Rate laws multiple reactions

Presto, a third-order rate law This multiplication should not be taken as representing a chemical event or as carrying such implications it is only a valid mathematical manipulation. Other similar transformations can be given,2 as when one multiplies by another factor of unity derived from the acid ionization equilibrium of HOC1. (The reader may show that this gives a second-order rate law.) These considerations illustrate that it is the rate law and not the reaction itself that has associated with it a unique order. 

How can one formulate a mechanism for a chain reaction when the rate law does not provide the composition of the transition state The process is an inexact one, but this guidance may be helpful. Factor the rate law mentally into components that suggest the multiplication of one reagent by one intermediate. Thus, for Eqs. (8-20)-(8-23), we might rewrite Eq. (8-24) to read... 

An unusual feature of a CSTR is the possibility of multiple stationary states for a reaction with certain nonlinear kinetics (rate law) in operation at a specified T, or for an exothermic reaction which produces a difference in temperature between the inlet and outlet of the reactor, including adiabatic operation. We treat these in turn in the next two sections. 

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... 

These solutions to the one-dimensional advection-diffusion model can be used to estimate reaction rate constants Ck) from the pore-water concentrations of S, if and s are known. More sophisticated approaches have been used to define the reaction rate term as the sum of multiple removals and additions whose functionalities are not necessarily first-order. Information on the reaction kinetics is empirically obtained by determining which algorithmic representation of the rate law best fits the vertical depth concentration data. The best-fit rate law can then be used to provide some insight into potential... 

Kinetics in polycrystals differ from those in solution phase, because in the former, the thermal reactions usually follow a nonexponential rate law, something that is attributed to a multiple-site problem. In contrast to a first-order reaction in solution, the rate constant of a nonexponential process in the solid state is time dependent molecules located in the reactive site will have decayed during the warmup procedure and/or the initial stage of the reaction at the given temperature. These considerations need to be taken into account when the decay of the intensity of the IR signals in a matrix at low temperature are used for kinetic measurements [70]. 

The reaction order approach for the LSV response to simple reaction mechanisms, eqns. (49)—(51), has already been described. These equations are applied directly to experimental data and rate laws are derived before a mechanism or a theoretical model is considered. Since RA and RB are separable during LSV analysis, the changes in RB as a function of CA can be observed directly from djf p/d log v [72], When RB is changing with changes in CA, this slope will not be linear over large intervals but will appear to be linear over small intervals of v. For the reaction order analysis, CA was defined as the concentration when RB is half-way between the limiting values, usually 1 and 2, i.e. 1.5. In terms of n, multiples of CA, there are again three distinct cases which must be satisfied by f(n). They are n = 1 (,RB = 1.5), n < 1 (RB = 1) and n> 1 (RB = 2). These requirements are satisfied by eqn. (65) and illustrates how RB varies with n. 

In the presence of multiple states, the right-hand-side term consists of sums, products, and nesting of elementary functions such as logy, expy, and trigonometric functions, called the S -system formalism [602]. Using it as a canonical form, special numerical methods were developed to integrate such systems [603]. The simple example of the diffusion-limited or dimensionally restricted homodimeric reaction was presented in Section 2.5.3. Equation 2.23 is the traditional rate law with concentration squared and time-varying time constant k (t), whereas (2.22) is the power law (c7 (t)) in the state differential equation with constant rate. 

As we shall see later in the book, there ate some instances in which it is timch more convenient to work in teims of the number of moles (Aa.Ab) or molar flow rates (F, Fg, etc.) rather than conveision. Membrane reactors and gas-phase multiple reactions are two such cases where molar flow rates rather than conversion are preferred. Consequently, the concentrations in the rate laws need to be expressed in terms of (he molar flow rates. We start by recalling and combining Equations (3-40) and (3-41) ... 

In analyzing the multiple reactions in Table 6-2, we cany out the procedure shown in Table 6-3 (not necessarily in exact order) when the rate Law is known for at least one species in each of the individual reactions. 

We can easily extend the concepts described in the preceding. section to polymerization reactions. In this section we show how the rate laws are formulated so that one can use the techniques developed in Chapter 6 for multiple reactions to determine the molecular weight distribution and other properties. In the material that follows we focus on free-radical polymerization. 

At this point one could use the techniques developed in Chapter 6 on multiple reactions to follow polymerization process. However, by using the PSSH, we can manipulate the rate law into a form that allows closed-form solutions for a number of polymerization reactions. 

Use bifurcation theory (Section 8.6.5 on the CD-ROM) to determine the possible regions for multiple steady states for the gas reaction with the rate law... 

Example 6-4 Stoichiometry and Rate Laws for Multiple Reactions... 

Some prefer to write the surface reaction rate in terms cf the fraction of the surface of sites covered (i.e.,/ ) rather than the number of sites C. s covered, the difference being the multiplication factor of the total site concentration, C, In any event, the final form of the rate law is the same because C, k, and so on, are all lumped... 

A The first step will be rate-limiting. It will determine the rate for the entire reaction because it is slower than the other steps. This step is a unlmolecular process with the rate given by answer A. Choice C would be correct if the reaction as a whole were one elementary step instead of three, but the stoichiometry of a reaction composed of multiple elementary steps cannot be used to predict a rate law. 

Many of the basic kinetic techniques of physical organic chemistry lose their usefulness when applied to radical reactions. Because of quantum-mechanical restrictions on spin multiplicity, reaction of a radical with a closed-shell molecule or ion generates another radical, with the radical species themselves being present in low concentration. For this reason, radical reactions are often chain reactions, whose kinetics are dominated by initiation or termination steps, with the rates of reactions generating most product frequently having little influence on the overall rate law. 

In summary, for many reactions involving multiple steps and pathways, the powers in the rate laws surprisingly agree with the stoichiometric coefficients. Consequently, to facilitate describing this class of reactions, we say a reaction/o//o>v.t an elementary rate law when the reaction orders are identical with the stoichiometric coefficients of the reacting species for the reaction as written. It is important to remember that the rate laws are determined by experimental observation They are a function of the reaction chemistry and not the type of reactor in which the reactions occur. Table 3-1 gives examples of rate laws for a number of reactions. 

Example 6-5 Stoichiometry aaj Rate Laws for Multiple Reactions Consider the following set of reactioas ... 

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