Rate expression Determination

The obvious challenge in the interpretation of the data is to find a suitable explanation for the independence of the third term of the rate law, Eq. (102), on the concentrations of HSO3 and 02. The rate expression determined experimentally could be modeled quantitatively by combining the following propagation steps with the uncatalyzed reaction mechanism ... 

Given a feed stream of known composition and a set of chemical reactions with known rate expressions, determine I, the set of all possible steady-state species concentrations... 

The preceding discussion may shed some light on why chain reactions occur, but the thoughtful reader will doubtless have asked by now how one determines which reactions are involved in a given chain. The answer is not simple, since reproduction of the form of a rate expression determined experimentally cannot... 

What rate expression results from this mechanism Is this expression consistent with the rate expression determined experimentally ... 

If the data are consistent with the indicated rate expression, determine values for k,., and the equilibrium constant for the reaction. 

One apparent problem with this argument relating to kf and is that it is only easily derived for a single-step reaction. In practice, many reactions have complex rate expressions (determined by experiment) where reaction orders are not related to reaction stoichiometry. Mechanisms may not be known, and where they are, they are often multi-step. Nevertheless, even in these cases, when the system is at equilibrium, we have a distinctive situation in which all the individual elementary steps of a reaction mechanism must be in equilibrium too (see Chapter 16 for the use of the idea of elementary steps in the context of reaction mechanisms). As a result, each elementary step of the reaction sequence can be treated as an equilibrium in itself Equilibrium constants for each step can then be expressed in terms of the forward and reverse rate constants found for each elementary step. Using the relationship we discussed in Chapter 7 for finding the overall equilibrium constant of a sequence of connected equilibria, it can be demonstrated that it is valid to evaluate the expression for K(, directly from the overall stoichiometric equation, even though the reaction may take place in several steps. 

Low temperatures strongly favor the formation of nitrogen dioxide. Below 150°C equiUbrium is almost totally in favor of NO2 formation. This is a slow reaction, but the rate constant for NO2 formation rapidly increases with reductions in temperature. Process temperatures are typically low enough to neglect the reverse reaction and determine changes in NO partial pressure by the rate expression (40—42) (eq. 13). The rate of reaction, and therefore the... 

Kinetic data provide information only about the rate-determining step and steps preceding it. In the hypothetical reaction under consideration, the final step follows the rate-determining step, and because its rate will not affect the rate of the overall reaction, will not appear in the overall rate expression. The rate of the overall reaction is governed by the second step, which is the bottleneck in the process. The rate of this step is equal to A2 multiplied by the molar concentration of intermediate C, which may not be directly measurable. It is therefore necessary to express the rate in terms of the concentrations of reactants. In the case under consideration, this can be done by recognizing that [C] is related to [A] and [B] by an equilibrium constant ... 

Because proton-transfer reactions between oxygen atoms are usually very fast, step 3 can be assumed to be a rapid equilibrium. With the above mechanism assume4 let us examine the rate expression which would result, depending upon which of the steps is rate-determining. 

These examples illustrate the relationship between kinetic results and the determination of reaction mechanism. Kinetic results can exclude from consideration all mechanisms that require a rate law different from the observed one. It is often true, however, that related mechanisms give rise to identical predicted rate expressions. In this case, the mechanisms are kinetically equivalent, and a choice between them is not possible on the basis of kinetic data. A further limitation on the information that kinetic studies provide should also be recognized. Although the data can give the composition of the activated complex for the rate-determining step and preceding steps, it provides no information about the structure of the intermediate. Sometimes the structure can be inferred from related chemical experience, but it is never established by kinetic data alone. 

The order of reactivity of the hydrogen halides is HI > HBr > HCl, and reactions of simple alkenes with HCl are quite slow. The studies that have been applied to determining mechanistic details of hydrogen halide addition to alkenes have focused on the kinetics and stereochemistry of the reaction and on the effect of added nucleophiles. The kinetic studies often reveal complex rate expressions which demonstrate that more than one process contributes to the overall reaction rate. For addition of hydrogen bromide or Itydrogen... 

TWo types of rate expressions have been found to describe the kinetics of most aromatic nitration reactions. With relatively unreactive substrates, second-order kinetics, first-order in the nitrating reagent and first-order in the aromatic, are observed. This second-order relationship corresponds to rate-limiting attack of the electrophile on the aromatic reactant. With more reactive aromatics, this step can be faster than formation of the active electrq)hile. When formation of the active electrophile is the rate-determining step, the concentration of the aromatic reactant no longer appears in the observed rate expression. Under these conditions, different aromatic substrates undergo nitration at the same rate, corresponding to the rate of formation of the active electrophile. 

Molecular bromine is believed to be the reactive brominating agent in uncatalyzed brominations. The brominations of benzene and toluene are first-order in both bromine and the aromatic substrate in trifluoroacetic acid solution, but the rate expressions become more complicated when these reactions take place in the presence of water. " The bromination of benzene in aqueous acetic acid exhibits a first-order dependence on bromine concentration when bromide ion is present. The observed rate is dependent on bromide ion concentration, decreasing with increasing bromide ion concentration. The detailed kinetics are consistent with a rate-determining formation of the n-complex when bromide ion concentration is low, but with a shift to reversible formation of the n-complex... 

This relationship permits the measurement of the ratio k /ky. The initial concentrations [B—X]o and [B—Y]o are known from the conditions of the experiment The reaction can be stopped at some point when some of both B—X and B—Y remain unreacted, or an excess of B—X and B—Y can be used so that neither is completely consumed when A—A has completely reacted. Determination of [B—X], and [B—Y], then provides the information needed to calculate k /ky. It is clear that the reactions being compared must be of the same order. If they were not, division of the two rate expressions would leave uncanceled concentration terms. 

The slope of a concentration-time curve to define the rate expression can be determined. However, experimental studies have shown the reaction cannot be described by simple kinetics, but by the relationship ... 

Here, we shall examine a series of processes from the viewpoint of their kinetics and develop model reactions for the appropriate rate equations. The equations are used to andve at an expression that relates measurable parameters of the reactions to constants and to concentration terms. The rate constant or other parameters can then be determined by graphical or numerical solutions from this relationship. If the kinetics of a process are found to fit closely with the model equation that is derived, then the model can be used as a basis for the description of the process. Kinetics is concerned about the quantities of the reactants and the products and their rates of change. Since reactants disappear in reactions, their rate expressions are given a... 

Table 3-11 gives the initial rate data [-d(B2Hg)/dt] reported for the gas phase reaetion of diborane and aeetone at 114°C BjHg -i-dMejCO —> 2(Me2CHO)2BH. If a rate expression is of the form Rate = PmcjCO determine n, m, and k. 

Reaction Rate Expression 191 Determine the order of the reaetion and the rate eonstant. Solution... 

Sinee step 2 is the rate-determining step, tlien the rate expression is ... 

Adesina has shown that it is superfluous to carry out the inversion required by Equation 5-255 at every iteration of the tri-diagonal matrix J. The vector y"is readily computed from simple operations between the tri-diagonal elements of the Jacobian matrix and the vector. The methodology can be employed for any reaction kinetics. The only requirement is that the rate expression be twice differentiable with respect to the conversion. The following reviews a second order reaction and determines the intermediate conversions for a series of CFSTRs. 

Consider the solution of Equation 6-170 for eaeh of the four types of rate expressions to determine the optimum temperature progression at any given fraetional eonversion X. ... 

A simpler phenomenological form of Eq. 13 or 12 is useful. This may be approached by using Eq. 4 or its equivalent, Eq. 9, with the rate constants determined for Na+ transport. Solving for the AG using Eqn. (3) and taking AG to equal AHf, that is the AS = 0, the temperature dependence of ix can be calculated as shown in Fig. 16A. In spite of the complex series of barriers and states of the channel, a plot of log ix vs the inverse temperature (°K) is linear. Accordingly, the series of barriers can be expressed as a simple rate process with a mean enthalpy of activation AH even though the transport requires ten rate constants to describe it mechanistically. This... 

Rate expressions have been determined by experiment for a large number of reactions. For the process... 

As we have seen, rate expressions for reactions must be determined experimentally. Once this has been done, it is possible to derive a plausible mechanism compatible with the observed rate expression. This, however, is a rather complex process and we will not attempt it here. Instead, we will consider the reverse process, which is much more straightforward. Given a mechanism for a several-step reaction, how can you deduce the rate expression corresponding to that mechanism ... 

Sometimes the rate expression obtained by the process just described involves a reactive intermediate, that is, a species produced in one step of the mechanism and consumed in a later step. Ordinarily, concentrations of such species are too small to be determined experimentally. Hence they must be eliminated from the rate expression if it is to be compared with experiment. The final rate expression usually includes only those species that appear in the balanced equation for the reaction. Sometimes, the concentration of a catalyst is included, but never that of a reactive intermediate. 

Strategy Write the rate expression for the rate-determining second step. This will involve the unstable intermediate, O. To get rid of [O], use the feet that reactants and products are in equilibrium in step 1, so forward and reverse reactions occur at the same rate. 

Determine the rate expression (reaction order) from—... 

Rate constant The proportionality constant in the rate equation for a reaction, 288 Rate-determining step The slowest step in a multistep mechanism, 308 Rate expression A mathematical relationship describing the dependence of reaction rate upon the concentra-tion(s) of reactant(s), 288,308-309 Rayleigh, Lord, 190... 

Before discussing such theories, it is appropriate to refer to features of the reaction rate coefficient, k. As pointed out in Sect. 3, this may be a compound term containing contributions from both nucleation and growth processes. Furthermore, alternative definitions may be possible, illustrated, for example, by reference to the power law a1/n = kt or a = k tn so that k = A exp(-E/RT) or k = n nAn exp(—nE/RT). Measured magnitudes of A and E will depend, therefore, on the form of rate expression used to find k. However, provided k values are expressed in the same units, the magnitude of the measured value of E is relatively insensitive to the particular rate expression used to determine those rate coefficients. In the integral forms of equations listed in Table 5, units are all (time) 1. Alternative definitions of the type... 

The appropriate rate expression may be substituted in these relations and comparison with observations allows E and n to be determined. 

Since the extent of radical occlusion varies from one precipitation polymerization to the next, it is nearly impossible to develop a generalized polymerization rate equation. As a result, rate expressions are most often determined from experi-... 

However, the mechanisms by which the initiation and propagation reactions occur are far more complex. Dimeric association of polystyryllithium is reported by Morton, al. ( ) and it is generally accepted that the reactions are first order with respect to monomer concentration. Unfortunately, the existence of associated complexes of initiator and polystyryllithium as well as possible cross association between the two species have negated the determination of the exact polymerization mechanisms (, 10, 11, 12, 13). It is this high degree of complexity which necessitates the use of empirical rate equations. One such empirical rate expression for the auto-catalytic initiation reaction for the anionic polymerization of styrene in benzene solvent as reported by Tanlak (14) is given by ... 

Similar expressions can be written for third-order reactions. A reaction whose rate is proportional to [A] and to [B] is said to be first order in A and in B, second order overall. A reaction rate can be measured in terms of any reactant or product, but the rates so determined are not necessarily the same. For example, if the stoichiometry of a reaction is2A-)-B—>C- -D then, on a molar basis, A must disappear twice as fast as B, so that —d[A]/dt and -d[B]/dr are not equal but the former is twice as large as the latter. 

There is a large amount of evidence for the Sn2 mechanism. First, there is the kinetic evidence. Since both the nucleophile and the substrate are involved in the rate-determining step (the only step, in this case), the reaction should be first order in each component, second order overall, and satisfy the rate expression... 

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