Rate laws for complex reactions

Rate equations become more complicated when an overall reaction process occurs in a sequence of steps. Let us analyze this possibility by considering a hypothetical two-step sequence. In the first step, A changes to B, and in the second, B changes to C. 

Problem Find an expression for the concentration of reactants as a function of time for a bimolecular reaction VJ A + VgB V(C. 

Solution The differential rate expression for a second order reaction of A with B is 

At any instant of time, the amount of A consumed relative to some initial concentration is [A] - [A]o, and likewise for B, the change is [B] - [B]q. The reaction stoichiometry requires that these changes divided by the stoichiometry numbers are equal  

With this relation, we substitute for [B] in the product fc[A][B] in the differential rate expression and obtain an expression involving only the concentration variable [A]. We can also substitute for [A] and obtain an expression for [B]. 

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Note that the kinetics of complicated reactions can be studied by observing the behavior of one reactant at a time. If the concentration of one reactant is much smaller than the concentrations of the others, then the amounts of those reactants present in large concentrations will not change significantly and can be regarded as constant. The change in concentration with time of the reactant present in a relatively small amount can then be used to determine the order of the reaction in that component. This technique allows us to determine rate laws for complex reactions. 

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 mechanistic assignment of terms in empirical rate laws for complexation and redox reactions of metal ions in aqueous solution acid dependences in perchlorate media. G. Davies, Coord. Chem. 

Several interesting observations have been made on this reaction. First, the rate of isomerization was found to be the same as the rate of dehydration. All attempts to dehydrate the starting complex by conventional techniques were found to lead to isomerization. On the basis of this and other evidence, the mechanism proposed involves the aquation in the complex followed by anation. In this process, water first displaces Cl- in the coordination sphere and then is displaced by the Cl-, possibly by an SN1 mechanism. A trigonal bipyramid transition state could account for the Cl- reentering the coordination sphere to give an cis product. The rate law for this reaction is of the form... 

The kinetics of the general enzyme-catalyzed reaction (equation 10.1-1) may be simple or complex, depending upon the enzyme and substrate concentrations, the presence/absence of inhibitors and/or cofactors, and upon temperature, shear, ionic strength, and pH. The simplest form of the rate law for enzyme reactions was proposed by Henri (1902), and a mechanism was proposed by Michaelis and Menten (1913), which was later extended by Briggs and Haldane (1925). The mechanism is usually referred to as the Michaelis-Menten mechanism or model. It is a two-step mechanism, the first step being a rapid, reversible formation of an enzyme-substrate complex, ES, followed by a slow, rate-determining decomposition step to form the product and reproduce the enzyme ... 

Outer-sphere (OS) reaction rates and rate laws can be defined for solvolysis of a given complex. Complex formation is defined as the reverse reaction— that is, replacement of solvent (S) by another ligand (L ). Following the arguments of Tobe, in aqueous solution the general rate law for complex formation (eliminating charge for simplicity). 

In reality however, situations also exist where a more complex form of the rate expression has to be applied. Among the numerous possible types of kinetic expressions two important cases will be discussed here in more detail, namely rate laws for reversible reactions and rate laws of the Langmuir-Hinshelwood type. Basically, the purpose of this is to point out additional effects concerning the dependence of the effectiveness factor upon the operating conditions which result from a more complex form of the rate expression. Moreover, without going too much into the details, it is intended at least to demonstrate to what extent the mathematical effort required for an analytical solution of the governing mass and enthalpy conservation equations is increased, and how much a clear presentation of the results is hindered whenever complex kinetic expressions are necessary. 

In this associative pathway, the rate law depends on the concentrations of both the complex ion and the entering ligand because both are present in the transition state. Therefore, the rate law for the reaction is... 

The earliest claims centered on complexes of the types [PdX(Et4-dien)F+ and [PtX(dien)]. The anomalous reactivity of certain nucleophiles, including hydroxide, and the observation of a two-term rate law for anation reactions (X = H2O) led to the postulate that when associative ligand replacement was very slow, then a dissociative reaction path could be seen to compete. Because of this, these sterically hindered molecules were often termed pseudooctahedral. This early evidence has been reviewed (6, 12), and it is now known that these apparent anomalies can be interpreted instead in terms of the usual associative pathways. For example, the original detection of a two-term rate law for the anation of [Pt(OH2)(dien)] is probably a consequence of examining the reaction as it approached equilibrium (142). 

The equation that expresses the mathematical relationship between the rate of a chemical reaction and the concentration of reactants is called a rate law. For example, the reaction B, which is a one-step reaction, has only one activated complex between reactants and products. The rate law for this reaction is expressed as... 

In summary, the rate law for a reaction relates reaction rate, the rate constant k, and the concentration of the reactants. Although the equation for a reaction conveys a great deal of information, it is important to remember that the actual rate law and order of a complex reaction can be determined only by experiment. 

Consider the reaction shown in which I- replaces the trans Br in complex 3. The rate law for this reaction was determined to be... 

Closure. After completing this chapter the reader should be able to describe the different types of multiple reactions (series, parallel, complex. and independent) and to select a reaction system that maximizes the selectivity. The reader should be able to write down and use the algorithm for solving CRE problems with multiple reactions. The reader should also be able to point out the major differences in the CRE algorithm for the multiple reactions from that for the single reactions, and then discuss why care must be taken when writing the rate law and stoichiometric Steps to account for the rate laws for each reaction, the relative rates, and the net rates of reaction. 

To continue with arguments based on kinetics the rate law for the reaction 37 of Cr2 (aq) with V(H20)6 is equation (81). A simple interpretation of this form of the rate law is that one of the reaction partners undergoes proton dissociation, and in this case b would be identified with the dissociation constant. This interpretation of the rate law can be dismissed because the value of b is too large to answer even for Ad of the more acidic partner. The alternative general interpretation is that the reaction involves two activated complexes of different compositions, and though the order in which they appear in the reaction sequence is not specified by the rate law (an important point recognized by Haim, and dealt with him by him in detail ) this particular issue does not affect the validity of the conclusions which will be reached on the matter of whether an inner- or outer-sphere path operates. Each mechanism requires an intermediate to be formed which contains one V, one Cr, less one proton and which has a charge of 4+. The values of the specific rates apd specific rates ratios which follow from the experimental rate law are quite unrealistic if... 

The empirical rate law for the reaction of oxalate with soluble and polymeric Mn02 over the pH range of 4-6 is non-integer for both oxalate and Mn02 which indicates that there are several electron transfer reaction pathways possible. Oxalate forms a precursor complex with Mn02 Pi or to reaction as observed by almost constant Mn02 absorbance... 

The reduction of the chromium(in) complex [(HaO)5CrN3] + by nitrous acid has been studied, the rate law for the reaction ... 

The complex [CoC03(NH3)5]" will react in acid solution to generate CO2. One might expect the rate law for this reaction to have the form... 

The rate law for these reactions is usually first-order in the metal complex and XY concentrations. The reaction mechanism depends on the nature of XY and can be broken into three generally recognized categories, as shown in Scheme 5.17. 

The rate law for the reaction given below has been determined in aqueous 0.10 M HCIO4 with Fe in excess. The rate is first-order in the Cr-benzyl complex concentration and independent of the Fe concentration. Suggest a mechanism consistent with this rate law. (Hint see 22iang and Jordan. )... 

In contrast with these findings for cobalt(u) complexes of octahedral configuration, the interaction of Oa with a cobalt(n) histidine complex has been investigated in strongly basic media. In solutions of IM-NaOH, the spectrum of the cobalt(n) species is consistent with that of a tetrahedral complex with four nitrogen donors. The rate law for the reaction is... 

Many years ago it was demonstrated that the rate law for the reaction of [Fe(5N02phen)3], and of analogous sulfonato-substituted complexes, with peroxodisulfate took the form shown in equation (17) ... 

Substitution. The kinetics of reactions of /ra/t -CPdClgCR SR ) ] complexes with a series of amines of various basicities and steric requirements have been studied in 1,2-dimethoxyethane at 298 K. In fact the first step is fast, and it is the second step whose kinetics have been examined here. Amine basicity is an important factor logarithms of second-order rate constants correlate linearly with amine pKa values in the absence of steric hindrance. Steric factors have a large effect when the incoming amine e.g. 2-methylpyridine) or the leaving sulphide (e.g. phenyl isopropyl sulphide) is bulky. The kinetic pattern for the reaction of [PdCl2(cod)] with 2,2 -bipyridyl in aqueous methanol has been interpreted by the reaction sequence shown in Scheme 3. The mechanism is dissociative with respect to the palladium centre. The rate law for the reaction of the [PdBr(dien)]+ cation with inosine has already been discussed in the introductory section to this chapter. 

One type of reaction where a palladium(n)-platinum(n) reactivity comparison is not possible is that of substitution at [Pd(OHa)4] +. Indeed substitution reactions of this cation provide a uniquely interesting case of complex formation from a square-planar aquo-cation. The rate law for the reaction of [Pd(OH2)4] + with chloride or bromide ion is... 

Rate laws for the reactions of square-planar dithiolatopalladium(n) complexes with uni- or bi-dentate nucleophiles are usually second-order in form, with a first-order dependence on each reactant. Rates of reaction of these palladium(n) complexes are much less than had been anticipated by extrapolation from reaction rates of similar nickel(ii) complexes. This lower-than-expected reactivity may be ascribed to the lower stability of five-coordinate palladium(n) complexes, which will be intermediates or transition states in these reactions. ... 

Iron. The rate law for the reaction of the S-bonded thiocyanato-complex [Fe(CN)5(SCN)] with hydroxide ion suggests two parallel paths. The first is a D or 5 Nl(lim) path, with an [Fe(CN)6] intermediate discriminating between hydroxide ion and thiocyanate ion. The second path is thought to involve the intermediacy either of [Fe(CN)5(OH)(SCN)] or of [Fe(CN)6(NCS)] . An S Nlcb process is, of course, precluded here. The reactions of optically... 

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