Quantitative Theory

Quantitative Theory. —The foregoing qualitative account raises a number of questions concerning the comparative rates of the various processes involved. For example, considering the low solubility of monomer in the aqueous serum, is diffusion of monomer to the polymer particle rapid enough to keep pace with the polymerization Is it reasonable to suppose that initiator radicals are efficiently captured by the polymer particles (or micelles when present) before other fates befall them What will be the average radical concentration in a polymer particle Smith and Ewart developed a theory which answers these questions and offers a basis for quantitative treatment of emulsion polymerization. The salient features of their theory are set forth below. 

Under conditions normally employed, the rate p of generation of radicals from the initiator is of the order of 10 per cc. per second, and the number of polymer particles is about 10 per cc. (10 to 10 per cc. would include nearly all cases). Hence if all of the initiator radi- 

The diffusion constant of a primary radical must be of the order of 10 cm.2 sec.- the radius r is about 5X10 cm., and as we have seen 1 10 per second. Hence ]ag l0 radicals per cc. But the radicals are being generated at a rate of 10 cc. sec. hence the average lifetime of a radical from generation to capture by a polymer particle will be only 10 sec. The rate of termination by reaction between two radicals in the aqueous phase at the calculated equilibrium concentration, 10 radicals per cc., will be given by 

The termination constants kt found previously (see Table XVII, p. 158) are of the order of 3 X10 1. mole sec. Conversion to the specific reaction rate constant expressed in units of cc. molecule sec. yields A f=5X10 . At the radical concentration calculated above, 10 per cc., the rate of termination should therefore be only 10 radicals cc. sec., which is many orders of magnitude less than the rate of generation of radicals. Hence termination in the aqueous phase is utterly negligible, and it may be assumed with confidence that virtually every primary radical enters a polymer particle (or micelle). Moreover the average lifetime of a chain radical in the aqueous phase (i.e., 10 sec.) is too short for an appreciable expectation of addition of a dissolved monomer molecule by the primary radical prior to its entrance into a polymer particle. 

We turn next to consideration of the concentration of radicals in a polymer particle. The rate of termination in a particle of volume V may be written 

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A quantitative theory of rate processes has been developed on the assumption that the activated state has a characteristic enthalpy, entropy and free energy the concentration of activated molecules may thus be calculated using statistical mechanical methods. Whilst the theory gives a very plausible treatment of very many rate processes, it suffers from the difficulty of calculating the thermodynamic properties of the transition state. 

Definitive examples of intrinsic non-RRKM dynamics for molecules excited near their unimolecular tluesholds are rather limited. Calculations have shown that intrinsic non-RRKM dynamics becomes more pronounced at very high energies, where the RRKM lifetime becomes very short and dissociation begins to compete with IVR [119]. There is a need for establishing quantitative theories (i.e. not calculations) for identifying which molecules and energies lead to intrinsic non-RRKM dynamics. For example, at thenual... 

Much of chemistry occurs in the condensed phase solution phase ET reactions have been a major focus for theory and experiment for the last 50 years. Experiments, and quantitative theories, have probed how reaction-free energy, solvent polarity, donor-acceptor distance, bridging stmctures, solvent relaxation, and vibronic coupling influence ET kinetics. Important connections have also been drawn between optical charge transfer transitions and thennal ET. 

Bueche was able to incorporate these ideas into a quantitative theory, the mathematical details of which need not concern us. The result is complex, but simplifies when applied to polymers of very large molecular weight. In this limit the Bueche theory predicts... 

In all these examples of Pasteur s principle in action, surprise was occasioned by the mismatch between initial quantitative theory and the results of accurate measurement, and the surprise led to the resolution of the paradox. The principle remains one of the powerful motivating influences in the development of materials science. 

Both spin-lattice and spin-spin relaxation depend on rates of molecular motion, for relaxation results from the interaction of fluctuating magnetic fields set up by nuclei in the spin system and in the lattice. A quantitative theory of this dependence was given by Bloembergen et al., who obtained... 

If we now transfer our two interacting particles from the vacuum (whose dielectric constant is unity by definition) to a hypothetical continuous isotropic medium of dielectric constant e > 1, the electrostatic attractive forces will be attenuated because of the medium s capability of separating charge. Quantitative theories of this effect tend to be approximate, in part because the medium is not a structureless continuum and also because the bulk dielectric constant may be an inappropriate measure on the molecular scale. Eurther discussion of the influence of dielectric constant is given in Section 8.3. 

The quantitative theory of ionic reactions, within the limitations of a continuum model of the solvent, is based on the Bom equation for the electrostatic free energy of transfer of an ion from a medium of e = 1 to the solvent of dielectric constant... 

No completely general and quantitative theory of the stereochemical activity of the lone-pair of electrons in complex halides of tervalent As, Sb and Bi has been developed but certain trends are discernible. The lone-pair becomes less decisive in modifying the stereochemistry (a) with increase in the coordination number of the central atom from 4 through 5 to 6, (b) with increase in the atomic weight of the central atom (As > Sb > Bi), and (c) with increa.se in the atomic weight of the halogen (F > Cl > Br > 1). The relative energies of the various valence-Ievel orbitals may also be an important factor the F(a) orbital of F lies well below both the s and the p valence... 

Concentration of Electrolyte Myer and Sievers"" applied the Donnan equilibrium to charged membranes and developed a quantitative theory of membrane selectivity. They expressed this selectivity in terms of a selectivity constant, which they defined as the concentration of fixed ions attached to the polymer network. They determined the selectivity constant of a number of membranes by the measurement of diffusion potentials. Nasini etal and Kumins"" extended the measurements to paint and varnish films. 

Langmuir (1916), whp put forward the fir quantitative theory of the adsorption of a gaS, assumed that a gas molecule condensing from the gas phase-would adhere to the surface fora short time before evaporating and that the condensed layer was only one atom or molecule thick. If 0 is the fraction of the surface area covered by adsorbed molecules at any time, the rate of desorption is proportional to 0 and equal to k 0 where is a constant at constant temperature. Similarly the rate of adsorption will be proportional to the area of bare surface and to the rate at which the molecules strike the surface (proportional to the gas pressurep). At equilibrium the rate of desorption equals the rate of adsorption... 

Kuchanov, S. Modern Aspects of Quantitative Theory of Free-Radical Copolymerization. Vol. 103, pp. 1-102. 

In the following, the traditional treatment of the rate equation (3) will be adopted, taking the preexponential factor as a constant. Evidently, no other procedure is available at present. Even if a quantitative theory of the outlined problems were available, mathematical difficulties would render it possible to present only selected computerized data. 

To set up a quantitative theory based on this qualitative picture, we need to know the rate at which molecules collide and the fraction of those collisions that have at least the energy Emin required for reaction to occur. The collision frequency (the number of collisions per second) between A and B molecules in a gas at a temperature T can be calculated from the kinetic model of a gas (Section... 

The strenuous efforts made in the study of CIDNP over the past few years have produced a basic quantitative theory of the effect. This has been extensively tested, chiefly in well-understood areas of radical chemistry, and found to be satisfactory. There still remains room for theoretical development and refinement, however, but the time is now ripe for the exploitation of the phenomenon in chemical studies. The... 

Palm, V. A., Osnovy Koiichestvennoi Teorii Organicheskikh Reaktsii (Principles of the Quantitative Theory of Organic Reactions), Izdatelstvo Khimiya, Leningrad, 1967, Ch. XII. 

Langmuir Quantitative theory of adsorption of gases on surfaces... 

The first ideas concerning a role of pairwise electrostatic interaction between ions were advanced in 1924 by Vladimir K. Semenchenko. A quantitative theory of the formation of ion pairs was formulated in 1926 by Niels Bjerrum. 

The full quantitative theory of NIS is somewhat involved and hence delegated to Appendix 2 (Part 111, 3 of CD-Rom ) that is based on [98, 7]. Here, we outline only the most important features. 

In Ref. 30, the transfer of tetraethylammonium (TEA ) across nonpolarizable DCE-water interface was used as a model experimental system. No attempt to measure kinetics of the rapid TEA+ transfer was made because of the lack of suitable quantitative theory for IT feedback mode. Such theory must take into account both finite quasirever-sible IT kinetics at the ITIES and a small RG value for the pipette tip. The mass transfer rate for IT experiments by SECM is similar to that for heterogeneous ET measurements, and the standard rate constants of the order of 1 cm/s should be accessible. This technique should be most useful for probing IT rates in biological systems and polymer films. 

From the above reasoning it may be concluded that the quantitative theory as it stands today gives the opportunity to provide an exhaustive description of the chemical structure of the products of free-radical copolymerization of any number of monomers m. 

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