Solvent-cage effects equation

Equation 6 indicates that the solvent strength, 6, is pressure-dependent, providing a potential route to improved selectivity and rate by "pressure-tuning the solvent. A number of attempts to demonstrate reactivity control in su rcritical CO2 for Diels-Alder (75-77) and organic photoreactions (78,79) have exhibited very small effects. Andrew and coworkers have recently demonstrated dramatic solvent cage effects on selectivity of a photo-Fries reaction close to the critical density.(80) More polar SCF s have shown more promising results control of esterification rates and polyester molecular weight distribution via enzymatic catalysis in fluoroform has been demonstrated. (81,82)... 

The cage effect was also analyzed for the model of diffusion of two particles (radical pair) in viscous continuum using the diffusion equation [106], Due to initiator decomposition, two radicals R formed are separated by the distance r( at / = 0. The acceptor of free radicals Q is introduced into the solvent it reacts with radicals with the rate constant k i. Two radicals recombine with the rate constant kc when they come into contact at a distance 2rR, where rR is the radius of the radical R Solvent is treated as continuum with viscosity 17. The distribution of radical pairs (n) as a function of the distance x between them obeys the equation of diffusion ... 

However, it is known, that in homolytical processes certaine influence on reaction rate has also so-called "cage effect", which is described by density of medium cohesion energy. That was confirmed by generalization of data concerning to influence of solvents upon decomposition rate of benzoyl peroxide [2] or oxidizing processes [3, 4], That is why the data analysis from work [1] is seemed as expedient by means of five parameter equation ... 

A more detailed analysis of the cage effect has been presented by Northrup and Hynes [103]. As before, the diffusion equation was used to describe the relative motion of radicals, but the effect of solvent structures on the radicals when in close proximity to each other was incorporated via the potential of mean force (see Chap. 2, Sect. 6.6 and Chap. 8, Sect. 2.6) and hydrodynamic respulsion (see Chap. 8, Sect. 2.5 and Chap. 9. Sect. 3). The solvent structure is due to short-range liquid structure. Solvent... 

In Sect. 2.1, the timescale over which the diffusion equation is not strictly valid was discussed. When using the molecular pair analysis with an expression for h(f) derived from a diffusion equation analysis or random walk approach, the same reservations must be borne in mind. These difficulties with the diffusion equation have been commented upon by Naqvi et al. [38], though their comments are largely within the framework of a random walk analysis and tend to miss the importance the solvent cage and velocity relaxation effects. 

Non-activated double bonds, e.g. in the allylic disulfide 1 (Fig. 10.2) in which there are no substituents in conjugation with the double bond, require high initiator concentrations in order to achieve reasonable polymerisation rates. This indicates that competition between addition of initiator radicals (R = 2-cyanoisopropyl from AIBN) to the double bond of 1 and bimolecular side reactions (e.g. bimolecular initiator radical-initiator radical reactions outside the solvent cage with rate = 2A t[R ]2 where k, is the second-order rate constant) cannot be neglected. To quantify this effect, [R ] was evaluated using the quadratic Equation 10.5 describing the steady-state approximation for R (i.e. the balance between the radical production and reaction). In Equation 10.5, [M]0 is the initial monomer concentration, k is as in Equation 10.4 (and approximately equal to the value for the addition of the cyanoisopropyl radical to 1-butene) [3] and k, = 109 dm3 mol 1 s l / is assumed to be 0.5, which is typical for azo-initiators (Section 10.2). The value of 11, for the cyanoisopropyl radicals and 1 was estimated to be less than Rpr (Equation 10.3) by factors of 0.59, 0.79 and 0.96 at 50, 60 and 70°C, respectively, at the monomer and initiator concentrations used in benzene [5] ... 

It is in this region of time that the random walk or diffusion equation analysis is least adequate. The prime failure of the random walk analysis is to ignore the oscillatory motion of molecules. The encounter pair separates from encounter but gets reflected back towards each other by the solvent cage after a time about equal to the period of oscillations in a solvent cage. To incorporate this effect, Noyes [265] used the approximate form of h(t)... 

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