Rate laws types

Kinetics There have been few comprehensive studies of the kinetics of selective oxidation reactions (31,32). Kinetic expressions are usually of the power-rate law type and are applicable within limited experimental ranges. Often at high temperature the rate expression is nearly first order in the hydrocarbon reactant, close to zero order in oxygen, and of low positive order in water vapor. Many times a Mars-van Krevelen redox type of mechanism is assumed to operate. 

Newsome3 analyzed the suitability of several rate equations for the HTS reaction on ferrochrome catalysts. The information analyzed pointed to a power rate law-type equation given in Equations 6.19 and 6.20,... 

The rate law type refers to the exponents treated as variables and allowed to vary in the data analysis procedure to analyze the (tr,T) data. 

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 observed rate law depends on the type of catalyst used with promoted iron catalysts a rather complex dependence on nitrogen, hydrogen, and ammonia pressures is observed, and it has been difficult to obtain any definitive form from experimental data (although note Eq. XVIII-20). A useful alternative approach... 

It is Langmuir-Hinshelwood in type, and the usually observed rate law is... 

Neither (A3.4.15) nor (A3.4.17) is of the fonn (A3,4,10) and thus neither reaction order nor a unique rate codficient can be defined. Indeed, the number of possible rate laws that are not of the fonu of (A3.4.10) greatly exceeds those cases following (A3.4.10). However, certain particularly simple reactions necessarily follow a law of type of (A3.4.10). They are particularly important from a mechanistic point of view and are discussed in the next section. 

In contrast to the bimoleciilar recombination of polyatomic radicals ( equation (A3.4.34)1 there is no long-lived intennediate AB smce there are no extra intramolecular vibrational degrees of freedom to accommodate the excess energy. Therefore, the fonnation of the bond and the deactivation tlirough collision with the inert collision partner M have to occur simultaneously (within 10-100 fs). The rate law for trimoleciilar recombination reactions of the type in equation (A3.4.47) is given by... 

Dilatant fluids (also known as shear thickening fluids) show an increase in viscosity with an increase in shear rate. Such an increase in viscosity may, or may not, be accompanied by a measurable change in the volume of the fluid (Metzener and Whitlock, 1958). Power law-type rheologicaJ equations with n > 1 are usually used to model this type of fluids. 

The observed rate law for this type of reaction is usually first order in each reactant. Extensive theoretical treatments have been perfomred, most notably by R. A. Marcus and N. S. Hush, details of which can be found in more specialized sources ... 

The second type of behaviour (Fig. 1.89) is much closer to that which one might predict from the regular cracking of successive oxide layers, i.e. the rate decreases to a constant value. Often the oxide-metal volume ratio (Table 1.27) is much greater than unity, and oxidation occurs by oxygen transport in the continuous oxide in some examples the data can be fitted by the paralinear rate law, which is considered later. Destructive oxidation of this type is shown by many metals such as molybdenum, tungsten and tantalum which would otherwise have excellent properties for use at high temperatures. 

Acid catalysis. Consider the reverse of the scheme written in Eqs.(10-37)-(10-38). Derive the rate law for the reverse reaction, and discuss different limiting forms as to the type of acid catalysis demonstrated. 

To construct an overall rate law from a mechanism, write the rate law for each of the elementary reactions that have been proposed then combine them into an overall rate law. First, it is important to realize that the chemical equation for an elementary reaction is different from the balanced chemical equation for the overall reaction. The overall chemical equation gives the overall stoichiometry of the reaction, but tells us nothing about how the reaction occurs and so we must find the rate law experimentally. In contrast, an elementary step shows explicitly which particles and how many of each we propose come together in that step of the reaction. Because the elementary reaction shows how the reaction occurs, the rate of that step depends on the concentrations of those particles. Therefore, we can write the rate law for an elementary reaction (but not for the overall reaction) from its chemical equation, with each exponent in the rate law being the same as the number of particles of a given type participating in the reaction, as summarized in Table 13.3. 

In Eq. (1.5) the surface coverage is given by 9c, and 9c is related to parameter X of Eq. (1.7). Equation (1.5) can be rewritten to show explicitly its dependence on gas-phase concentration. Equation (1.17a) gives the result. This expression can be related to practical kinetic expressions by writing it as a power law as is done in Eq. (1.18b). Power-law-type rate expressions present the rate of a reaction as a function of the reaction order. In Eq. (1.17b) the reaction order is m in H2 and —n in CO. 

Figure 22.1 A. Schema for a physiologically based pharmacokinetic model incorporating absorption in the stomach and intestines and distribntion to various tissues. B. Each organ or tissue type includes representation of perfusion (Q) and drug concentrations entering and leaving the tissue. Fluxes are computed by the product of an appropriate rate law, and permeable surface area accounts for the affinity (e.g., lipophilic drugs absorbing more readily into adipose tissue). Clearance is computed for each tissue based on physiology and is often assumed to be zero for tissues other than the gut, the liver, and the kidneys.

Table 15-1 summarizes the parameters for the experimental kinetics of the three simplest types of rate laws. 

Horne has studied the kinetics of exchange in aqueous perchlorate media at temperatures down to —78 °C by the isotopic method ( Fe) and dipyridyl separation. The same rate law in these ice media as in aqueous solution was observed, although the acid dependence was small. Horne concluded that the same exchange mechanism occurs in solid and liquid solvent. Evidence for a Grotthus-type mechanism has been summarised. ... 

These oxidations have attracted wide interest and both specialised and comparative studies have been published. The rate laws which are summarised in Table 14 are not distinctive although the acidity dependence of the V(V) oxidations of some substrates suggests by analogy that a pinacol-type of oxidation may occur cf. V(V)-pinacol complexes, p. 388), viz. 

However both groups agree on the stoichiometries, which are all of the type 2 Cr(n) 1 reductant molecule, and on the rate laws, which are generally... 

In order to test rate laws, a must be determined as a function of time using an appropriate experimental technique. If the reaction involves the loss of a volatile product as shown in Eq. (8.1), the extent of reaction can be followed by determining the mass loss either continuously or from sample weight at specific times. Other techniques are applicable to different types of reactions. After a has been determined at several reaction times, it is often instructive to make a graph of a versus time before the data are analyzed according to the rate laws. As will be shown later, one can often eliminate some rate laws from consideration because of the general shape of the a versus t curve. 

Reactions in which a gas or liquid reacts with the surface of a solid are rather common processes in inorganic chemistry. The product that forms as a layer on the surface of the solid may impede the other reactant from contacting the solid. There are several types of behavior that depend on how the product layer affects the mobility of reactants, but in this instance, we will assume that the rate is inversely proportional to the thickness of the product layer. When the rate law is written in terms of the thickness of the product layer, x, the result is... 

In another type of reaction, the penetration of the mobile reactant varies as 1/x3, which gives rise to a so-called cubic rate law of the form... 

As will be described later, a common and important type of reaction that involves the oxidation of metals during corrosion processes sometimes follows a rate law of this form. 

As shown by Eq. (8.15), the reaction is a "two-thirds" order, but that does not involve the concept of molecularity. Since the surface area is a maximum at the beginning of the reaction, the rate is maximum at that time and decreases thereafter. A rate law of this type is known as a deceleratory rate law. As will be shown later, there are several rate laws that show this characteristic. 

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