Steps, elementary quasi-equilibrium

To construct a composite event from a detailed sequence, one can proceed as in Example 2.5 and represent the concentrations of intermediates in the controlling elementary step by quasi-equilibrium expressions (see Eq. 2.5-2b). However, it is simpler to construct the composite event in the form of Eq. (2.4-1) then the results of Section 2.4 can be applied directly. Thus, the intermediate nO l in the controlling step is producible from I nC ig) — H2( ) via the first two quasi-equilibrium reactions, whereas the intermediate fCjl is producible from l- -iC g) — H2( ) via the last two quasi-equilibrium reactions. With these substitutions into Eq. (2.5-41). we get... 

Reaction rates for the acid catalyzed elementary steps in hydrocracking can be expressed as follows when the metal catalyzed (de)-hydrogenation reactions are in quasi equilibrium ... 

In this approximation we assume that one elementary step determines the rate while all other steps are sufficiently fast that they can be considered as being in quasi-equilibrium. If we take the surface reaction to AB (step 3, Eq. 134) as the rate-determining step (RDS), we may write the rate equations for steps (1), (2) and (4) as ... 

We will list the elementary steps and decide which is rate-limiting and which are in quasi-equilibrium. For ammonia synthesis a consensus exists that the dissociation of N2 is the rate-limiting step, and we shall make this assumption here. With quasi-equilibrium steps the differential equation, together with equilibrium condition, leads to an expression for the coverage of species involved in terms of the partial pressures of reactants, equilibrium constants and the coverage of other intermediates. 

In the following we shall examine a model system for the car catalyst where CO and NO react to yield the more environmentally friendly products CO2 and N2. The reaction shall be split up into the following elementary steps, which all are assumed to be in quasi-equilibrium, except step 2, which is assumed to be the rate-limiting step ... 

When the overall rate of a multistep reaction is determined solely by a single elementary step whose rate is extremely small compared with the rates of the other elementary steps, the multistep reaction is called the reaction of a single rate-determining step. In such a multistep reaction, as shown in Fig. 7-11 (a), all the elementary steps except for the rate-determining step are cmisidered to be in quasi-equilibrium. Note that the multistep reaction of a sin le rate-determining step is rather uncommon in practice. 

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 ... 

Theoretical treatments for the analysis of complex electrode reactions in terms of elementary steps can be made, as in general chemical kinetics, by the steady state [3, 5, 7] or the quasi-equilibrium methods [4, 5, 47]. 

Quasi-equilibrium treatments, on the other hand, assume that all elementary steps prior to the rds are almost in equilibrium, i.e. they can occur sufficiently fast not to alter significantly their equilibrium conditions under net charge flow at the interface. For this assumption to be valid, the elementary step rate coefficients must be at least ten times larger than that of the rds. 

Consider a micellar solution at equilibrium that is subject to a sudden temperature change (T-jump). At the new temperature the equilibrium aggregate size distribution will be somewhat different and a redistribution of micellar sizes will occur. Aniansson and Wall now made the important observation that when scheme (5.1) represents the kinetic elementary step, and when there is a strong minimum in the micelle size distribution as in Fig. 2.23(a) the redistribution of micelle sizes is a two-step process. In the first and faster step relaxation occurs to a quasi-equilibrium state which is formed under the constraint that the total number of micelles remains constant. Thus the fast process involves reactions in scheme (5.1) for aggregates of sizes close to the maximum in the distribution. This process is characterized by an exponential relaxation with a time constant Tj equal to... 

In summary, it can be seen for the three-step reaction scheme of this example that the net rate of the overall reaction is controlled by three kinetic parameters, KTSi, that depend only on the properties of the transition states for the elementary steps relative to the reactants (and possibly the products) of the overall reaction. The reaction scheme is represented by six individual rate constants /c, and /c the product of which must give the equilibrium constant for the overall reaction. However, it is not necessary to determine values for five linearly independent rate constants to determine the rate of the overall reaction. We conclude that the maximum number of kinetic parameters needed to determine the net rate of overall reaction is equal to the number of transition states in the reaction scheme (equal to three in the current case) since each kinetic parameter is related to a quasi-equilibrium constant for the formation of each transition state from the reactants and/or products of the overall reaction. To calculate rates of heterogeneous catalytic reactions, an addition kinetic parameter is required for each surface species that is abundant on the catalyst surface. Specifically, the net rate of the overall reaction is determined by the intrinsic kinetic parameters Kf s as well as by the fraction of the surface sites, 0, available for formation of the transition states furthermore, the value of o. is determined by the extent of site blocking by abundant surface species. 

An important point concerning the set of reactions 5.2 is worth noting. As already mentioned, it is generally assumed in such reactions that the net rate of generation of an intermediate can be set to zero. In another approach, if an equilibrium step is involved, we use the quasi-equilibrium approximation, in which the ratio of the forward to backward rates of a rapid elementary step is set equal to unity. In either case, it must be remembered that the fundamental... 

Ri R2, the rates can be denoted by the vectors shown in Figure 2.2. The difference R+i — R-i = R+2 — R-2> where the indices -I- and — refer to the forward and backward rates of the elementary steps, respectively. However, since R+i and R i are large, their ratio approaches one, R+i/R-i 1, which implies that the quasi-equilibrium hypothesis can be applied R+i/R i = fc+iCA/fc-iCR = KiCa/cr = l,thatis,iTi = cr/ca, which is the well-known equilibrium expression for an elementary step. 

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