Rate laws single

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... 

A variation on the use of pseudo-ordered reactions is the initial rate method. In this approach to determining a reaction s rate law, a series of experiments is conducted in which the concentration of those species expected to affect the reaction s rate are changed one at a time. The initial rate of the reaction is determined for each set of conditions. Comparing the reaction s initial rate for two experiments in which the concentration of only a single species has been changed allows the reaction order for that species to be determined. The application of this method is outlined in the following example. 

The rate of a process is expressed by the derivative of a concentration (square brackets) with respect to time, d[ ]/dt. If the concentration of a reaction product is used, this quantity is positive if a reactant is used, it is negative and a minus sign must be included. Also, each derivative d[ ]/dt should be divided by the coefficient of that component in the chemical equation which describes the reaction so that a single rate is described, whichever component in the reaction is used to monitor it. A rate law describes the rate of a reaction as the product of a constant k, called the rate constant, and various concentrations, each raised to specific powers. The power of an individual concentration term in a rate law is called the order with respect to that component, and the sum of the exponents of all concentration terms gives the overall order of the reaction. Thus in the rate law Rate = k[X] [Y], the reaction is first order in X, second order in Y, and third order overall. 

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. 

When a reaction proceeds in a single elementary step, its rate law will mirror its stoichiometry. An example is the rate law for O3 reacting with NO. Experiments show that this reaction is first order in each of the starting materials and second order overall NO + 03- NO2 + O2 Experimental rate = i [N0][03 J This rate law is fully consistent with the molecular view of the mechanism shown in Figure 15-7. If the concentration of either O3 or NO is doubled, the number of collisions between starting material molecules doubles too, and so does the rate of reaction. If the concentrations of both starting materials are doubled, the collision rate and the reaction rate increase by a factor of four. 

Isolating isoprene as the only reactant whose concentration changes simplifies the general rate law to a form that can be tested against the single concentration rate laws ... 

Moser and Gratzel reported that the absorption signals of Ij could be seen immediately after a laser flash on an I -containing a-Fe203 sol. The quantum yield was 80%. In a subsequent reaction, 1 disappeared according to a second order rate law 2 Ij 21 - - I2. The authors pointed out that single crystals and polycrystalline... 

The above general, idealised rate law is compatible with at least three different potential pathways for nitration one-step, concerted pathway [1] that involves a single transition state (5),... 

Despite the authority apparent in its name, no single rate law describes how quickly a mineral precipitates or dissolves. The mass action equation, which describes the equilibrium point of a mineral s dissolution reaction, is independent of reaction mechanism. A rate law, on the other hand, reflects our idea of how a reaction proceeds on a molecular scale. Rate laws, in fact, quantify the slowest or rate-limiting step in a hypothesized reaction mechanism. 

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 the simplest cases of reactive transport, a species sorbs according to a linear isotherm (Chapter 9), or reacts kinetically by a zero-order or first-order rate law. There is a single reacting species, and only one reaction is considered. In these cases, the governing equation (Eqn. 21.1 or 21.2) can be solved analytically or numerically, using methods parallel to those established to solve the groundwater transport problem, as described in the previous chapter (Chapter 20). 

The oxidation rate should, in principle, be described by a law using a rate constant independent of pH, as long as a single reaction mechanism is involved. The rate law (28.4) is unusual in that the rate varies with the concentration of the Mn11 component, rather than an individual species. If we hypothesize that the catalytic activity is promoted by a surface complex >MnOMnOH, a slightly different form of the rate law may be appropriate. Since the surface complex would... 

In this case, we find we can match the experimental results using a single value for the rate constant, but an initial catalyst mass that depends on pH, as shown in Figure 28.3. Here, we take the rate constant to be 1013 molal-4 s-1, and use initial catalyst masses of 6 mg at pH 9.5, 5 mg at pH 9.3, and 0.6 mg at pH 9. This form of the rate law seems more satisfactory than the previous form, since we might reasonably expect the amount of catalyst, perhaps an MnC03 colloid, initially present to be greatest in the experiments conducted at highest pH, but the rate constant to be invariant. 

The primary use of chemical kinetics in CRE is the development of a rate law (for a simple system), or a set of rate laws (for a kinetics scheme in a complex system). This requires experimental measurement of rate of reaction and its dependence on concentration, temperature, etc. In this chapter, we focus on experimental methods themselves, including various strategies for obtaining appropriate data by means of both batch and flow reactors, and on methods to determine values of rate parameters. (For the most part, we defer to Chapter 4 the use of experimental data to obtain values of parameters in particular forms of rate laws.) We restrict attention to single-phase, simple systems, and the dependence of rate on concentration and temperature. It is useful at this stage, however, to consider some features of a rate law and introduce some terminology to illustrate the experimental methods. 

In this chapter, we describe how experimental rate data, obtained as described in Chapter 3, can be developed into a quantitative rate law for a simple, single-phase system. We first recapitulate the form of the rate law, and, as in Chapter 3, we consider only the effects of concentration and temperature we assume that these effects are separable into reaction order and Arrhenius parameters. We point out the choice of units for concentration in gas-phase reactions and some consequences of this choice for the Arrhenius parameters. We then proceed, mainly by examples, to illustrate various reaction orders and compare the consequences of the use of different types of reactors. Finally, we illustrate the determination of Arrhenius parameters for die effect of temperature on rate. 

This chapter provides an introduction to several types of homogeneous (single-phase) reaction mechanisms and the rate laws which result from them. The concept of a reaction mechanism as a sequence of elementary processes involving both analytically detectable species (normal reactants and products) and transient reactive intermediates is introduced in Section 6.1.2. In constructing the rate laws, we use the fact that the elementary steps which make up the mechanism have individual rate laws predicted by the simple theories discussed in Chapter 6. The resulting rate law for an overall reaction often differs significantly from the type discussed in Chapters 3 and 4. 

The presence (or absence) of pore-diffusion resistance in catalyst particles can be readily determined by evaluation of the Thiele modulus and subsequently the effectiveness factor, if the intrinsic kinetics of the surface reaction are known. When the intrinsic rate law is not known completely, so that the Thiele modulus cannot be calculated, there are two methods available. One method is based upon measurement of the rate for differing particle sizes and does not require any knowledge of the kinetics. The other method requires only a single measurement of rate for a particle size of interest, but requires knowledge of the order of reaction. We describe these in turn. 

To develop a kinetics model (i.e., a rate law) for the reaction represented in 9.1-1, we focus on a single particle, initially all substance B, reacting with (an unlimited amount... 

The single particle acts as a batch reactor in which conditions change with respect to time, This unsteady-state behavior for a reacting particle differs from the steady-state behavior of a catalyst particle in heterogeneous catalysis (Chapter 8). The treatment of it leads to the development of an integrated rate law in which, say, the fraction of B converted, /B, is a function oft, or the inverse. 

A reaction which follows power-law kinetics generally leads to a single, unique steady state, provided that there are no temperature effects upon the system. However, for certain reactions, such as gas-phase reactions involving competition for surface active sites on a catalyst, or for some enzyme reactions, the design equations may indicate several potential steady-state operating conditions. A reaction for which the rate law includes concentrations in both the numerator and denominator may lead to multiple steady states. The following example (Lynch, 1986) illustrates the multiple steady states... 

As for a single-stage CSTR, the volume of each stage is obtained from the material balance around that stage, together with the rate law for the system. Thus, for the ith stage, the material-balance equation (14.3-5) becomes... 

The liquid-phase reaction A - B + C takes place in a single-stage CSTR The rate law for... 

Our treatment of chemical kinetics in Chapters 2-10 is such that no previous knowledge on the part of the student is assumed. Following the introduction of simple reactor models, mass-balance equations and interpretation of rate of reaction in Chapter 2, and measurement of rate in Chapter 3, we consider the development of rate laws for single-phase simple systems in Chapter 4, and for complex systems in Chapter 5. This is... 

Simple integrated rate laws for single reactants allow us to express the rate of reaction as a function of time. These are summarised in Table 10.2. 

The single most important factor that determines the rate of a reaction is concentration — primarily the concentrations of the reactants, but sometimes of other species that may not even appear in the reaction equation. The relation between the rate of a reaction and the concentration of chemical species is termed the rate law it is the cornerstone of reaction mechanisms. The rate law alone allows much insight into the mechanism. This is usually supplemented by an examination of other factors which can also be revealing. (For these, see Chap. 2)... 

The kineticist should always strive to get as complete (and accurate ) a rate law as conditions will allow. The mechanism that is suggested to account for the rate law is, however, a product of the imagination, and since it may be one of several plausible mechanisms, it might very well turn out to be incorrect. Indeed, it is impossible to prove any single mechanism but so much favorable data may be amassed for a mechanism that one can be fairly certain of its validity. 

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