The Encounter Rate Criterion

As in the case of hydrogen exchange discussed in chapter 2 (Section l.F), it is possible to apply the Smoluchowski equation [Eq. (2.1)] to calculate the encounter rate. For nitration of heteroaromatic substrates, Ridd took a fixed value of 6 for the ratio of the ions (63JCS4204), thus simplifying the expression to Eq. (3.17). Since values of ii at various temperatures are well known, the calculation of A(enc.) at various acidities 

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Any attempt to calculate partial rate factors for the i-oxides being The encounter rate criterion ( 8.2.3) is ambiguous for these compounds. 

Pyridone is one of the compounds for which the encounter rate criterion for choosing between nitration via the free base or by the conjugate acid is ambiguous ( 8.2.3). 

The encounter rate criterion is applicable in those cases where the concentration of a free base in a strongly acid solution is so low that even if every molecular collision resulted in reaction, the calculated reaction rate would be lower than that measured. This limiting rate, the encounter rate, is given by Eq. (2.1), where is the viscosity of the medium, k is Boltzmann s constant, N is Avogadro s number, and rA and rB are the radii of the ions. 

The above mechanistic criteria almost invariably allow a firm conclusion to be made concerning the species reacting in heteroaromatic nitration. Model compound studies appear to provide the best evidence, and the rate profile approach can be very successful, whereas activation parameters and the encounter rate criterion give, by themselves, less useful information. However, use of a single criterion can often give equivocal evidence, and a firm conclusion frequently requires several criteria used together. 

However, the theory of Section II in this chapter gives the exact criterion that diffusion theory for the encounter rate is valid when a in Eq. (2.33) is large than unity. For potentials without barriers the factor IF(0)/IF(oo) is close to unity, and the criterion for applicability of the diffusion theory is that the mean free path Dj/C8kT/nfi) be much smaller than the particle size (1 + b)/4-However, for large barriers the criterion is similar to that of Verwey and Overbeek, except that SR is the half-width of the potential barrier at an energy of only kT/2 below the maximum. This is much less than the full width of the barrier, and for reasonable values of the parameters in aqueous solution it can be shown that DLVO theory breaks down for equal-size particles with R, > 150 nm. However, when hydrodynamic interaction between particles is introduced in the formula for in terms of and it is concluded ... 

Recently kinetic data have become available for the nitration in sulphuric acid of some of these hydroxy compounds (table 10.3). For 4-hydroxyquinoline and 4-methoxyquinoline the results verify the early conclusions regarding the nature of the substrate being nitrated in sulphuric acid. Plots of log Q against — (Lf + logioflHao) fo " these compounds and for i-methyl-4-quinolone have slopes of i-o, i-o and 0-97 at 25 C respectively, in accord with nitration via the majority species ( 8.2) which is in each case the corresponding cation of the type (iv). At a given acidity the similarity of the observed second-order rate constants for the nitrations of the quinolones and 4-methoxy-quinoline at 25 °C supports the view that similarly constructed cations are involved. Application of the encounter criterion eliminates the possibilities of a... 

The flow of treated brine depends on the capacity of the plant and on the concentrations of brine into and out of the cells. It depends less strongly on other operating conditions. As a rule of thumb from our design examples, we can estimate the flow of membrane-plant treated brine in m hr as 0.5 times the plant capacity in tons of chlorine per day. Using the more frequently encountered rating in terms of tpd NaOH, we estimate the diameter required to meet the criterion for weir overflow rate as... 

In the previous chapter, we discussed liquid polymer systems. These exist in specific states selected by the laws of equilibrium thermodynamics. The rules that control structure formation during crystallization are different and this is an important point Structure formation is here governed by kinet-ical criteria rather than by equilibrium thermodynamics. What does this mean Indeed, here we encounter a new criterion The structure that develops at a given temperature is that with the maximum rate of development rather than that with the lowest free energy. As a consequence, treatment of... 

The derivation of a similarly atypical rate expression is required for the simulation of the electrochemical behavior encountered in electrohydrodimerization studies. In these studies, the variation of the bulk concentration of the olefin (e.g., ethyl cinnamate, diethyl furmarate) reveals that there is a concentration dependence to the reaction order associated with the dimerization of the electrogenerated radical ion [33]. This variation in apparent reaction order with concentration can only be attributed to a two-step mechanism [25] involving two independent rate or equilibrium processes. A mechanism that meets this criterion and appears to fit the electrochemical data is the preequilibrium mechanism [36] in which the electrogenerated radical ions first engage in an equilibrium dimerization before the rate-determining ring closure of the dimer takes place. Symbolically, this mechanism may be written ... 

Certain equilibrium states of thermodynamic systems are stable to small fluctuations others are not. For example, the equilibrium state of a simple gas is stable to all fluctuations, as are most of the equilibrium states we will be concerned with. It is possible, however, to carefully prepare a subcooled liquid, that is, a liquid below its normal solidiflcation temperature, that satisfies the equilibrium criteria. This is an tin-.stable equilibrium. state because the slightest disturbance, such as tapping on the. side of the containing ve.s.sel, will cause the liquid to freeze. One sometimes encounters mixtures that, by the chemical reaction equilibrium criterion (see Chapter 13). should react however, the chemical reaction rate is so small as to be immeasurable at the temperature of interest. Such a mixture can achieve a state of thermal equilibrium that is stable with respect to small fluctuations of temperature and pressure. If, however, there is a sufficiently large, but temporary, increase in temperature. so that die rate of the chemical reaction is appreciable for some period of time) and then the system... 

There are a standard series of Noise Rating (NR) Curves (Figure 20.4) which are stylised forms of the loudness response curves. These NR curves are often used as a criterion for noise control, and as such are internationally accepted. Other criteria may also be encountered. 

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