Applying Rate Laws

A3 Avrami—Erofeev three-dimensional growth of nuclei [-ln(l -a)]V3 

Deceleratory a-time curves based on geometrical models  

FIGURE 7.3 Avarami-Erofeev plots of the data shown in Table 7.2. 

In Chapter 8, it will be shown that most of the rate laws shown in Table 7.2 can be put in the form of a composite rate law involving three exponents. We will also describe the difficulties associated with attempts to determine these exponents from (a,i) data. While the discussion up to this point has set forth the basic principles of sohd state reactions, their application to specific 

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Upon applying rate law to these reactions, the six differential equations can be written as... 

Ref. 205). The two mechanisms may sometimes be distinguished on the basis of the expected rate law (see Section XVni-8) one or the other may be ruled out if unreasonable adsorption entropies are implied (see Ref. 206). Molecular beam studies, which can determine the residence time of an adsorbed species, have permitted an experimental decision as to which type of mechanism applies (Langmuir-Hinshelwood in the case of CO + O2 on Pt(lll)—note Problem XVIII-26) [207,208]. 

The apparent activation energy is then less than the actual one for the surface reaction per se by the heat of adsorption. Most of the algebraic forms cited are complicated by having a composite denominator, itself temperature dependent, which must be allowed for in obtaining k from the experimental data. However, Eq. XVIII-47 would apply directly to the low-pressure limiting form of Eq. XVIII-38. Another limiting form of interest results if one product dominates the adsorption so that the rate law becomes... 

Complex chemical mechanisms are written as sequences of elementary steps satisfying detailed balance where tire forward and reverse reaction rates are equal at equilibrium. The laws of mass action kinetics are applied to each reaction step to write tire overall rate law for tire reaction. The fonn of chemical kinetic rate laws constmcted in tliis manner ensures tliat tire system will relax to a unique equilibrium state which can be characterized using tire laws of tliennodynamics. 

The condition [B]o = [A]o applied to the reaction with v = k[A][B] does not distinguish it from either of the cases v = k[A]2 or v = k[B]2. That is to say, the seeming mathematical simplification of Eq. (2-15) as compared with Eq. (2-20) comes at the expense of more explicit information about the rate law that is otherwise obtained from a single experiment. 

Now we shall apply this to different rate laws. For first-order kinetics either of two forms can be used ... 

Certain features of the reaction schemes manifest themselves in the rate law in a regular way. These features guide the investigator to one or more mechanisms consistent with the data. The same considerations allow one to reject certain alternatives. Listed here is a set of rules, or more properly clues, that are useful guides to the correct scheme. Each is accompanied by examples as to how they can be applied. 

When the initiation and termination reactions are the reverse of one another, the kinetic form is usually simpler than when the two are independent. Also, the transition-state composition follows directly from the rate law, which is why the term well-behaved is applied. Imagine, for example, that the termination step in the system most recently presented was the recombination of two sulfate radical ions rather than Eq. (8-38) ... 

Also, the rates of the propagation steps are equal to one another (see Problem 8-4). This observation is no surprise The rates of all the steps are the same in any ordinary reaction sequence to which the steady-state approximation applies, since each is governed by the same rate-controlling step. The form of the rate law for chain reactions is greatly influenced by the initiation and termination reactions. But the chemistry that converts reactant to product, and is presumably the matter of greatest importance, resides in the propagation reactions. Sensitivity to trace impurities, deliberate or adventitious, is one signal that a chain mechanism is operative. 

In the mechanisms considered so far, there have only been one or two intermediates. In a chain reaction, a highly reactive intermediate reacts to produce another highly reactive intermediate, which reacts to produce another, and so on (Fig. 13.19). In many cases, the reaction intermediate—which in this context is called a chain carrier—is a radical, and the reaction is called a radical chain reaction. In a radical chain reaction, one radical reacts with a molecule to produce another radical, that radical goes on to attack another molecule to produce yet another radical, and so on. The ideas presented in the preceding sections apply to chain reactions, too, but they often result in very complex rate laws, which we will not derive. 

This rate law has been found to apply. It has been noted that the 2 in Sn2 stands for bimolecular. It must be remembered that this is not always the same as second order (see p. 291). If a large excess of nucleophile is present-—for example, if it is the solvent—the mechanism may still be bimolecular, though the experimentally determined kinetics will be first order ... 

Since the pressure build up is primarily due to the evolution of CO as MDI is being decomposed to carbodiimide, the thermodynamic relationship PV = nRT may be applied to convert the pressure profiles to plots of moles of CO2 generated vs. time. This is shown for the 225 °C isotherm in Figure 3. The theoretical curve obtained through the application of zero-order kinetics is also shown in this plot and the data seem to be well accommodated by this rate law throughout the majority of the run. 

When asked to determine a rate law and rate constant, we must determine the order of the reaction. The rate law for this reaction may contain the concentrations of NO2, F2, and NO2 F raised to powers x, y, and z that must be determined Rate = k [N02] [F2] [NO2 F] Because the rate law contains more than one species, we need to use either isolation or initial rates to determine the orders of reaction. In the experiments whose data are shown, initial rate data are obtained for various combinations of initial concentrations. We apply the ratios of these initial rates to evaluate the orders. 

If mechanism (A) applied the Cr(VI)+V(IV) system would be anomalous when compared with the Cr(VI) + Fe(II) and Ce(IV) + Cr(III) reactions which have similar rate laws and Cr(V) -> Cr(IV) transformations as rate-controlling steps. Apart from this there are other good reasons for rejecting mechanism (+). At 25 °C, K is 10 ° and k is 0.56 l.mole sec , allowing At2 to be calculated as... 

The observed rate law, which applies only to (15) in 0.5 M sodium acetate buffer in the pH range 4.18-5.05 is... 

Applying a steady-state approximation to [H02 ] the mechanism leads to the rate law... 

If it is assumed that the rate-determining step is reaction (116), that reactions (118) and (119) are much more rapid than (115) and (116) and that the steady-state hypothesis can be applied, the resulting rate law corresponds exactly to the experimental one, provided that a = = 2 and k — k i. At the... 

This is the important Hill-Langmuir equation. A. V. Hill was the first (in 1909) to apply the law of mass action to the relationship between ligand concentration and receptor occupancy at equilibrium and to the rate at which this equilibrium is approached. The physical chemist I. Langmuir showed a few years later that a similar equation (the Langmuir adsorption isotherm) applies to the adsorption of gases at a surface (e g., of a metal or of charcoal). 

As was discussed earlier in this chapter, the concept of a reaction order does not apply to a crystal that is not composed of molecules. However, there are numerous cases in which the rate of reaction is proportional to the amount of material present. We can show how this rate law is obtained in a simple way. If the amount of material at any time, t, is represented as W and if we let W0 be the amount of material initially present, the amount of material that has reacted at any time will be equal to (W0 — W). In a first-order reaction the rate is proportional to the amount of material. Therefore, the rate of reaction can be expressed as... 

This is the rate law that applies when the product layer is protective in nature. 

The kinetic models just described are only a few of those that have been found to represent reactions in solids. Moreover, it is sometimes observed that a reaction may follow one rate law in the early stages of the reaction, but a different rate law may apply in the later stages. Because many of the rate laws that apply to reactions in solids are quite different from those encountered in the study of reactions... 

Although we have shown several kinetic models for reacting solids, none specifically applies to a reaction between two solids. A rate law that was developed many years ago to model reacting powders is known as the Jander equation, and it is written as... 

In an extension of this study, the dissociation rates for the copper complexes of the corresponding 16- and 17-membered N5-macrocycles have been studied (Hay, Bembi, McLaren Moodie, 1984). Similar rate laws to that just given for the 15-membered ring complex also apply in these cases. It was found that there is an increase in rate as the ring size increases the respective kH constants being 0.049 dm6 mol-2 s 1 (15-membered), 4.85 dm6 mol 2 s 1 (16-membered) and 1.18 x 103 dm6... 

Do the kinetic rate constants and rate laws apply well to the system being studied Using kinetic rate laws to describe the dissolution and precipitation rates of minerals adds an element of realism to a geochemical model but can be a source of substantial error. Much of the difficulty arises because a measured rate constant reflects the dominant reaction mechanism in the experiment from which the constant was derived, even though an entirely different mechanism may dominate the reaction in nature (see Chapter 16). 

In geochemical kinetics, the rates at which reactions proceed are given (in units such as mol s-1 or mol yr 1) by rate laws, as discussed in the next section. Kinetic theory can be applied to study reactions among the species in solution. 

In this chapter we consider the problem of the kinetics of the heterogeneous reactions by which minerals dissolve and precipitate. This topic has received a considerable amount of attention in geochemistry, primarily because of the slow rates at which many minerals react and the resulting tendency of waters, especially at low temperature, to be out of equilibrium with the minerals they contact. We first discuss how rate laws for heterogeneous reactions can be integrated into reaction models and then calculate some simple kinetic reaction paths. In Chapter 26, we explore a number of examples in which we apply heterogeneous kinetics to problems of geochemical interest. 

Different reaction mechanisms can predominate in fluids of differing composition, since species in solution can serve to promote or inhibit the reaction mechanism. For this reason, there may be a number of valid rate laws that describe the reaction of a single mineral (e.g., Brady and Walther, 1989). It is not uncommon to find that one rate law applies under acidic conditions, another at neutral pH, and a third under alkaline conditions. We may discover, furthermore, that a rate law measured for reaction with deionized water fails to describe how a mineral reacts with electrolyte solutions. 

In this chapter we construct a variety of kinetic reaction paths to explore how this class of model behaves. Our calculations in each case are based on kinetic rate laws determined by laboratory experiment. In considering the calculation results, therefore, it is important to keep in mind the uncertainties entailed in applying laboratory measurements to model reaction processes in nature, as discussed in detail in Section 16.2. 

According to Knauss and Wolery (1988), this rate law is valid for neutral to acidic solutions a distinct rate law applies in alkaline fluids, reflecting the dominance of a second reaction mechanism under conditions of high pH. 

Jin, Q. and C. M. Bethke, 2003, A new rate law describing microbial respiration. Applied and Environmental Microbiology 69,2340-2348. 

In subsequent chapters, we may have to consider forms other than this straightforward power-law form the effects of T and composition may not be separable, and, for complex systems, two or more rate laws are simultaneously involved. Nevertheless, the same general approaches described here apply.)... 

The simple theories of reaction rates involve applying basic physical chemistry knowledge to calculate or estimate the rates of successful molecular encounters. In Section 6.3 we present important results from physical chemistry for this purpose in subsequent sections, we show how they are used to build rate theories, construct rate laws, and estimate the values of rate constants for elementary reactions. 

We eliminate cNC>3 (not allowed in the final rate law) by applying the stationary-state hypothesis to NO, rNOj = 0 (and subsequently to NO) ... 

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