Solid state rate equations

Answer During an actual firing schedule the rate constant / is a function of time. What this means is that all of the solid-state rate equations (Jander, Ginstling-Brounshtein, etc.) which are normally applied for constant k can be used, but the product kt must be replaced by an average product designated kt and given by the equation... 

When reactants are distributed between several phases, migration between phases ordinarily will occur with gas/liquid, from the gas to the liquid] with fluid/sohd, from the fluid to the solid between hquids, possibly both ways because reactions can occur in either or both phases. The case of interest is at steady state, where the rate of mass transfer equals the rate of reaction in the destined phase. Take a hyperbohc rate equation for the reaction on a surface. Then,... 

There have been few discussions of the specific problems inherent in the application of methods of curve matching to solid state reactions. It is probable that a degree of subjectivity frequently enters many decisions concerning identification of a best fit . It is not known, for example, (i) the accuracy with which data must be measured to enable a clear distinction to be made between obedience to alternative rate equations, (ii) the range of a within which results provide the most sensitive tests of possible equations, (iii) the form of test, i.e. f(a)—time, reduced time, etc. plots, which is most appropriate for confirmation of probable kinetic obediences and (iv) the minimum time intervals at which measurements must be made for use in kinetic analyses, the number of (a, t) values required. It is also important to know the influence of experimental errors in oto, t0, particle size distributions, temperature variations, etc., on kinetic analyses and distinguishability. A critical survey of quantitative aspects of curve fitting, concerned particularly with the reactions of solids, has not yet been provided [490]. 

Thus the key experimental observation Equation (7.11), is satisfied in presence of spillover. When an external overpotential AUWR is applied, with a concomitant current, I, and O2 flux I/2F, although UWR is not fixed anymore by the Nemst equation but by the extremally applied potential, still the work function Ow will be modified and Equations (7.11) and (7.12), will remain valid as long as ion spillover is fast relative to the electrochemical charge transfer rate I/2F.21 This is the usual case in solid state electrochemistry (Figs. 7.3b, 7.3d) as experimentally observed (Figs. 5.35, 5.23, 7.4, 7.6-7.9). 

This equation gives (0) = 0, a maximum at =. /Km/K2, and (oo) = 0. The assumed mechanism involves a first-order surface reaction with inhibition of the reaction if a second substrate molecule is adsorbed. A similar functional form for (s) can be obtained by assuming a second-order, dual-site model. As in the case of gas-solid heterogeneous catalysis, it is not possible to verify reaction mechanisms simply by steady-state rate measurements. 

Note also that we have just introduced the concepts of nuclei and nucleation in our study of solid state reaction processes. Our next step will be to examine some of the mathematics used to define rate processes in solid state reactions. We will not delve into the precise equations here but present them in Appendices at the end of this chapter. But first, we need to examine reaction rate equations as adapted for the solid state. 

Three types of rate equations are shown here. These rate equations ean be used for quite complieated reactions, but a specific method or measurement approach is needed. How we do this is critical to determining accurate estimation of the progress of a solid state reaction. We will discuss suitable methods in another chapter. We now return to the subject of nucleation so that we can apply the rate equations given above to specific cases. First, we examine heterogeneous processes. 

Let us now turn to diffusion in the general case, without worrying about the exact mechanism or the rates of diffusion of the various species. As an example to illustrate how we would analyze a diffusion-limited solid state reaction, we use the general equation describing formation of a compound with spinel (cubic) structure and stoichiometry ... 

These rate equations C2in be used for quite complicated reactions, but a specific method or approach is needed. Many authors have tried to devise methods for obtaining rate constants and orders of reaction for given solid state reactions. None have been wholly successful, except for Freeman and Carroll (1948). 

The adsorption rate of a certain substance on a surface of a solid state is described by equation of the type... 

The most common reaction exhibited by coordination compounds is ligand substitution. Part of this chapter has been devoted to describing these reactions and the factors that affect their rates. In the solid state, the most common reaction of a coordination compound occurs when the compound is heated and a volatile ligand is driven off. When this occurs, another electron pair donor attaches at the vacant site. The donor may be an anion from outside the coordination sphere or it may be some other ligand that changes bonding mode. When the reaction involves an anion entering the coordination sphere of the metal, the reaction is known as anation. One type of anation reaction that has been extensively studied is illustrated by the equation... 

As a result, nitroalkene (42a) is almost completely isomerized into N-oxide (47a), which is stable up to 100°C (167). Isomerization proceeds smoothly both in solutions and the solid state. The isomerization rate is approximated by the first-order equation and depends on the polarity of the solvent (isomerization in hexane occurs 70 times more slowly than that in ethanol). 

However, we have only considered a steady state solid model in Equation (8.4). Consider the effect if we add the transient term (Equation (3.40)), the rate of enthalpy per unit area in the CV,... 

Figure 18. A simple bistable pathway [96], Left panel The metabolite A is synthesized with a constant rate vi and consumed with a rate vcon V2(A) + V3(A), with the substrate A inhibiting the rate V3 at high concentrations (allosteric regulation). Right panel The rates of vsyn vi const. and vcon V2 (A) + V3(A) as a function of the concentration A. See text for explicit equations. The steady state is defined by the intersection of synthesizing and consuming reactions. For low and high influx v, corresponding to the dashed lines, a unique steady state A0 exists. For intermediate influx (solid line), the pathway gives rise to three possible solutions of A0. The rate equations are specified in Eq. 67, with parameters 0.2, 3 2.0, Kj 1.0, and n 4 (in arbitrary units).

Andreasen et al. [86] also found that ball milling increased the rate constant, k, in the JMAK equation (Sect. 1.4.1), of reaction (Rib) in solid state but virtually had no effect on the rate constant of reaction (R2). They also showed that the reaction constant, k, of reaction (Rib) in solid state increases with decreasing grain size of ball-milled LiAlH within the range 150-50 mn. Andreasen et al. concluded that the reaction (Rib) in solid state is limited by a mass transfer process, e.g., long range atomic diffusion of Al while the reaction (R2) is limited by the intrinsic kinetics (too low a temperature of decomposition). In conclusion, one must say that ball milling alone is not sufficient to improve the kinetics of reaction (R2). A solution to improvement of the kinetics of reaction (R2) could be a suitable catalytic additive. 

In this equation, aua represents the product of the coefficient of electron transfer (a) by the number of electrons (na) involved in the rate-determining step, n the total number of electrons involved in the electrochemical reaction, k the heterogeneous electrochemical rate constant at the zero potential, D the coefficient of diffusion of the electroactive species, and c the concentration of the same in the bulk of the solution. The initial potential is E/ and G represents a numerical constant. This equation predicts a linear variation of the logarithm of the current. In/, on the applied potential, E, which can easily be compared with experimental current-potential curves in linear potential scan and cyclic voltammetries. This type of dependence between current and potential does not apply to electron transfer processes with coupled chemical reactions [186]. In several cases, however, linear In/ vs. E plots can be approached in the rising portion of voltammetric curves for the solid-state electron transfer processes involving species immobilized on the electrode surface [131, 187-191], reductive/oxidative dissolution of metallic deposits [79], and reductive/oxidative dissolution of insulating compounds [147,148]. Thus, linear potential scan voltammograms for surface-confined electroactive species verify [79]... 

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