First-order reactions heterogeneous

An interesting method, which also makes use of the concentration data of reaction components measured in the course of a complex reaction and which yields the values of relative rate constants, was worked out by Wei and Prater (28). It is an elegant procedure for solving the kinetics of systems with an arbitrary number of reversible first-order reactions the cases with some irreversible steps can be solved as well (28-30). Despite its sophisticated mathematical procedure, it does not require excessive experimental measurements. The use of this method in heterogeneous catalysis is restricted to the cases which can be transformed to a system of first-order reactions, e.g. when from the rate equations it is possible to factor out a function which is common to all the equations, so that first-order kinetics results. 

For catalytic reactions carried out in the presence of a heterogeneous catalyst, the observed reaction rate could be determined by the rate of mass transfer from the bulk of the reaction mixture and the outer surface of the catalyst particles or the rate of diffusion of reactants within the catalyst pores. Consider a simple first order reaction its rate must be related to the concentration of species S at the outer surface of the catalyst as follows ... 

Two important ways in which heterogeneously catalyzed reactions differ from homogeneous counterparts are the definition of the rate constant k and the form of its dependence on temperature T. The heterogeneous rate equation relates the rate of decline of the concentration (or partial pressure) c of a reactant to the fraction / of the catalytic surface area that it covers when adsorbed. Thus, for a first-order reaction,... 

Riesenfeld and Bohnholtzer and Riesenfeld and Schumacher used ozone concentrated by liquefaction and distillation. From their kinetic measurements they conclude that a reaction of the second order and one of the first order take place simultaneously at quite low pressures, 6-60 mm. Hg the first order reaction predominates. The velocity constants of the second order reaction are not influenced by the total pressure, while those of the first order reaction appear to be inversely proportional to the total pressure. The figures given show that the first order reaction at the lower pressures is considerably influenced by the surface, and is quite probably a heterogeneous reaction, though the authors themselves do not consider this to be definitely shown. The decomposition appears to be rather sensitive to catalysts such as dust particles. 

Given any complex system of heterogeneous catalytic first order reactions the mass balance on a differential volume element of the reactor at the height h yields the following system of differential equations for the j-th reaction component i) for the bubble phase... 

In summary, the expression for the net rate of particle transport through the interaction boundary layer takes the same form as a first-order, reversible, heterogeneous chemical reaction, provided that f/>in lI and — mn2 are large compared to kT, min 6, — /trnox — 82 ... 

Heterogeneously catalyzed reactions are usually studied under steady-state conditions. There are some disadvantages to this method. Kinetic equations found in steady-state experiments may be inappropriate for a quantitative description of the dynamic reactor behavior with a characteristic time of the order of or lower than the chemical response time (l/kA for a first-order reaction). For rapid transient processes the relationship between the concentrations in the fluid and solid phases is different from those in the steady-state, due to the finite rate of the adsorption-desorption processes. A second disadvantage is that these experiments do not provide information on adsorption-desorption processes and on the formation of intermediates on the surface, which is needed for the validation of kinetic models. For complex reaction systems, where a large number of rival reaction models and potential model candidates exist, this give rise to difficulties in model discrimination. 

Free radicals formed in solution, unless generated by the decomposition of a substance which produces two radicals in close proximity, seldom dimerize. This is because there are usually facile first order and pseudo first order reaction pathways open to these reactive intermediates. Due to the heterogeneity of the electrode process, intermediates are formed in relatively high concentration at the electrode solution interface. If the dimerization of the intermediates is an energetically favourable process, conditions can be optimized so that yields of dimer are high. 

In most gas-solid heterogeneous catalyst systems, the effect of pressure often is correlated with an adsorption model of the Langmuir-Hinshelwood type. The over-all rate constant for the first order reaction is related to the adsorption model constants by... 

Figure 12-5 (a) Effectiveness factor plot for nth-order kinetics spherical catalyst particles (from Mass Transfer in Heterogeneous Catalysis, hy C. N. Satterfield, 1970 reprint edition Robert E. Krieger Publishing Co., 1981 reprinted by permission of the author), (b) First-order reaction in different pellet geometries (from R. Aris, Introduction to the Analysis of Chemical Reactors, 1965, p. 131 reprinted by permission of Prentice-Hall, Englewood Cliffs, NJ)... 

Ft,k rate constant for a homogeneous (heterogeneous) first order reaction m/s... 

Since chemical adsorption is an exothermic process, the surface coverage decreases with an increase in temperature. The rate of a heterogeneous reaction is proportional to the coverage. As a result the reaction can be treated as a first-order reaction for weakly adsorbed gaseous species. For the strongly adsorbed cases the reaction is the zeroth order because the reaction rate is independent of the partial pressure. For the intermediate case the reaction rate may be a fraction. 

The present paper (just as previous ones, references 2-6) considers the thermal mechanism as being responsible for the formation of DSs. In other words, the factor of nonlinearity is here the exponential dependence of the reaction heat generation intensity on temperature, which is the commonest in chemistry, and the concentration-velocity relation corresponds to the linear case of a first-order reaction. Consideration of the chemically simplest case aims at forming a basis of the theory of DS in heterogeneous catalysis and its further development by consistently complicating the kinetic law of a reaction and introducing into the model nonlinearities (feedbacks) of both... 

Many different types of interfacial boundaries can be probed by SECM. The use of the SECM for studies of surface reactions and phase transfer processes is based on its abilities to perturb the local equilibrium and measure the resulting flux of species across the phase boundary. This may be a flux of electrons or ions across the liquid/liquid interface, a flux of species desorbing from the substrate surface, etc. Furthermore, as long as the mediator is regenerated by a first-order irreversible heterogeneous reaction at the substrate, the current-distance curves are described by the same Eqs. (34) regardless of the nature of the interfacial process. When the regeneration kinetics are more complicated, the theory has to be modified. A rather complete discussion of the theory of adsorption/desorption reactions, crystal dissolution by SECM, and a description of the liquid/liquid interface under SECM conditions can be found in other chapters of this book. In this section we consider only some basic ideas and list the key references. 

Models of parallel pseudo first-order reactions consider the case when two interactions with different rate constants proceed simultaneously. Such situations can be attributed to different kinds of receptor sites or to different states of the analyte [8,11]. In the first case the model can describe heterogeneity of the sensor surface the second may concern a macromolecular analyte that can be present in various conformations, protonation states, etc. Besides two sets of rate constants, the models also require specification of proportion p between the two fractions of the receptor or analyte. For the model considering two kinds of receptors, the following equations are obtained ... 

Studies of tethered electroactive species are less sensitive to pinholes than experiments with solution reactants and blocking layers, although heterogeneity and roughness of the substrate and film defects can still play a role. The rate constant, k, in this case has units of a first-order reaction (s ). Rate constants can be determined by a voltammetric method as described earlier for electroactive monolayers (Section 14.3.3). In addition potential-step chronoamperometry can be employed, in which case the current follows a simple exponential decay (88, 90, 91) ... 

Reoxidation of the cosubstrate at an appropriate electrode surface will lead to the generation of a current that is proportional to the concentration of the substrate, hence the coenzyme can be used as a kind of mediator. The formal potential of the NADH/NAD couple is - 560 mV vs. SCE (KCl-saturated calomel electrode) at pH 7, but for the oxidation of reduced nicotinamide adenine dinucleotide (NADH) at unmodified platinum electrodes potentials >750 mV vs. SCE have to be applied [142] and on carbon electrodes potentials of 550-700 mV vs. SCE [143]. Under these conditions the oxidation proceeds via radical intermediates facilitating dimerization of the coenzyme and forming side-products. In the anodic oxidation of NADH the initial step is an irreversible heterogeneous electron transfer. The resulting cation radical NADH + looses a proton in a first-order reaction to form the neutral radical NAD, which may participate in a second electron transfer (ECE mechanism) or may react with NADH (disproportionation) to yield NAD [144]. The irreversibility of the first electron transfer seems to be the reason for the high overpotential required in comparison with the enzymatically determined oxidation potential. 

Equation (4.4.9b) may be compared with the boundary condition for a mixed heterogeneous reaction k c] = D dcldy),. It is evident that Rj = 1 corresponds to a first-order reaction, whereas if R c , there is a correspondence as well for reactions other than first order. For simplicity, let us consider the perfectly rejecting case of R = 1. Here, the appropriate dimensionless parameter characterizing the boundary condition is the ratio of the permeation velocity to the diffusion velocity... 

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