Pre-equilibrium condition

The assumption that the rate of consumption of the intermediate in the slow step is insignificant relative to its rates of formation and decomposition in the first step is called a pre-equilibrium condition. A pre-equilibrium arises when an intermediate is formed and sustained in a rapid formation reaction and its reverse. The calculation of the rate law is then much simpler. For instance, if we propose that the first step in the NO oxidation gives rise to a pre-equilibrium, we would write... 

Figure 7.8. A typical plot of the concentrations of A, B, and C ingredients for the pre-equilibrium conditions given in Example 7.6...

A second common approximation is the steady-state condition. That arises in the example if A is fast compared with in which case [7] remains very small at all times. If [J] is small then d[I /dt is likely to be approximately zero at all times, and this condition is commonly invoked as a mnemonic in deriving the differential rate equations. The necessary condition is actually somewhat weaker (9). For equations 22a and b, the steady-state approximation leads, despite its different origin, to the same simplification in the differential equations as the pre-equilibrium condition, namely, equations 24a and b. 

The pre-equilibrium condition describes situations in which reaction intermediates are formed and removed in steps prior to the rate-determining one. If the formation and removal of the intermediate in prior steps is rapid, an equilibrium concentration is established. If the intermediate appears in the rate-determining step, its relatively slow reaction in that step does not change its equilibrium concentration. The pre-equilibrium condition is sometimes called a fast equilibrium. 

This is essentially the classic case of the steady-state approximation and pre-equilibrium conditions for a two-step reaction with a reversible step. See Espenson, J. H. Chemical Kinetics and Reaction Mechanisms, 2nd ed. McGraw-Hill New York, 1995 pp 77-90. 

The pre-equilibrium condition describes situations in which reaction intermediates are formed and removed in steps prior to the rate-determining one. 

In this solvent the reaction is catalyzed by small amounts of trimethyl-amine and especially pyridine (cf. 9). The same effect occurs in the reaction of iV -methylaniline with 2-iV -methylanilino-4,6-dichloro-s-triazine. In benzene solution, the amine hydrochloride is so insoluble that the reaction could be followed by recovery. of the salt. However, this precluded study mider Bitter and Zollinger s conditions of catalysis by strong mineral acids in the sense of Banks (acid-base pre-equilibrium in solution). Instead, a new catalytic effect was revealed when the influence of organic acids was tested. This was assumed to depend on the bifunctional character of these catalysts, which act as both a proton donor and an acceptor in the transition state. In striking agreement with this conclusion, a-pyridone is very reactive and o-nitrophenol is not. Furthermore, since neither y-pyridone nor -nitrophenol are active, the structure of the catalyst must meet the conformational requirements for a cyclic transition state. Probably a concerted process involving structure 10 in the rate-determining step... 

They argued that pre-equilibria to form Cl+ or S02C1+ may be ruled out, since these equilibria would be reversed by an increase in the chloride ion concentration of the system whereas rates remained constant to at least 70 % conversion during which time a considerable increase in the chloride ion concentration (the byproduct of reaction) would have occurred. Likewise, a pre-equilibrium to form Cl2 may be ruled out since no change in rate resulted from addition of S02 (which would reverse the equilibrium if it is reversible). If this equilibrium is not reversible, then since chlorine reacts very rapidly with anisole under the reaction condition, kinetics zeroth-order in aromatic and first-order in sulphur chloride should result contrary to observation. The electrophile must, therefore, be Cli+. .. S02CI4- and the polar and non-homolytic character of the transition state is indicated by the data in Table 68 a cyclic structure (VII) for the transition state was considered as fairly probable. 

The pre-equilibrium and the steady-state approximations are two different approaches to deriving a rate law from a proposed mechanism, (a) For the following mechanism, determine the rate law by the steady-state approximation, (c) Under what conditions do the two methods give the same answer (d) What will the rate law become at high concentrations of Br ... 

Upon addition of a solution of sulfuric acid in D20 the reaction of A-acetoxy-A-alkoxyamides obeys pseudo-unimolecular kinetics consistent with a rapid reversible protonation of the substrate followed by a slow decomposition to acetic acid and products according to Scheme 5. Here k is the unimolecular or pseudo unimolecular rate constant and K the pre-equilibrium constant for protonation of 25c. Since under these conditions water (D20) was in a relatively small excess compared with dilute aqueous solutions, the rate expression could be represented by the following equation ... 

Case B is very common and can also be worked out easily. It is seen that the barriers for both the forward and backward reaction of (1) are much lower than the barrier for (2). We are dealing with a fast pre-equilibrium and a ratedetermining reaction (2) ki, k i k2 (concentrations omitted). It is also referred to as Curtin-Hammett conditions in U S. literature it refers to the kinetics of a system of a number of rapidly equilibrating species or conformations, each one of which might undeigo a different conversion, but all that counts is the global, lowest barrier, as that is the direction the system takes. 

Several variants of separation methods based on dialysis, ultrafiltration, and size exclusion chromatography have been developed that work under equilibrium conditions. Size exclusion chromatography especially has become the method of choice for binding measurements. The Hummel-Dreyer method, the vacancy peak method, and frontal analysis are variants that also apply to capillary electrophoresis. In comparison to chromatographic methods, capillary electrophoresis is faster, needs only minimal amounts of substances, and contains no stationary phase that may absorb parts of the equilibrium mixture or must be pre-equilibrated. 

This simple model predicts that the observed kinetics is determined by the rate of fragmentation only when the reverse process is much slower than counterdiffusion (i.e. when k f under activation control. On the other hand, for an endergonic fragmentation it is expected that k f k and /Cobs = The reaction now is described as a pre-equilibrium... 

Provided the proton transfer is rapid and reversible, the proton transfer process can be treated as a pre-equilibrium. At a constant [OH ] the system will still possess first-order kinetics even when the amount of substrate in the form of the conjugate base is large and the stationary state condition is not required. The rate constant kobs takes on the dependence. obs = k2Khy[On]/(l + Xhy[OH-])... 

Tables 8 and 9 and Figure 5 show that the rate data for hydrolysis of nitrone are almost identical to that for the hydrolysis of oxaziridine under all conditions of acidity at 24.2 °C. This evidence confirms that the salt of both oxaziridine and nitrone has the same kinetics on addition to water and forms products at a rate greater than that of unprotonated oxaziridine or nitrone. The decreasing rate at higher acidities is due to decreasing water activity in the acid media and is well explained by the Bunnett and Bunnett-Olsen criteria of the mechanism. The presented evidence57 is consistent with the mechanism outlined in Scheme 3, for example, a rapid protonation pre-equilibrium of nitrone (II) and oxaziridine (I) to form a common intermediate (HI) followed by slow nucleophilic attack by water and rapid decomposition to benzaldehyde and t-butylhydroxylamine.

By comparing these two treatments, find the conditions under which the pre-equilibrium would be set up. 

This corresponds to the steady state analysis (a), where most of the intermediate is removed by the reverse reaction and very little by reaction. It is only under these conditions that the pre-equilibrium is likely to be set up and maintained throughout reaction see Section 8.4. 

SN1 reactions, such as hydrolyses and substitution by anions of tertiary halogen-oalkanes, are good examples of reactions with a reversible first step. In these a carbonium ion is produced, which then reacts with water or anions. Chapter 6, Problem 6.5, illustrates the different rate expressions found after applying the steady state treatment, and after assuming a pre-equilibrium where the equilibrium lies very far to the left, i.e. where K is very small, and only a very, very small amount of R+ is present. It also looks at the conditions under which the amount of R+ present in a steady state could approximate to a pre-equilibrium. The discussion did not include the situation where K is not small. 

Under these conditions the steady state and the pre-equilibrium state are indistinguishable. However, the conditions under which they become indistinguishable are different in each, i.e. 

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