Deviations from Ideal Kinetics

A given polymerization conforming strictly to the reaction scheme represented by Eqs. (6.3) to (6.7) is commonly considered as an ideal polymerization. Any system deviating from this pattern of reaction is to be considered as a cas e iwnideal polymerization. The ideal behavior requires constancy of the term R /[l] [M] ), as according to Eq. (6.24) it is expressed as 

Practical free-radical polymerizations often deviate from Eq. (6.126) because the assumptions made in the ideal kinetic scheme are not fuUy satisfied by the actual reaction conditions or because some of these assumptions are not valid. For example, according to the ideal kinetic scheme that leads to Eq. (6.126), the initiation rate (Rf) and initiator efficiency (/) are independent of monomer concentration in the reaction mixture and primary radicals (i.e., radicals derived directly from the initiator) do not terminate kinetic chains, thoughrifi reality R may depend on [M], as in the case of cage effect (see Problem 6.7) and, at high initiation rates, some of the primary radicals may terminate kinetic chains (see Problem 6.25). Moreover, whereas in the ideal kinetic scheme, both kp and kt are assumed to be independent of the size of the growing chain radical, in reahty k[ may be size-dependent and diffusion-controlled, as discussed later. 

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The optimal ratio between hydrolysis and polymerization velocities can be obtained not only by changing sulphuric acid concentration, but also by changing initiate s (ammonium persulfate (APS)) concentration. It s concentration doesn t affect the process velocity much (order of reaction s initiation rate is 0.24), but it greatly affects polymer s molecular mass -much more than it can expected according to classical concepts [3], We think that detected deviations from ideal kinetics exist because processes of polymerization and hydrolysis of AN occur simultaneously which results in formation of a more reactive monomer - AA. 

Deviations from ideal kinetics due to size-dependence and diffusion control of termination produce relatively weak effects at low conversions. However, at high conversions these effects are very significant in most radical polymerizations. Thus, instead of the reaction rate falling with time, as would be expected from Eq. (6.24) since the monomer and initiator concentrations decrease with conversion, an exact opposite behavior is observed in many polymerizations where the rate of polymerization increases with time. A typjgal example of this phenomenon is shown in Fig. 6.10 for the polymerization of methyl methacrylate in benzene solution at 50°C (Schulz and Haborth, 1948). An acceleration is observed at relatively high monomer concentrations and the curve for the pure monomer shows a drastic autoacceleration in the polymerization rate. This type of behavior observed under isothermal conditions is referred to as the gel effect. It is also known as the Tromsdorff effect or Norrish-Smith effect in honor of the early researchers in this field. ... 

In the literature two types of effects leading to deviations from ideal kinetics are treated separately cooperative effects per se, e.g. changes in affinity or activity of an enzyme with changes in the concentration and allosteric effects, binding of the substrate at a site other then the active site affects affinity or binding, by stabilizing the high affinity (R) state of the other active site. 

The view that decreases with radical size has received further support. It has been shown that the effect can lead to deviations from ideal kinetics and to serious overestimates of primary radical termination. Non-ideal kinetics in solution polymerizations have, however, been attributed to this type of termination. It has been considered for the case of tributyltin methacrylate and for retarded polymerizations. A new procedure for studying the process has been described. 

The expression d In a/d In c is known as the thermodynamic factor and is a special case of the Wagner factor (or thermodynamic enhancement factor) which plays an important role for the kinetic properties of electrodes. This term indicates the deviation from ideality of the mobile component. For ideal systems this quantity becomes 1 and comparison with Pick s first law yields... 

Ideal gases obey the ideal gas law at all temperatures and pressures. However, there are no ideal gases, only real gases. Real gases deviate from ideal behavior most strongly at high pressures and/or low temperatures. So, where do the basic tenets of Kinetic Molecular Theory fail ... 

In order to explain deviations from ideal behaviour, it is necessary to modify the kinetic theory of gases. The following two postulates of the kinetic theory do not appear to hold good under all conditions. Let us examine them more critically. 

For ideal radical polymerization to occur, three prerequisites must be fulfilled for both macro- and primary radicals, a stationary state must exist primary radicals have to be for initiation only and termination of macroradicals only occur by their mutual combination or disproportionation. The rate equation for an ideal polymerization is simple (see Chap. 8, Sect. 1.2) it reflects the simple course of this chain reaction. When the primary radicals are deactivated either mutually or with macroradicals, kinetic complications arise. Deviations from ideality are logically expected to be larger the higher the concentration of initiator and the lower the concentration of monomer. Today termination by primary radicals is an exclusively kinetic problem. Almost nothing has been published on the mechanism of radical liberation from the aggregation of other initiator fragments and from the cage of the... 

In the next sections the quantitative effect of reactions (1), (3), (4), and (5) which can be considered as various deviations from ideal systems, on kinetics, molecular weights, and polydispersities, will be presented. The magnitude of only one variable will be changed each time to demonstrate clearly the effect of slow initiation, termination, transfer, and slow exchange on the polymerization rates and properties of the resulting polymers. 

Figure 12-15 A molecular interpretation of deviations from ideal behavior, (a) A sample of gas at a low temperature. Each sphere represents a molecule. Because of their low kinetic energies, attractive forces between molecules can now cause a few molecules to stick together. (b) A sample of gas under high pressure. The molecules are quite close together. The free volume is now a much smaller fraction of the total volume.

First, the system deviates from ideality as there is a finite rate of mass transfer of solute molecules across the chromatographic interface. The contribution to the overall HETP arising from this kinetic control of the sorption-desorption process increases with increasing mobile phase flow-rate. 

The selectivity and conversion of a chemical reactor depend not only on the kinetics of the reaction but also on the time for which the reaction partners are available for the reaction, that is, the hydrodynamic behavior. By determining the residence-time behavior of a reactor, one can determine its deviation from ideal hydrodynamic behavior (boundary cases plug flow and complete backmixing) and hence decide with which reactor model the real reactor can best be modeled. 

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