Eigen curvature

Bronsted and Pedersen [20] indicated that the rate constant for proton transfer from acid to a base cannot continue to increase in accord with a linear Bronsted law but must be limited by an encounter rate. This prediction was confirmed by Eigen s school [21] who showed that changed from 1 to zero as the p/f of the donor acid fell below that of the acceptor base (Fig. 5). Eigen [21] considered the following scheme (sometimes called the Eigen mechanism) for proton transfer from HX to Y where reactions in brackets occur in the encounter complex (Eqn. 28). The overall rate constants are given in Eqns. 29 and 30. 

Diffusion control occurs if the reactions inside the encounter complex are faster than the rate of encounter complex formation or decomposition (i.e. Ac b- or 

Acb a Acb b)- Under the latter conditions Eqns. 29 and 30 collapse to Eqns. 31 and 32  

The meaning of at this stage might appear to have little relevance this is not so - the unit value for proton transfer between heteroatoms at A K 0 indicates that the transition state of the rate-limiting step is close to products as indeed it might be for an endothermic reaction. At ApK 0, = 0 and the transition state would be predicted to be close to reactants in agreement with fact for this exothermic reaction. 

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Table 2 includes data for proton transfer between heteroatoms low values of AG and w for these systems are consistent with classical Eigen curvature. As a general rule it should be noted that intrinsically fast reactions (low AG ) will exhibit sharp curvature. 

B14 Cell B14 contains the value of the pressure-curvature eigenvalue Ar, with the name eigen contained in A14. The value or the eigenvalue is entered iteratively until the nondimensional axial inlet velocity uj = — 1. In this spreadsheet the analyst is expected to watch the value of inlet velocity as the iteration proceeds and adjust the eigenvalue accordingly. 

The value of e is a measure of the anisotropy of p (r) at p. A direction has been assigned to e, namely the direction of the soft curvature given by the eigen vector associated with a,. This direction is called the major axis of 280. It is normally indicated by a double-headed arrow. 

Eigen (1964) found that a plot of ApR against the rate constant for proton transfer between acetylacetone and a series of bases gave a curved plot. It should be noted, however, that Eigen s explanation for curvature is quite different from the one based on Marcus theory and the reactivity-selectivity principle. The curvature discussed by Eigen is attributed to a change from a rate-determining proton transfer to a diffusion controlled reaction which is independent of the catalyst p. 

It has been common practice to equate the value of )3 with the degree of proton transfer in the transition state /3 values close to 0 are taken to be indicative of reactant-like transition states and those close to 1 of product-like transition states. Any value outside these limits is inconsistent with this practice. Early investigators were only able to follow reactions within a limited rate constant range. With the development of fast reaction techniques (Eigen, 1964 Caldin, 1964) the predicted (Br nsted and Pedersen, 1923) curvature of the plots was fully established (cf. Bell and Lidwell, 1940). Pronounced curvature is in fact seen for fast proton transfers in DMSO (see p. 156). 

Another approach used to interpret curvature of Br0nsted plots has been given by Murdoch (1972). This model, which incorporates Marcus theory, shows that the diffusive steps (10a, c) of the three-stage Eigen mechanism can also influence curvature. It is shown mathematically that increased difficulty of diffusion has the same... 

Fig. 36. General acid-catalysed aminolysis of isocyanic acid [121] Eigen type curvature consistent with diffusion limiting proton transfer. CA, chloroacetic acid Dabco, l,4-diazabicyclo-(2,2,2)-octane AC, acetic add AN, anilinium ion PM, iV-propargylmorpholinium ion CEM, 2-chloroethylmorpholinium ion, MeM, iV-methylmorpholinium ion EG, ethyl glycinate BOR, boric acid MBA, methyl j8-alaninate ET, ethylammonium ion Q, quinuclidinium ion PIP, piperidinium ion ACET, acetamidinium ion Gu, guanidium ion.

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