Coupled rate laws

Taken together, Eqs. 1.1 and 1.2 constitute a sequential chemical reaction to form bicarbonate from dissolved carbon dioxide  

An Arrhenius plot of a zeroth-order reaction rate coefficient (normalized to unit surface itiv i atul the unit cell) for the dissolution of a variety of silicate minerals (data from B. J. Wood and I V, Walther, Rates of hydrothermal reaction. Science 222 413 (19K3). See Section 3.1 for additional discussion of rale coefficients for dissolution reactions. 

The sequential reactions in Eqs. 1.48 and 1.50 are special cases of the two abstract reaction schemes 

For the sequential reaction in Eq. 1.51, the set of rate equations generated through these simplifying assumptions is 

Mathematical solutions of coupled rate equations are available for a variety til special cases,16 but approximate solutions informed by experimental data concerning the relative rates of contributing reactions are more the rule. For the iructions in Eq. 1.48, as an example, it is known2,7 that the second reaction comes to equilibrium very much faster than the first and that, in the first icaclion, the forward rate is much smaller than the backward rate. Thus the rate o formation of bicarbonate from the hydration of C02 is limited by the rate of lot mation of true carbonic acid (at pH 8). With respect to Eqs. 1.53a and I s tc, this means that, on the time scale of formation of species C (H2CO ), the iale of increase of the concentration of species D (OH ) is nil. Moreover, the i onccnlralion of species B (11,0) is effectively constant in aqueous solution and 

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For the reactions in Eq. 1.50, it is known5 that the first reaction comes to equilibrium much more quickly than the second and that in the second reaction the forward rate is much larger than the backward rate. As in the C02 hydration reaction, the concentration of water is effectively constant (species E in Eq. 1.52). Thus the rate of inner-sphere complex formation from the outer-sphere complex intermediate species limits the overall rate of the reaction in Eq. 1.8. The impact of these experimental facts on the coupled rate laws in Eq. 1.53a and 1.54c is to reduce them to a single equation ... 

Systems for which there is only a single kinetic process are the exception, not the rule. In general there is a spectrum of relaxation times. As long as these are well separated the individual decay constants are easily measured. However, deducing the coupled rate laws and the rate constants is more difficult. If the kinetic processes proceed at similar rates considerable sophistication in data analysis is needed to obtain reliable values of the decay constants. 

It is not always possible to solve kinetic problems using simple approaches. Even if the mechanism only involves a few steps the data may not readily suggest the coupled rate laws. Chemical intuition is required to propose a plausible mechanism which then must be tested. The natural procedure is to integrate the mechanistic rate equations and to compare the predicted concentrations with the observed ones. If a set of kinetic parameters can be found that satisfactorily accounts for the data, the mechanism is a possible one. Such calculations always involve extensive numerical work. Thus reasonable starting points are needed, even if modern high-speed computers are used. 

The system of coupled differential equations that result from a compound reaction mechanism consists of several different (reversible) elementary steps. The kinetics are described by a system of coupled differential equations rather than a single rate law. This system can sometimes be decoupled by assuming that the concentrations of the intennediate species are small and quasi-stationary. The Lindemann mechanism of thermal unimolecular reactions [18,19] affords an instructive example for the application of such approximations. This mechanism is based on the idea that a molecule A has to pick up sufficient energy... 

To trace a reaction path incorporating redox kinetics, we first set a model in redox disequilibrium by disabling one or more redox couples, then specify the reaction in question and the rate law by which it proceeds. To model the progress of Reaction 17.1, for example, we would disable the redox couple between vanadyl and vanadate species. In a model of the oxidation of Fe++ by manganite (MnOOH), we would likely disable the couples for both iron and manganese. 

Using the modified thermodynamic database, we simulate reaction over 300 minutes in a fluid buffered to a pH of 7. We prescribe a redox disequilibrium model by disabling redox couples for chromium and sulfur. We set 10 mmolal NaCl as the background electrolyte, initial concentrations of 200 (imolal for CrVI and 800 innolal for H2S, and small initial masses of Cr2C>3 and S(aq). Finally, we set Equation 17.29 as the rate law and specify that pH be held constant over the simulation. 

A complex reaction requires more than one chemical equation and rate law for its stoichiometric and kinetics description, respectively. It can be thought of as yielding more than one set of products. The mechanisms for their production may involve some of the same intermediate species. In these cases, their rates of formation are coupled, as reflected in the predicted rate laws. 

These rate laws are coupled through the concentrations. When combined with the material-balance equations in the context of a particular reactor, they lead to uncoupled equations for calculating the product distribution. For a constant-density system in a CSTR operated at steady-state, they lead to algebraic equations, and in a BR or a PFR at steady-state, to simultaneous nonlinear ordinary differential equations. We demonstrate here the results for the CSTR case. 

Kinetic schemes involving sequential and coupled reactions, where the reactions are either first-order or pseudo-first order, lead to expressions for concentration changes with time that can be modeled as a sum of exponential functions where each of the exponential functions has a specific relaxation time. More complex equations have to be derived for bimolecular reactions where the concentrations of reactants are similar.19,20 However, the rate law is always related to the association and dissociation processes, and these processes cannot be uncoupled when measuring a relaxation process. 

Although more fundamental approaches are used in the science of chemical reaction engineering to account for the diffusion/reaction coupling, we rather propose the explanation restricted to rate laws of first order with respect to hydrogen and based on intuition. 

The above 3H experiments do not give a direct measure of inversion in the [Co(en)2((S)-AAOMe)]3+ reactant, although they do establish that it is fast under the coupling conditions (seconds), while in the dipeptide product [Co(en)2((S)-AA-(S)AA OMe)]3+ it is slow (hours). But this much was already known, in a qualitative way, from the published literature (6, 7). To minimize the problem it was essential to determine the rate laws and rate constants for both epimerization and aminolysis and to investigate differences between the A- and A-reactants. The following experiments established the rate laws for these two processes and outline the difficulties associated with using... 

As well as deceleratory reactions, kineticists often find that some chemical systems show a rate which increases as the extent of reaction increases (at least over some ranges of composition). Such acceleratory, or autocatalytic, behaviour may arise from a complex coupling of more than one elementary kinetic step, and may be manifest as an empirically determined rate law. Typical dependences of R on y for such systems are shown in Figs 6.6(a) and (b). In the former, the curve has a basic parabolic character which can be approximated at its simplest by a quadratic autocatalysis, rate oc y(l - y). 

The rate laws and hence the mechanisms of chemical reactions coupled to charge transfer can be deduced from LSV measurements. The measurements are most applicable under conditions where the charge transfer can be considered to be Nernstian and the homogeneous reactions are sufficiently rapid that dEv/d log v is a linear function, i.e. the process falls into the KP or purely kinetic zone. In the 1960s and 1970s, extensive... 

In Section 4.4.2 some concepts were developed which allow us to quantitatively treat transport in ionic crystals. Quite different kinetic processes and rate laws exist for ionic crystals exposed to chemical potential gradients with different electrical boundary conditions. In a closed system (Fig. 4-3a), the coupled fluxes are determined by the species with the smaller transport coefficient (c,6,), and the crystal as a whole may suffer a shift. If the external electrical circuit is closed, inert (polarized) electrodes will only allow the electronic (minority) carriers to flow across AX, whereas ions are blocked. Further transport situations will be treated in due course. 

Rivera Islas et al. proposed an alternative approach, which should be generally employed for the study of nonlinear systems and was believed to respond better to the complexity of the Soai reaction than the consideration of single rate laws [69]. In such an attempt a priori approximations are usually avoided, all possible species are considered, the velocity of equilibria is especially taken into account, and the coupling of chemically realistic reaction steps is not disregarded. On the other hand, a larger number of variables and parameters have to be handled. Hence such an approach can only be conducted numerically but, in the best case, it can mimic the mechanics of the real system because of its similar coupled and multistep design. 

Formulating appropriate rate laws for CO adsorption, OH adsorption and the reaction between these two surface species, a set of four coupled ordinary differential equations is obtained, whereby the dependent variables are the average coverages of CO and OH, the concentration of CO in the reaction plane and the electrode potential. In accordance with the experiments, the model describes the S-shaped I/U curve and thus also bistability under potentiostatic control. However, neither oscillatory behavior is found for realistic parameter values (see the discussion above) nor can the nearly current-independent, fluctuating potential be reproduced, which is observed for slow galvanodynamic sweeps (c.f. Fig. 30b). As we shall discuss in Section 4.2.2, this feature might again be the result of a spatial instability. 

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