equilibrium conditions. 

In most electrochemical reactions, except very fast diffusion-controlled processes, the adsorption of reactants is a relatively fast step compared with succeeding electron transfer steps and can be considered in quasi-equilibrium. A knowledge of reactant adsorption behavior is necessary for interpretation of the mechanism of the reaction. Equilibrium adsorption studies are directed toward the evaluation of the surface concentration of reactants in relation to the electrode potential, the temperature, the activity of reactants, and other species in the bulk and the energy of adsorption as a function of the partial coverage 0. Study of the surface coverage by adsorbed intermediates can in some cases give additional information to the kinetic approach. Determination of adsorbed intermediates would indicate the path which the reaction follows. 

Figure 10 Theoretical variations at a fixed potential of the HER current, the surface coverage 6jj, and the bulk fractional concentration with the H adsorption free energy AGadg for two HER mechanisms (a) electroadsorption step in quasi-equilibrium (a case where the rate-determining step changes from electrocombination to chemical combination) (b) electroadsorption coupled with chemical combination. The symmetry factors are all taken equal to 1/2. The represented case is for Langmuir type H adsorption.

Figure 12.7 Pt content counted in Ptj Ru, j /Ru(0001) surface alloys as a function of initial Pt coverage. Dotted line, no Pt loss dashed/solid line, quasi-equilibrium of the Pt contents in the two outermost layers as calculated from calculated segregation energies (see Ref [8] for details).

Adsorption microcalorimetry of N2 and Ar at 77K was carried out with an equipment described by Rouquerol (ref. 8) and which associates quasi equilibrium adsorption volumetry with isothermal low temperature microcalorimetry (using Tian Calvet heat flow-meters) so that two curves are continuously recorded (heat flow and quasi equilibrium pressure) as a function of the amount of gas introduced into the systems. Continuous plots of the adsorption isotherm and of the derivative enthalpy of adsorption h vs surface coverage may easily be doived (refs. 4,7). 

Figure 2.5a shows a snapshot from a Ru(0 001) surface with a small coverage of adsorbed O atoms at 300K. The O atoms are randomly distributed and move around like in a Brownian motion with a mean residence time (at 300K) of 60 ms at a certain adsorption site. However, due to the weak attraction between two adatoms with a minimum at a distance of 2flo ( o = lattice constant of the substrate), at higher coverages a separation into two phases, namely, a quasi-gaseous and a quasi-crystalline phase, takes place (Fig. 2.5b) [9]. Under present conditions, the two phases are in equilibrium with each other, a situation that is rationalized by the phase diagram depicted in Fig. 2.6a. In our case, the horizontal scale (composition) denotes the concentration of occupied sites (i.e., overall coverage 0). As long as 0 is small, we...

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