The Rate Determining Step

A unique method for the study of the Ti ion coordination state and the composition of the active species is, in principle, the ESR spectroscopy. Unfortunately, as reported in Chapter 5.1, it has not been possible to identify with this method any species having the requirements of active centers. 

The material presented in this chapter is quite difficult to account for. It is known that Mg/Ti catalysts are able to produce isotactic polypropylene even without donor. Zambelli and Corradini have shown that it is not necessary for the donor and aluminum alkyl to make up part of the ligands which constitute the environment of the active center. 

Moreover, application of the above law to the formation rates of isotactic and atactic fractions showed that the overall rate equation is the result of two equations characterized by different values of kA (200 1 mol-1 for the isospecific centers and 40 1 mol 1 for the non-specific centers). Thus, the kinetic behavior of the polymerization was rationalized on the basis of a two-center polymerization model. Furthermore, based on an approximate estimate of the partition function of the transition state involving propagating chain and coordinated monomer, monomer insertion was proposed as the rate determining step. 

A comparison between the productivity trends of binary and ternary catalysts in the presence of an external donor proves to be quite interesting. While the atactic fraction productivity has a similar behavior for both catalysts, the binary catalyst shows a decrease of isotactic productivity with increased D/Al ratio the ternary catalyst, on the other hand, shows a maximum (see Fig. 40). This is still in agreement with the Burfield model. In fact, these results can be accounted for by assuming that the internal donor gives rise to the formation of a new type of isospecific center, characterized by kA and kD values different from those present in binary catalysts. The presence of a maximum in productivity is consistent with the formation of complexes between the aluminum alkyl and the donor and the resulting modification of the adsorption equilibrium, as suggested by Burfield1S6 . 

It is difficult to obtain reliable values for the various adsorption constants since the aluminum alkyl and the donor apparently also influence the catalyst deactivation with reaction time. Nevertheless, it has been found that qualitatively kA kA for binary catalysts. This result appears to be more reliable than that reported by Keii et al., viz. kA kA, since this would be in contrast with the greater Lewis acidity which should characterize atactic centers as compared to isotactic ones. This discrepancy is most likely due to the fact that Keii took ternary catalysts into consideration. In that case, the equilibria of complexing between aluminum alkyl, internal base, and active centers are superimposed on organoaluminum adsorption. 

We now suppose that the second step in the reaction we are considering is very fast, so that whenever an I molecule is formed, it decays rapidly into P. We can use the condition /ct, to write and k - k k. Equation 7.13 then 

However, we need to be alert to the possibihty that a rate-determining step may also stem from the low concentration of a crucial reactant or catalyst and need not correspond to the step with the highest activation barrier. A rate-determining 

The following reaction is one of the early steps of glycolysis Case study 43)  

Might the phosphorylation of F6P be rate determining under these 

Strategy Compare the value of the reaction quotient, Q (Section 4.2), with the equilibrium constant. If Q K, the reaction step is far from equiUbrium and it is so slow that it maybe rate deterrnining. 

Equation (3.48) represents the most general expression which emerges from the suggested mechanism it contains all the rate constants and makes no assumption about irreversibility. The drawback is that it requires us to know six rate constants. 

What would happen, however, if we were also told that k 2 = 0 and k2 is very small compared with kj, k, k, and fca We will use Eq. (3.47) for our argument since it presents the rate constants individually. 

What is the physical meaning of the conditions we have inserted First, by making k 2 = 0 we postulate that the step in Eq. (3.26) is in practice irreversible. Second, by making k2 much smaller than the remaining velocity constants we postulate that the step in Eq. (3.26) is much slower than the others. In fact the restrictions we have placed on kj and k 2 have characterized the step in Eq. (3.26) as slow and irreversible we have made it the rate-determining step. 

A rate-determining step is one which is essentially irreversible and has a small forward velocity constant compared to the velocity constants of the other steps. All the steps preceding a rate-determining step must be reversible with small equilibrium constants, otherwise the concentration of reactive intermediates will build up. The concentration of the latter formed by reactions prior to the slow step will have their equilibrium values the speed of the back reaction rather than the rate of the slow step maintains their concentrations at the required low level. 

Once a rate-determining step has been identified, the derivation of a rate law is relatively simple and does not require overt use of the SSA. Thus 

Consider the series reaction A— B— C. If the first step is very much slower than the second step, the rate of formation of C is controlled by the rate of the first step, which is called the rate-determining step (rds), or rate-limiting step, of the reaction. Similarly, if the second step is the slower one, the rate of production of C is controlled by the second step. The slower of these two steps is the bottleneck in the overall reaction. This flow analogy, in which the rate constants of the separate steps are analogous to the diameters of necks in a series of funnels, is widely used in illustration of the concept of the rds. 

Strictly speaking, the flow analogy is valid only for consecutive irreversible reactions, and it can be misleading if reverse reactions are significant. Even for irreversible reactions the rds concept has meaning only if one of the reactions is much slower than the others. For reversible reactions the free energy reaction coordinate diagram is a useful aid. In Fig. 5-10, for example, the intermediate 1 is unstable with respect to R and P, and its formation (the kf step) is the rds of the overall reaction. 

When the overall reaction includes more than two elementary steps, the situation may not be easy to analyze. The product of the nth step is the reactant of the (n -I- l)st step, but in order for these two states to be represented by the same free energy they must have the same composition this means that the stoichiometric composition must be constant throughout the entire series of reactions. Suppose that it has been possible to construct the free energy reaction coordinate. Murdoch gives this method for identifying the rds  

Divide the reaction into sections, the first section beginning with R and ending at the first intermediate (1,) more stable than R. The second section begins at the end of the first section and ends at the next intermediate more stable than 1 . Continue until P is reached. 

Calculate the energy difference between the transition state of highest energy in each section and the initial energy of the section. 

Usually, the proton transfer step in an acid—base catalyzed reaction is fast if it takes place between oxygen or nitrogen atoms. The reaction step may be slow and rate-determining if the proton is transferred either from or to a carbon atom. 

Two conditions must be met if the proton transfer step is to be slow and rate-determining (a) k must be smaller than the rate coefficient of a diffusion-controlled reaction, and (b) fe i must be smaller than ku. The first condition is fulfilled in most proton transfer processes to or from carbon. It may be met also in a proton transfer among oxygen, sulfur, or nitrogen atoms if either the acidity of the proton donor or the basicity of the acceptor is extremely low. The second condition, e.g. 1kn, can be fulfilled if the intermediate is sufficiently unstable with respect to decomposition toward the products. In several examples, ku and ft, are of the same order of magnitude, e.g. k [ ku, and the proton transfer step is partially rate-determining. For these cases, the theoretical rate equation must be derived with the aid of the stationary state method Vol. 2, pp. 352-354). 

The molecularity of a reaction is simply the number of different molecules (or atoms or ions) that are involved in the rate-determining step. It equals 1 for a unimolecular reaction, 2 for a bimolecular reaction, and 3 for a termolecular reaction 

In electrode kinetics, at least for reactions with not more than three to four consecutive steps for one act of the overall reaction, there is usually one step that—when the reaction is running in the steady state—determines the rate of the whole reaction. Obviously, if one is to design a catalyst, this is the reaction, the rate of which has to be increased. Let me illustrate this—as Henry Eyring used to do—with a homely model. 

If one wanted to make the overall reaction A — D go faster, one would have to speed up passage through the road building, that is, accelerate the r.d.s., B — C. 

I need hardly say that similar things apply to molecules in reactions. When we say we are going to try to speed up the r.d.s., we are talking about catalysis. 

There is still something to say about A B when there is a holdup for B C, the r.d.s. One can imagine an eager 80 mph driver approaching the man with the red flag who is going to slow him down to 10 mph while he or she trickles along the one narrow lane still open. 

Naturally, there are many happy drivers who do not know about the holdup between B and C and when they arrive at B and are slowed down to 10 mph, some of them, the majority, start going back to A to see if it is possible to find another way to D that escapes the holdup at B. 

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The treatment may be made more detailed by supposing that the rate-determining step is actually from species O in the OHP (at potential relative to the solution) to species R similarly located. The effect is to make fi dependent on the value of 2 and hence on any changes in the electrical double layer. This type of analysis has permitted some detailed interpretations to be made of kinetic schemes for electrode reactions and also connects that subject to the general one of this chapter. 

B. Reaction within the Adsorbed Film as the Rate-Determining Step... 

The above equations can apply when the rate-determining step is first order even though the complete reaction mechanism is complicated. Thus for the reac-... 

Some early observations on the catalytic oxidation of SO2 to SO3 on platinized asbestos catalysts led to the following observations (1) the rate was proportional to the SO2 pressure and was inversely proportional to the SO3 pressure (2) the apparent activation energy was 30 kcal/mol (3) the heats of adsorption for SO2, SO3, and O2 were 20, 25, and 30 kcal/mol, respectively. By using appropriate Langmuir equations, show that a possible explanation of the rate data is that there are two kinds of surfaces present, 5 and S2, and that the rate-determining step is... 

Fig. 1. The rate-determining step in the neutral hydrolysis of paramethoxy-phenyl dichloroacetate. In the reactant state (a) a water molecule is in proximity of the carbonyl carbon after concerted proton transfer to a second water molecule and electron redistribution, a tetrahedral intermediate (b) is formed.

Melander first sought for a kinetic isotope effect in aromatic nitration he nitrated tritiobenzene, and several other compounds, in mixed acid and found the tritium to be replaced at the same rate as protium (table 6.1). Whilst the result shows only that the hydrogen is not appreciably loosened in the transition state of the rate-determining step, it is most easily understood in terms of the S 2 mechanism with... 

A primary isotope effect /ch/ d of 6.4 (extrapolated for 35 C) is observed for the metalation and the methylation of 171b when the C-5 position is deuterated. This value is in excellent agreement with the primary isotope effect of 6.6 reported for the metalation of thiophene (392) and it confirms that the rate-determining step is the abstraction by the base of the acidic proton. 

With the potential energies shown on a common scale we see that the transition state for formation of (CH3)3C is the highest energy point on the diagram A reaction can proceed no faster than its slowest step which is referred to as the rate determining step In the reaction of tert butyl alcohol with hydrogen chloride formation of the... 

The rate determining step m the S l mechanism is dissociation of the alkyloxo mum ion to the carbocation... 

Secondary and tertiary alcohols react with hydrogen halides by a mech anism that involves formation of a carbocation intermediate m the rate determining step... 

Assuming that the rate determining step in the reaction of cyclohexanol with hydrogen bro mide to give cyclohexyl bromide is unimolecular write an equation for this step Use curved arrows to show the flow of electrons... 

Like the reaction of tert butyl alcohol with hydrogen chloride step 2 m which tert butyloxonium ion dissociates to (CH3)3C and water is rate determining Because the rate determining step is ummolecular the overall dehydration process is referred to as a ummolecular elimination and given the symbol El... 

Because the rate determining step involves two molecules—the alkyloxonium ion and water—the overall reaction is classified as a bimolecular elimination and given the sym bol E2... 

What are the implications of the effects of the various halide leaving groups Because the halogen with the weakest bond to carbon reacts fastest Ingold concluded that the carbon-halogen bond breaks m the rate determining step The weaker the carbon-halogen bond the easier it breaks... 

The alkyl halide m this case 2 bromo 2 methylbutane ionizes to a carbocation and a halide anion by a heterolytic cleavage of the carbon-halogen bond Like the dissoci ation of an aUcyloxonmm ion to a carbocation this step is rate determining Because the rate determining step is ummolecular—it involves only the alkyl halide and not the base—It is a type of El mechanism... 

Both steps m this general mechanism are based on precedent It is called elec trophilic addition because the reaction is triggered by the attack of an acid acting as an electrophile on the rr electrons of the double bond Using the two rr electrons to form a bond to an electrophile generates a carbocation as a reactive intermediate normally this IS the rate determining step... 

Hughes and Ingold interpreted second order kinetic behavior to mean that the rate determining step is bimolecular that is that both hydroxide ion and methyl bromide are involved at the transition state The symbol given to the detailed description of the mech anism that they developed is 8 2 standing for substitution nucleophilic bimolecular... 

As we have seen the nucleophile attacks the substrate m the rate determining step of the Sn2 mechanism it therefore follows that the rate of substitution may vary from nucleophile to nucleophile Just as some alkyl halides are more reactive than others some nucleophiles are more reactive than others Nucleophilic strength or nucleophilicity, is a measure of how fast a Lewis base displaces a leaving group from a suitable substrate By measuring the rate at which various Lewis bases react with methyl iodide m methanol a list of then nucleophihcities relative to methanol as the standard nucleophile has been compiled It is presented m Table 8 4... 

The Stvfl mechanism is an ionization mechanism The nucleophile does not participate until after the rate determining step has taken place Thus the effects of nucleophile and alkyl halide structure are expected to be different from those observed for reactions pro ceedmg by the 8 2 pathway How the structure of the alkyl halide affects the rate of Stvfl reactions is the topic of the next section... 

How does the rate determining step of the mechanism respond to this property of the solvent ... 

Step 1 Sulfur tnoxide attacks benzene m the rate determining step... 

Each act of proton abstraction from the a carbon converts a chiral molecule to an achi ral enol or enolate ion The sp hybridized carbon that is the chirality center m the start mg ketone becomes sp hybridized m the enol or enolate Careful kinetic studies have established that the rate of loss of optical activity of sec butyl phenyl ketone is equal to Its rate of hydrogen-deuterium exchange its rate of brommation and its rate of lodma tion In each case the rate determining step is conversion of the starting ketone to the enol or enolate anion... 

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