Leffler parameters

The (La3+( OCH3))2 and 9 Zn2+( OCH3) promoted methanolyses of the phos-phonates (22)17e and phosphonothioates (23)17g generally follow the same sort of trends as the phosphates and phosphorothioates discussed above so they need not be discussed in great detail here. Analysis of the linear Bronsted plots for the phospho-nates 22 gives the relationships shown in Equations (27)-(29) which shows the common trend that the j8lg observed for the metal-catalyzed reactions are greater than that of the methoxide reactions. Since the Leffler parameter, a, for the La3+-catalyzed... 

Reactions of a wide range of substituted phenyl acetates with six a-effect nucleophiles have revealed little or no difference, compared with phenolate nucleophiles, in the values of the Leffler parameters. As a result, the case for a special electronic explanation of the a-effect is considered unproven. Studies of the kinetics and mechanism of the aminolysis and alkaline hydrolysis of a series of 4-substituted (21) and 6-substituted naphthyl acetates (22) have revealed that, for electron-withdrawing substituents, aminolysis for both series proceeds through an unassisted nucleophilic substitution pathway. 

IE. Leffler, Parameters for the Description of Transition States, Science, 1953,117, 340. 

The Leffler parameter a has been applied to the effect of point mutations of amino acid residues on the binding step of tyrosyl- RNA synthetase, the enzyme responsible for catalysing formation of tyrosyl- RNA during protein biosynthesis. The logarithm of the rate constant for the reaction is linearly dependent on the logarithm of the equilibrium constant (Equation 26) for a range of directed point mutations of amino acid residues close to the proposed active site. ... 

Extensive theoretical analyses of the compensatory enthalpy-entropy relationship were first carried out by Leffler and later by Leffler and Grunwald, Exner, and Li. The empirical linear relationship between the thermodynamic or activation parameters AH and AS) directly leads to Eq. 11, where the proportional coefficient p, or the slope of the straight line in Figure 9, has a dimension of temperature. Merging Eq. 11 into the differential form of the Gibbs-Helmholtz Eq. 12 gives Eq. 13 ... 

The relationship between rates and equilibria was first proposed by Bronsted and Pedersen (1924) and subsequently treated by Bell (1936) and Evans and Polanyi (1938) it is shown in (108). Some 30 years ago Leffler (1953) provided (108) with a molecular basis by suggesting that the parameter a reflects... 

Figure 7. Mittag-Leffler model for the rebinding of CO to Mb, after a photodissociation. Data from Austin et al. [108]. The temperature dependence of To follows the Arrhenius law shown in the insert, with the parameters xm = 3.4 x 10 10 s and E, = 1470 K.

Reactivity-selectivity relationships are obtainable in carbene chemistry provided an independent selectivity parameter is found. Under such circumstances the relative rate data then serve as a measure of relative reactivity. This has been done with a number of Hammett studies. Thus the addition of dichlorocarbene to compounds [ 1 ] — [3] (listed in Table 15) indicates that a reactivity-selectivity relationship is obtained using p as a measure of selectivity. A more reactant-like transition state is obtained for the more reactive substrate in accordance with the Leffler- Hammond postulate. 

Equation (43), if strictly valid, implies that rate constants of reactions of the given series depend only on one instead of two parameters. (The strong resemblance of this problem to that of the Hammett equation was considered by Leffler and Grunwald, 1963, and by Ritchie and Sager, 1964.) In a general case, the correlation may be given by... 

We derive the extrathermodynamic relationships as a basis for the definition and understanding of the parameters for QSAR. This derivation is similar to that of Grunwald and Leffler Cl). 

It is convenient to use Fox functions (generalized Mellin-Barnes integras) when solving equations with fractional derivatives because Laplace and Fourier transformations for Fox functions may be expressed via Fox functions with given parameters. The connection of Mittag-Leffler functions with Fox functions is as follows [216,217] ... 

Therefore, in the present context, the Kramers escape rate can be best understood as playing the role of a decay parameter in the Mittag-Leffler functions governing the highly nonexponential relaxation behavior of the system. 

Equation (156) illustrates how anomalous relaxation influences the complex susceptibility arising from the slowest relaxation mode in the noninertial limit. Therefore, in anomalous relaxation the Kramers escape rate can be best understood as playing the role of a decay parameter in the Mittag-Leffler function governing the relaxation behavior of the system. We remark that the slowest decay mode, the parameters of which are determined by Eq. (156), will... 

The parameter is the bond length, y, at any point in the reaction map divided by the bond length in the product (rj,(product)) ( Jrc is the value of ry in the reaction complex. The terms r, (r Jrc nd q, for the bond fission component, x, are similarly defined. Since t can be estimated from Leffler s a parameters (see later) it provides a useful experimental index of tightness in a reaction where substantial change is suffered by two bonds. 

Thus in the Hammett format Peq = Pf - Pr and in the normalised (Leffler) format Ugq = a, - a,. The parameter Ugq is unity and Uf and should both therefore be numerically less than 1 because of the negative sign of a,.. 

Parameters from J.E. Leffler and E. Grunwald, John Wiley, New York, 1963. T.W. Bentley and G. Llewellyn, Progr. Phys. Org. Chem., 1990,17, 121. 

Leffler [10] had proposed that the slope of an empirical linear correlation between log k and log K (i.e. of AG4 vs. ACj j,) could be identified as a parameter measur-... 

Figure 2.4 Effect of increasing reactivity ko of the carbonyl compound on the parameter 0 (redrawn from Leffler and Grunwald, 1963).

For the solution of the fractional-order differential equations, most effective and easy analytic methods were developed based on the formula of the Laplace transform method of the Mittag-Leffler function in two parameters for further details see Refs. [19,38]. 

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