Encounter frequency

A minor component, if truly minute, can be discounted as the reactive form. To continue with this example, were KCrQ very, very small, then the bimolecular rate constant would need to be impossibly large to compensate. The maximum rate constant of a bimolecular reaction is limited by the encounter frequency of the solutes. In water at 298 K, the limit is 1010 L mol-1 s"1, the diffusion-controlled limit. This value is derived in Section 9.2. For our immediate purposes, we note that one can discount any proposed bimolecular step with a rate constant that would exceed the diffusion-controlled limit. 

If the rate of a reaction is governed by the encounter frequency, it is said to be diffusion-controlled. This frequency imposes an upper limit on the rate of reaction that can be evaluated by the use of Fick s laws of diffusion. The mathematical expression of this phenomenon was first presented by von Smoluchowski.2 We shall adopt a simple approach,3,4 although more rigorous derivations have been given.5... 

In solution, the triplet biradical 14b dimerizes, and the dimeric products are formed with strong chemically induced nuclear polarization. The absolute rate of the dimerization at 146 K, as monitored in viscous solution by ESR spectroscopy, is just about that predicted by the spin-corrected encounter frequency under those conditions. The cycloaddition of the triplet with a typical alkene, acrylonitrile, also can be followed in this way. 

Accordingly the increase in excimer/molecular fluorescence yield ratio with temperature in this region (Fig. 9) reflects the corresponding increase in encounter frequency dm[M] of excited and unexcited molecules if t/([M]rP)/ dT 0, and the temperature coefficient is related to the activation energy for viscous flow of the solvent Ed. 

In systems where quenching is much smaller than that predicted by diffusion-controlled encounter frequencies, the reason for inefficiency may be that either a heat of activation or an entropy of activation is necessary. The dependence of Ksv on solvent viscosity then disappears. For example, bromobenzene is a weak quencher for fluorescence of aromatic hydrocarbons, the quenching constant being nearly the same in hexane as in viscous paraffins. 

The maximum value for the bimolecular rate constant occurs when the activation energy act is zero and the steric factor is 1. The rate is then said to be diffusion-controlled, and it is equal to the encounter frequency of the molecules. Assuming that the reacting molecules are uncharged spheres of radius rA and rB, the encounter frequency may be calculated as... 

Calculations suggest that the diffusion-controlled encounter frequency of an enzyme and a substrate should be about 109 s-1 M l. The observed values in Table 4.4 tend to fall in the range of 106 to 108 s 1 M l. The faster ones are close to... 

Electrostatic forces 179-181, 325-327 Electrostatic strain and stability 74 Enamines 76, 77 Enantiomer 245 Encounter complex 159 Encounter frequency 164,166 Enediols 251, 252... 

Synzymes may achieve rate enhancements via binding and proximity effects. Such effects can occur when two reactive partners are bound within the same nanospace, thus increasing their relative encounter frequencies (i.e., the effective concentrations) (Fig. 13.1a) [4]. Some synzymes participate as reagents, and these catalysts feature structurally distinct substrate-binding site(s), together with a cat-alytically effective site(s) (Fig. 13.1b). 

These systems are imperfect models for molecular recognition, since both the potential functions and energy distributions that describe the interactions in MESA are different from those at the molecular level. Moreover, the encounter frequencies between objects in MESA (10 3—10-2 s-1) are much smaller than those between molecules (102-103 s, for micromolar concentrations). Despite these differences, our model manages to exhibit the salient characteristics of molecular recognition assembly depends on the shapes and interfacial properties of the faces that recognize one another. 

Reactions in Solution Molecular motion in liquids is diffusional in place of free flight but the concept of activation energy and stearic requirements survive. Molecules have to jostle their way through the solvent and so the encounter frequency is drastically less than in a gas. Since a molecule migrates only slowly into the region of a possible reaction partner, it also migrates only slowly away from it. 

The frequency with which A molecules in a solution will encounter B molecules is this frequency multiplied by ns, the mole fraction of B. For very dilute solutions ub = Nb/Nb, the ratio of the molecular densities of B to S molecules. But l/Ns, the volume per solvent molecule, can be written as t ab, with y determined by the packing factor for the lattice. By substituting these relationships in Eq, (XV.2.2) we can write for the encounter frequency of A and B in such a lattice s... 

A summary is made of the general classes of phenomena which can influence the reactivity of functional groups at heterogeneous interfaces, and potential pitfalls are pointed out in the reliance upon molecular analogy. Experimental results are reviewed pertaining to the thermodynamic and kinetic encounter frequencies of reagents on crosslinked polystyrenes and the chemisorption of olefins on oxide-free surfaces of elemental carbon. 

Reaction Crosslink Density Functionalization (mmole/gm) Encounter Frequency (sec"1)... 

Any model which is to be consistent with these results must include a broad spectrum of encounter frequencies for reaction partners at different relative locations in the network. The free energy of activation for these motions are related to the same factors previously discussed which govern the equilibrium properties. For the 4% crosslinked material there appears to be some fraction of substituents which are mutually inaccessible on any practical time scale. The majority of reactive pairs appear to be governed by encounter frequencies on the order of 10 sec l, more than ten orders of magnitude slower than for the molecular analogs in solution. 

In some instances, hydrous metal oxide surfaces may simply concentrate reactants in a small space, leading to increased encounter frequency. Adsorbed species diffuse in two dimensions along the surface, while dissolved species diffuse in three dimensions. For comparable diffusion coefficients, encounter frequencies between reactants are increased when the dimensions of the system are reduced (Hardt, 1979 Adam and Delbruck, 1968),... 

Transport is an integral component of all reaction systems. In well-mixed homogeneous solutions, the concentrations of all reactants and products are the same throughout the system, and there is no net movement of chemicals in space. The role of mass transport becomes evident only when chemical reactions are extremely fast. Diffusion determines the encounter frequency of reacting molecules and sets an upward limit on overall rates of reaction. (For example, for a diffusion-controlled bimolecular reaction in water the reaction rate constant is on the order of 1010 to 1011 M 1 s"1.) Mass transport plays a pronounced role in surface chemical reactions, since net movement of reactants (from solution to the surface) and products (from the surface to solution) often takes place. 

The rates of important processes in macromolecular solutions are often influenced or controlled by the binary diffusional encounter frequency of reactants. 

The frequency with which two reactive species encounter one another in solution represents an upper bound on the bimolecular reaction rate. When this encounter frequency is rate limiting, the reaction is said to be diffusion controlled. Diffusion controlled reactions play an important role in a number of areas of chemistry, including nucleation, polymer and colloid growth, ionic and free radical reactions, DNA recognition and binding, and enzyme catalysis. 

Encounter frequencies between solute molecules are lower than collision frequencies in the gas phase, because the translational motion is hindered by the bulk of solvent molecules. Conversely, once reactants have encountered each other in solution, they... 

The change in the rate constants with temperature for the reactions of ribonuclease (RNase) with hydrated electrons and OH radicals was measured. The RNase molecule unfolds reversibly at elevated temperatures exposing sites particularly reactive towards hydrated electrons. The theoretical treatment leads to an estimate of the encounter frequencies for differently shaped macromolecules with small radiolytically produced solvent radicals. The derived encounter frequencies are compared with experimentally determined rate constants. Values as high as 1013 M"1 secr1 are understandable. 

T n recent experiments (3) we have found that the value of rate constants for reactions between e m and certain protein molecules such as ribonuclease (RNase) depends on the conformation of the molecule. For ribonuclease it was found that the rate constant increases upon unfolding of the molecule. This increase can partly be ascribed to exposure of the hidden disulfide bridges but could also partly be caused by an increase in encounter frequency of the hydrated electron with the unfolded molecule. 

The expression derived by Debye for the encounter frequency in reactions between spherical reactants can not be applied to molecules that differ markedly from spherical shape, such as the rodshaped collagen. In this paper the theory applied by Debye has been extended to reactions between a small spherical molecule and a cylindrical macromolecule. 

An explanation of these results will be attempted by first dealing with the problem of encounter frequency of reactants in general and then discussing the special case of RNase. 

Here v is the encounter frequency, Ri and R2 are the radii of the two different reacting particles in cm., Di and D2 are the diffusion constants for the two different molecules in cm.2 sec.-1 and N is Avogadro s number. This expression cannot be applied to macromolecules which are not spherical in shape. 

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