Rate Determining Step — Quasi-Equilibrium

The rate expression is now obtained as follows. Starting with the rate determining step  

This enables the elimination of the unknown quantities 0a and 0b from (3.10). The remaining 0 can be eliminated by use of the site balance (3.5). 

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

If other species also adsorb on the active sites they occupy these catalytic centres and hence lower the reaction rate. The effect Of these inhibitors should be included in the rate expression and can be summarised in Eqn. (3.17) for the reaction under consideration  

Here Ka, Kb and K represent the adsorption equilibrium constants of the components A, B and I, respectively, if the surface reaction is rate determining k represents here the apparent (observed) overall reaction rate constant 

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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 ... 

By utilizing this we can now express the coverages of all the relevant intermediates and the overall rate in terms of equilibrium constants of the steps in quasi-equilibrium, the pressures of the reactants and the products, and the rate constant of the rate-determining step. 

It is important to realize that the assumption of a rate-determining step limits the scope of our description. As with the steady state approximation, it is not possible to describe transients in the quasi-equilibrium model. In addition, the rate-determining step in the mechanism might shift to a different step if the reaction conditions change, e.g. if the partial pressure of a gas changes markedly. For a surface science study of the reaction A -i- B in an ultrahigh vacuum chamber with a single crystal as the catalyst, the partial pressures of A and B may be so small that the rates of adsorption become smaller than the rate of the surface reaction. 

Gomez-Sainero et al. (11) reported X-ray photoelectron spectroscopy results on their Pd/C catalysts prepared by an incipient wetness method. XPS showed that Pd° (metallic) and Pdn+ (electron-deficient) species are present on the catalyst surface and the properties depend on the reduction temperature and nature of the palladium precursor. With this understanding of the dual sites nature of Pd, it is believed that organic species S and A are chemisorbed on to Pdn+ (SI) and H2 is chemisorbed dissociatively on to Pd°(S2) in a noncompetitive manner. In the catalytic cycle, quasi-equilibrium ( ) was assumed for adsorption of reactants, SM and hydrogen in liquid phase and the product A (12). Applying Horiuti s concept of rate determining step (13,14), the surface reaction between the adsorbed SM on site SI and adsorbed hydrogen on S2 is the key step in the rate equation. 

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. 

Rate determining step (cont.) electrocatalysis and, 1276 methanol oxidation, 1270 in multistep reactions, 1180 overpotential and, 1175 places where it can occur, 1260 pseudo-equilibrium, 1260 quasi equilibrium and, 1176 reaction mechanism and, 1260 steady state and, 1176 surface chemical reactions and, 1261 Real impedance, 1128, 1135 Reciprocal relation, the, 1250 Recombination reaction, 1168 Receiver states, 1494 Reddy, 1163... 

Figure 9. Visualisation of the rate determining step and quasi-equilibrium steps.

Various rate expressions may be derived from a kinetic model, depending on the assumptions of quasi-equilibrium and rate determining step. Experimental data should provide information which expression describes best the rate dependency. 

Fig. 3.2. Visualisation of the quasi-equilibrium and rate determining steps. The lengths of the arrows are proportional to the rates of the relevant steps.

We can now proceed to obtain the kinetic equation for the same reaction in a somewhat more complicated case, when the second step (Eq. 9F) is assumed to be the rate determining step, and the first step (Eq. 8F) is at quasi-equilibrium. 

In Fig. 51 we show calculated Tafel lines for reaction 191 at quasi-equilibrium followed by reaction 201 as the rate-determining step. All lines were calculated for 0 ranging from 1x10 to (1 - 0) = 10" from the full equation, without neglecting the preexponential terms in 0. The value of / used to calculate each line is shown. Note the dependence of the region of linearity of the Tafel lines at intermediate... 

Now we come to the concept of quasi-equilibrium. If there is a distinct rate-determining step in a reaction sequence, then all other steps before and after it must be effectively at equilibrium. This comes about because the overall rate is, by definition, very slow compared to the rate at which each of the other steps could proceed by itself, and equilibrium in these steps is therefore barely disturbed. To see this better, consider the specific example given earlier for chlorine evolution. Assume, for the sake of argument, that the values of the exchange current density i for steps 8F and 9F are 250 and 1.0 mA/cm, respectively. Assume now that we apply a current density of 0.5 mA/cm. We can calculate the overpotential corresponding to each step in the sequence, using Eq. 6E, namely... 

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. 

In Sections 3.2 and 3.4, the approach based on one rate-determining step with the other steps in quasi-equilibrium was applied to simplify the derivation of rate expressions. A further simplification of the rate expressions is obtained when the product concentrations or partial pressures are negligibly small, i.e., at low conversions of a pure reactant feed stream. Similarly, at high conversions, when nearly only product is present, the rate expressions can also be simplified. Applying e single-site model, Eqns. (3.17)-(3.19), for the reaction A B at low conversion, the following expressions (often called "initial rate expressions") are obtained, provided that the feed contains pure A. 

The first is by means of the assumption that the formation and decomposition of a particular transition state limits the rate of the overall reaction and that any steps prior to the rate-determining step characterized by that transition state are at quasi-equilibrium. This allows the concentrations of any intermediates involved in the rds to be expressed as a function of potential [/(U)] and hence a kinetic expression for the potential dependence of the rds can be formulated in terms of its Tafel slope (6) and transfer coefficient (a), formally defined as b = dVId In i = RTIaF. 

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... 

Possible Rate-Determining Steps (with Preceding Quasi-Equilibrium Steps Where Present) for the Oxidation of Propane and Platinum in Phosphoric Acid, 80°C-150°C and 0.30-5.0 V (Reversible Hydrogen Scale) (Work by G. Stoner)... 

Figure 3.7. Illustration of quasi-equilibrium and rate determining step concepts.

Thus, it is a metal-deficit, p-type semiconductor in a wide range of sulfur activities. Only at very low pS2 near the Mn/MnS equilibrium it is a metal-excess, n-type semiconductor with doubly ionized interstitial cations and quasi-free electrons [57, 58]. The growth of MnS proceeds by outward diffusion of cations, being the rate-determining step of manganese sulfidation. The low nonstoichiometry is the reason the MnS growth is several orders of magnitude slower than that of other transition metal sulfides [59, 60]. 

Because all the steps are reversible, some further clarification is needed with respect to the rate-determining step. It can be imagined that the forward and reverse rates of four of the five steps are much larger than those of the fifth step. In fact, they are so much larger that the forward and reverse rates of each step may be assumed to be nearly equal. In other words, these four steps may be assumed to be in quasi-equilibrium. Note, however, that the net rate is still the same for all the steps, that is, r(net) = (ri+ - ri ) St (r2+ - r2 )... = (75+ - r ). For the fifth step, we have the additional fact that r + r, while for the other... 

It can be seen that, if there is a rate-determining step, only the rate constants of that step enter into the rate equation. The rate constants of the other steps in quasi-equilibrium appear only as ratios which, as is already known, are equal to the equilibrium constants of these equilibrated steps. This is a considerable simplification of the kinetic problem, since, at least in principle, equilibrium constants are more easily arrived at than rate constants. Even when this is not so, the number of arbitrary constants in the rate equation is reduced considerably. 

If in a closed sequence it is possible under a given set of conditions to recognize a rate-determining step (subscript f), all the other steps being in quasi-equilibrium, the free-energy difference for the overall reaction AG will be proportional to the free-energy difference for the rate-determining step AG,. 

At high temperatures where the rate of the reaction is still slow but measurable, the rate-determining step is (2) the first step is in quasi-equilibrium and thus ... 

According to the concept of a rate-determining step, if there is one, all other steps in a sequence are in quasi-equilibrium. Thus these other steps must be reversible. Consider now a closed sequence where all steps are irreversible ... 

The equation shows protonation of an adsorbed TOA molecule at the interface. This first step is in quasi-stationary equilibrium, which is different from the two following rate determining steps ... 

Most electrode reactions encountered in the field of corrosion involve the transfer of more than one electron. Such reactions take place in steps, of which the slowest, called the rate-determining step, abbreviated RDS, determines the overall reaction rate. In simple cases, one can identify the rate-determining step by an analysis of the measured Tafel slopes. In the so-called quasi-equilibrium approach one assumes that with the exception of the rate-limiting step, all other steps are at equilibrium. This greatly simplifies the mathematical equations for the reaction rate. More realistic approaches require numerical simulation and shall not be discussed here. To illustrate the quasi equilibrium approach to the study of multi-step electrode reactions we shall look at proposed mechanisms for the dissolution of copper and of iron. 

A relatively long sequence of steps, frequently encountered in practice, evidently requires quite a number of rate equations. In many cases one of the steps is intrinsically much slower than the others. A steady state is established in which the rates of the other steps adapt to the rate of this step — it is the rate determining step. For steady state conditions only one rate equation will suffice to describe the process. All the other steps will be in quasi equilibrium. The rate determining step may change with the operating conditions so that care has to be taken when using this concept. The change will be revealed by shifts in the product distribution. 

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