Rate laws, approximate reactant concentration

The rate-determining step approximation is made to determine a rate law for a mechanism in which one step occurs at a rate substantially slower than any others. The slow step is a bottleneck, and the overall reaction rate cannot be larger than the slow step. The rate law for the rate-determining step is written first. If a reaction intennediate appears as a reactant in this step, its concentration term must be eliminated from the rate law. The final rate law only has concentration terms for reactants and products. 

The order of the rate law with respect to the three reactants can be determined by comparing the rates of two experiments in which the concentration of only one of the reactants is changed. For example, in experiment 2 the [H+] and the rate are approximately twice as large as in experiment 1, indicating that the reaction is first-order in [H+]. Working in the same manner, experiments 6 and 7 show that the reaction is also first-order with respect to [CaHeO], and experiments 6 and 8 show that the rate of the reaction is independent of the [I2]. Thus, the rate law is... 

Power law expressions are useful as long as the approximate orders of reactant concentration are constant over a particular concentration course. A change in the order of the reaction corresponds to a change in the surface concentration of a particular reactant. A low reaction order usually implies a high surface concentration, a low reaction order, and a low surface reaction of the corresponding adsorbed intermediates. In order to deduce (Eq. (1.17b)) the rate of surface carbon hydrogenation, the power law of Eq. (1.18) has been used. 

Valuable information on mechanisms has been obtained from data on solvent exchange (4.4).The rate law, one of the most used mechanistic tools, is not useful in this instance, unfortunately, since the concentration of one of the reactants, the solvent, is invariant. Sometimes the exchange can be examined in a neutral solvent, although this is difficult to find. The reactants and products are however identical in (4.4), there is no free energy of reaction to overcome, and the activation parameters have been used exclusively, with great effect, to assign mechanism. This applies particularly to volumes of activation, since solvation differences are approximately zero and the observed volume of activation can be equated with the intrinsic one (Sec. 2.3.3). 

For each kinetic scheme in Scheme 9.4, the rate law obtained by applying the Bodenstein approximation to the intermediate (I) is presented and, for this discussion, we consider that the reactant R is the component whose concentration can be easily monitored. The reactions are all expected to be first order in [R], but the first-order rate constants show complex dependences on [X] and, in two cases, also on [Y]. All the rate laws contain sums of terms in the denominator, and the compositions of the transition structures for formation and destruction of the intermediate are signalled by the form of the rate law when each term of the denominator is separately considered. This pattern is general and can be usefully applied in devising mechanisms compatible with experimentally determined rate laws even for much more complex situations. 

If the mechanism is not known in detail, the kinetic terms may be replaced by empirically-determined rate laws, i.e., by approximations to the reaction rate term that typically will be some (non-linear) polynomial fit of the observed rate to the concentrations of the major species in the reaction (reactants and products). Such empirical rate laws have limited ranges of validity in terms of the experimental operating conditions over which they are appropriate. Like other polynomial fitting procedures, these representations can rapidly go spectacularly wrong outside their range of validity, so that they must be used with great care. If this care is taken, however, empirical rate equations are of great value. 

The steady-state approximation is a more general method for solving reaction mechanisms. The net rate of formation of any intermediate in the reaction mechanism is set equal to 0. An intermediate is assumed to attain its steady-state concentration instantaneously, decaying slowly as reactants are consumed. An expression is obtained for the steady-state concentration of each intermediate in terms of the rate constants of elementary reactions and the concentrations of reactants and products. The rate law for an elementary step that leads directly to product formation is usually chosen. The concentrations of all intermediates are removed from the chosen rate law, and a final rate law for the formation of product that reflects the concentrations of reactants and products is obtained. 

In spherical coordinates, the dimensional mass transfer equation with radial diffusion and first-order irreversible chemical reaction exhibits an analytical solution for the molar density profile of reactant A. If the kinetics are not zeroth-order or first-order, then the methodology exists to find the best pseudo-first-order rate constant to match the actual rate law and obtain an approximate analytical solution. The concentration profile of reactant A in the liquid phase must satisfy... 

The second approach starts with an idea of possible mechanism, leading to a theoretical kinetic equation formulated in terms of concenhations of adsorbed reactants and intermediate species use of the steady-state principle then leads to an expression for the rate of product formation. Concentrations of adsorbed reactants are related to the gas-phase pressures by adsorption equations of the Langmuir type, in a way to be developed shortly the final equation, the form of which depends on the location of the slowest step, is then compared to the Power Rate Law expression, which if a possibly correct mechanism has been selected, will be an approximation to it. A further test is to try to fit the results to the theoretical equation by adjusting the variable parameters, mainly the adsorption coefficients (see below). If this does not work another mechanism has to be tried. 

In kinetic studies of the hydrogenation of aromatic hydrocarbons, the dependence of rate upon reactant pressures has usually been expressed in Power Rate Law formulations, that is, by orders of reaction that are simple exponents of the pressures. These as we have seen (Section 5.2) are at best approximations to more fundamental expressions based on concentrations of adsorbed species," " although they may well represent results over the limited range in which measurements were made. The Langmuir-Hinshelwood formalism has however sometimes been used, and heats of adsorption of the reactants in their reactive states derived from the temperature-dependence of their adsorption coefflcients. ... 

D22.3 The determination of a rate law is simplified by the isolation method in which the concentrations of all the reactants except one are in large excess. If B is in large excess, for example, then to a good approximation its concentration is constant throughout the reaction. Although the true rate law might be V = if[A][B]. we can approximate [B] by [Bfo and write... 

In a dissociative mechanism, the metal ion dissociates X in the first step to form a five-coordinate intermediate and then adds Y in the slow step to form the substituted product The rate law for the D mechanism is given by Equation (17.25) and depends on the concentration of the intermediate [MLj]. Applying the steady-state approximation to [MLj], as shown in Equation (17.26), we obtain the overall rate law in terms of only the reactant concentrations given by Equation (17.27). 

Pseudo-First-Order Reactions Under certain circumstances, second-order reactions can sometimes be approximated as first-order reactions. For example, consider a second-order reaction that depends on the concentrations of two different reactants (each to the first order). If one of the reactant concentrations is much larger than the other reactant concentration, then it will remain essentially constant (only slightly depleted) during the reaction process while the concentration of the other reactant is fully consumed. In this situation, the second-order rate law can be rewritten as a pseudo-first-order rate law. As an example, consider a second-order reaction that is first order with respect to two reactants A and B. The rate law for this reaction is... 

The rate laws developed in this chapter all assume that the reactions go to completion. While this a reasonable assumption for many reactions, there are many others where the reaction proceeds only partway and an equilibrium state is reached where considerable concentrations of both the reactant and product species remain. The rate expressions in this chapter can be modified to approximately account for this by incorporating a mathematical offset into the rate equation for example. 

The important theoretical expressions for fcc above have been derived from a model of hard-sphere molecules in a continuous medium. Intermolecular forces between reactant molecules have been neglected. When the reactants are ionic or polar, there will be long-range Coulombic interactions between them. For reactions between ions, we stated in Chapter 2 (Section 2.5.3) an expression for the value of the rate constant at low concentrations, and noted some reactions between oppositely-charged ions that have rate constants in approximate agreement with it. We also noted that for several such reactions the effect of added inert ions follows approximately the Debye-Hiickel limiting law. 

The steady-state approximation helps us relate the rate law determined from the RDS with the rate law as determined from experiment. Recall that the rate law of a mechanism is dictated by the stoichiometry of the RDS, but how can we know if this rate law is consistent with the experimental rate law We can use the fact that the preceding step(s) is/are at equilibrium to derive a rate law in terms of the original reactants (whose amounts or concentrations we can measure). There are two ways to do this. First, we will adopt a simplified approach. If the first step in the above two-step process is in fact in equilibrium, then we can write an equilibrium constant expression for it... 

This rate law can be expressed in terms of the reactant concentrations using the preequilibrium approximation ... 

Kinetic Equations 3-143 and 3-153 are obeyed by nucleides undergoing radioactive decay, where the rate constant k, is large and k2 is small. The reactant A is converted rapidly into the intermediate B, which slowly forms C. Figure 3-13b shows plots of the exponentials C-M and e-M and of their difference. Since k2 is small, the exponential e-k2t shows a slow decay while e klt shows a rapid decline. The difference of e-k2t-e-kl is shown by the dashed line in Figure 3-13b. The concentration of B is (Equation 3-143) equal to this difference multiplied by CAO (since kt k2). Therefore, the concentration of B rapidly rises to the value of CAO and then slowly declines. The rise in concentration C then approximately follows the simple first-order law. Conversely, when k, is small and k2 is large (k2 kj), the concentration of B is given by Equation 3-143 ... 

As seen in Table 2.1, the overall order of an elementary step and the order or orders with respect to its reactant or reactants are given by the molecularity and stoichiometry and are always integers and constant. For a multistep reaction, in contrast, the reaction order as the exponent of a concentration, or the sum of the exponents of all concentrations, in an empirical power-law rate equation may well be fractional and vary with composition. Such apparent reaction orders are useful for characterization of reactions and as a first step in the search for a mechanism (see Chapter 7). However, no mechanism produces as its rate equation a power law with fractional exponents (except orders of one half or integer multiples of one half in some specific instances, see Sections 5.6, 9.3, 10.3, and 10.4). Within a limited range of conditions in which it was fitted to available experimental results, an empirical rate equation with fractional exponents may provide a good approximation to actual kinetics, but it cannot be relied upon for any extrapolation or in scale-up. In essence, fractional reaction orders are an admission of ignorance. 

FIGURE 5.4 Schematic illustration of the active gas corrosion of Ti by HCl(g) when controlled by the diffusion of HCl(g) to the surface. Under diffusion control, the reaction rate is limited by the rate at which the HCl(g) reactant can diffuse across the diffusion zone (of thickness 5) to the surface. At steady state, the concentration profile across the diffusion zone is typically approximated as linear, enabling the diffusion flux to be calculated using a straightforward solution of Pick s first law. 

When condition (dai/dt)react (dai/dt)rei is not satisfied, the perturbation of equilibrium distribution is substantial. However, in this case realization of the quasi-steady-state condition is possible, provided the overall rate dai/dt is low compared to partial rates (daj/dt)rei and (dai/dt)i.eact Then, the microscopic kinetic equation can be solved by the quasi-steady-state approximation. The approximation implies that the non-equilibrium distribution functions depend on time implicitly via the total concentration of reactants rather than explicitly. This also means that the macroscopic reaction rates are low compared to microscopic reaction and relaxation rates. Since the distribution functions in this approximation depend on the total concentration only, the reaction rates, according to Eq. (8.50) also depend on the total concentration. Hence, we come to macroscopic kinetic equations that involve only the total concentration of reactants and certain combinations of microscopic rate constants that have the meaning of macroscopic constants. Note that these macroscopic equations need not be consistent with the macroscopic kinetic law as, besides elementary reactive processes, they involve unreactive processes. 

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