Surface Reaction Rate Expressions

A number of classic rate expressions are commonly used to characterize heterogeneous reactions. These include expressions for the Langmuir adsorption isotherm, competitive 

For example, the Langmuir adsorption isotherm specifically describes adsorption of a single gas-phase component on an otherwise bare surface. When more than one species is present or when chemical reactions occur, the functional form of the Langmuir adsorption isotherm is no longer applicable. Thus, although such simple functional expressions are very useful, they are not generally extensible to describe arbitrarily complex surface reaction mechanisms. 

In the Surface Chemkin formalism, surface processes are written as balanced chemical reactions governed by the law of mass-action kinetics. The framework was developed to provide a very general way to describe heterogeneous processes. In this section many of the standard surface rate expressions are introduced. The connection between these common forms and the explicit mass-action kinetics approach is shown in each case. 

The Langmuir adsorption isotherm describes the equilibrium between a single-component gas A and adsorbed species A(s) at a surface [237]. The expression relates the fraction of the surface 6a covered by adsorbed species as a function of the partial pressure pa exposed to the surface. The usual form of the Langmuir adsorption isotherm is 

At low pressures, the coverage of adsorbed species increases linearly with. However, as the overpressure of A gets large, the amount of adsorbed A(s) begins to saturate. That is, the coverage begins to approach 6a = 1, which is a monolayer, and (in this model) further adsorption cannot take place. (The BET isotherm, discussed in Section 11.4.6, describes multilayer adsorption.) 

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The objective of this exercise is to use the data to develop a global surface-reaction rate expression. The data, as well as some useful auxiliary information, are reported in spreadsheet form in the file StagCatData.xls. Some summary information is also included in Table 17.12. 

By combining surface-reaction rate laws with the Langmuir expressions for surface coverages, we can obtain Langmuir-Hinshelwood (LH) rate laws for surface-catalyzed reactions. Although we focus on the intrinsic kinetics of the surface-catalyzed reaction, the LH model should be set in the context of a broader kinetics scheme to appreciate the significance of this. 

As shown in Example 22-3, for solid particles of the same size in BMF, the form of the reactor model resulting from equation 22.2-13 depends on the kinetics model used for a single particle. For the SCM, this, in turn, depends on particle shape and the relative magnitudes of gas-film mass transfer resistance, ash-layer diffusion resistance and surface reaction rate. In some cases, as illustrated for cylindrical particles in Example 22-3(a) and (b), the reactor model can be expressed in explicit analytical form additional results are given for spherical particles by Levenspiel(1972, pp. 384-5). In other f l cases, it is convenient or even necessary, as in Example 22-3(c), to use a numerical pro-... 

Here the pseudo-homogeneous rate r is related to the surface reaction rate r" through the area of active catalyst per unit volume of reactor. Assuming further a plug-flow regime, the integration of the mass balance equation for this simple rate expression gives an expression for CO conversion ... 

Theoretical rate calculations. Statistical mechanics permits one in principle to compute reaction-rate expressions from first principles if one knows the potential energy surface over which the reaction occurs, and quantum mechanics permits one to calculate this potential energy surface. In Chapter 4 we consider briefly the theory of reaction rates from which reaction rates would be calculated. In practice, these are seldom simple calculations to perform, and one needs to find a colleague who is an accomplished statistical mechanic or quantum mechanic to do these calculations, and even then considerable computer time and costs are usually involved. 

Here the problem of formulating a reaction rate expression is much more difficult because there are many atoms involved, and consequently the statistical mechanics and quantum mechanics are much more complex. We wiU consider the forms of rate expressions for surface- and enzyme-catalyzed processes in Chapter 7, but fundamental theories are usually not obtainable. 

It is interesting to consider the temperature dependence of the reaction rates predicted by these limiting expressions, which are contained in the effective rate coefficients. The true surface reaction rate coefficient has the temperature dependence... 

We have thus far written unimolecular surface reaction rates as r" = kCAs assuming that rates are simply first order in the reactant concentration. This is the simplest form, and we used it to introduce the complexities of external mass transfer and pore diffusion on surface reactions. In fact there are many situations where surface reactions do not obey simple rate expressions, and they frequently give rate expressions that do not obey simple power-law dependences on concentrations or simple Arrhenius temperatures dependences. 

The surface reaction rate is assumed to be a unimolecular chemical reaction AS — B S, which has the rate expression... 

These rate expressions are for Langmuir-Hinshelwood kinetics, which are the simplest forms of surface reaction rates one could possibly find We know of no reactions that are this simple. LH kinetics requires several assumptions ... 

As with catalytic reactions, our task is to develop pseudohomogeneous rate expressions to insert into the relevant mass-balance equations. For ary multiphase reactor where reaction occurs at the interface between phases, the reactions are pritnarily surface reactions (rate r ), and we have to find these expressions as functions of concentrations and rate and transport coefficients and then convert them into pseudohomogeneous expressions,... 

An important feature of sohds reaction is that there are essential mass transport steps of reactants and products either in convective boundary layers or within the reacting sohds. These can cause sohds reaction processes to be mass transfer limited where the surface reaction rate coefficient does not appear in the reaction rate expression. 

We frequently do not have reliable reaction rate expressions for chain reactions, but we can compensate for this lack by designing and operating the reactor to manage the overall course of the reaction by properly dealing with mixing, mass transfer, promoters and inhibitors, and the presence of surfaces. 

The CO oxidation reaction occurs rapidly at room temperature and below. As an example, on a Au(lll) surface at 250 K with onear unity exposed to a constant CO pressure of Pco = 2 x 10 Torr, the reaction rate expressed as turn-over frequency (TOF [molecules CO2 (Au atom s)" )/ is approximately 2.5 x 10-3 immediately after the reaction has been initiated and then declines at a relatively constant rate reaching a value of 3 x 10" after the reaction has proceeded approximately 800 seconds. Reaction order... 

The writers have found in their laboratory that invariably after a certain burnoff (depending upon the reactor, temperature, and sample), a subsequent extended period of constant reaction rate, expressed in grams of carbon reacting per unit time, is attained. In this bumoff region, there obviously is equilibrium between the rate of formation of the surface-oxygen complex and its removal with a carbon atom. It is felt that this is the reaction rate most characteristic of a given temperature and should be used in kinetic calculations. In principle, Wicke (31) concurs with this reasoning and reports reactivity data only after the sample has attained a total surface area which is virtually constant. 

The reaction rate expressed in terms of surface concentrations provides the relationship between Cs and CL. From the definition of the effectiveness factor, we may express the required equality of mass transfer and reaction rates as... 

In Eqn. 5.3-1, rj is the effectiveness factor of the catalyst with respect to the dissolved gaseous reactant and the temperature of the outer surface. The rate of reaction within the catalyst pores is comprised in rj. R is the reaction rate expressed in moles of gaseous reactant, A, per unit of bubble-free liquid, per unit of time. Reaction is irreversible. In equation (1) it has not been assumed that the gas is pure gas A, its concentration in the bubbles being Cg. Also, Henry s law for the gas is assumed and written as in Eqn. 5,3-4. Using Henry s law, Eqn. 5.3-4, the intermediate concentrations (Cs, CL) can be eliminated using the above system of equations. This provides an expression of the global rate in terms of an apparent constant, ko, that contains the various kinetic and mass transfer steps. Therefore, the observed rate can be written as ... 

The predictive equation for the rate of deposition of ZnS (equation 40) is obtained by substituting the surface concentrations given by equations 36 and 37 into the reaction rate expression, equation 39 ... 

Finally the resulting rate expression for the surface reaction rate determining is given by (3.14). 

The form of the resulting expression differs from the gas phase reaction rate expressions due to the presence of a denominator which represents the reduction in rate due to adsorption phenomena and of which the individual terms represent the distribution of the active sites among the possible surface complexes and vacancies. These expressions are termed the Langmuir-Hinshelwood-Hougen-Watson (LHHW) rate expressions. 

Therefore, Equations 8.48 and 8.49 can be combined into one equation for concentration only. The effectiveness factor for the case considered can be calculated with the technique described in Chapter 7. It is important to stress that the effectiveness factor changes along the reactor because parameters of the reaction rate expression, Equation 6.18, e and a depend on the surface concentration and temperature. The calculated modified effectiveness factors for nonisothermal first-order reaction at different conversions = (1- CAJCa) are shown in Figure 8.9 versus the ratio of inner and outer diameters of the hollow cylinder. The parameters chosen for the calculation are ... 

If the deposition process is a first-order reaction having Arrhenius temperature dependence, the surface reaction rate, S, can be expressed as the product of the surface impingement rate and a reaction probability, ( ). In terms of the gas molar density and reactant mole fraction this is... 

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