Nonsteady-state kinetics

The radical concentration is zero at the very start of the polymerization and reaches the steady-state level only after a finite time. During this intervening ndnsteady period, the rate of change of the concentration of radicals will be given by the difference of the rates of their formation and termination, i.e., 

The advantage of photoinitiation lies in that the generation of radicals can be commenced (or stopped) instantly by turning on (or off) light to the polymerization cell and, moreover, time needed for temperature equilibration (when thermal initiators are used) can be avoided. Using Eq. (6.61) for the rate of photoinitiation, Eq. (6.79) becomes 

For mathematical convenience, Flory (1953) defined a new parameterr, called the average lifetime of a growing radical under steady-state conditions, as No. of radicals present at steady state 

of radicals disappearing per unit time at steady state 

Multiplying both the numerator and denominator of Eq. (6.83) by icp[M], is more conveniently expressed as (Ghosh, 1990 Odian, 1991)  

of radicals present at steady state No. of radicals disappearing per unit time at steady state 

Multiplying both the numerator and denominator of Eq. (6.83) by is more conveniently 

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They open a perspective for performing a new class of controlled nonsteady-state kinetic experiments, which can be termed joint kinetic experiments. ... 

With the TAP approach reactions also occur at a vacuum-solid interface. However, unlike the surface science approach, the solid surface in a TAP experiment is generally a disordered technical material such as industrial catalyst particles that are generally studied in flow reactors. The objective of a TAP experiment is to measure nonsteady-state kinetic properties and track how changes in reaction conditions and sample composition alter these properties. Incrementally increasing or decreasing the amount of a single component while simultaneously making kinetic measurements charts the correspondence between a component... 

SLOW EXHAUSTION OF INITIATOR, NONSTEADY-STATE KINETICS ... 

R Zipp, NFH Ho. Nonsteady state model of absorption of suspensions in the GI tract Coupling multi-phase intestinal flow with blood level kinetics. Pharm Res 10 S210, 1993. 

The bulk polymerization of acrylonitrile in this range of temperatures exhibits kinetic features very similar to those observed with acrylic acid (cf. Table I). The very low over-all activation energies (11.3 and 12.5 Kj.mole-l) found in both systems suggest a high temperature coefficient for the termination step such as would be expected for a diffusion controlled bimolecular reaction involving two polymeric radicals. It follows that for these systems, in which radicals disappear rapidly and where the post-polymerization is strongly reduced, the concepts of nonsteady-state and of occluded polymer chains can hardly explain the observed auto-acceleration. Hence the auto-acceleration of acrylonitrile which persists above 60°C and exhibits the same "autoacceleration index" as at lower temperatures has to be accounted for by another cause. 

Handling rate equations for complex mechanisms. While steady-state rate equations can be derived easily for the simple cases discussed in the preceding sections, enzymes are often considerably more complex and the derivation of the correct rate equations can be extremely tedious. The topological theory of graphs, widely used in analysis of electrical networks, has been applied to both steady-state and nonsteady-state enzyme kinetics 45-50 The method employs diagrams of the type shown in Eq. 9-50. Here... 

The catalytic oxidation of CO on iridium has not been as extensively studied as with palladium and platinum. However, as for these metals, both steady-state (40, 54, 124, 199-202) and nonsteady-state investigation (124, 200-203) have been carried out on both polycrystalline and single crystal surfaces. As the results are for the most part very similar to those obtained on palladium and platinum surfaces, only those results that shed additional light on the kinetics and mechanism basic to the reaction will be emphasized here. 

T.P Labuza and M. Saltmarch, Kinetics of browning and protein quality loss in whey powders during steady state and nonsteady state storage conditions. J. Food Sci., 1982, 47, 92-96, 113. 

Vorkenshtein and Gordshtein were the first to apply graph theory to investigation of the kinetics of nonsteady state processes (3,31). We present here a short summary of their method. 

The steady-state assumption that is helpful in simplifying the analysis of free-radical kinetics is not valid in many, if not most, cationic polymerizations, which proceed so rapidly that steady-state is not achieved. Some of these reactions (e.g., isobutylene polymerization by AICI3 at -100°C) are essentially complete in a matter of seconds or minutes. Even in slower polymerizations, the steady-state may not be achieved if > Rt- The expressions given above can only be employed if there is assurance that steady-state conditions exist, at least during some portion of the overall reaction. Steady state is implied if Rp is constant with conversion, except for changes due to decreased monomer and initiator concentrations. A more rapid decline in Rp with time than what is expected or an increase in Rp with time would signify a nonsteady state. Thus many of the experimental expressions reported in the literature to describe the kinetics of specific cationic polymerizations are not valid since they are based on data where steady-state conditions do not apply. 

Taking into account the dependence of free ions/ion pairs equilibrium on the chain length, analysis of the nonsteady-state portions of kinetic curves, yields the following equation for the induction period ... 

The kinetics of sorption of a penetrant into a polymer film of thickness i serves to illustrate problems of nonsteady-state diffusion. At t < 0, C = 0 for all x whereas, at t > 0, the surfaces at x = 0 and i assume the value C, which after enough time is the value achieved for all x, that is, equilibrium. If M. denotes the total amount of permeant that has entered the film at time t, and M , denotes the amount at equilibrium = A A Cg, then the solution to Equation 8 for these boundary conditions is... 

Studies of pore waters have become a standard tool for understanding the biogeochemical processes that influence sediments, and considerable efforts have been invested during the past several decades to develop techniques to collect samples, evaluate whether vertical profiles exhibit artifacts introduced during collection and handling, and develop approaches to model the observed profiles and obtain quantitative estimates of reaction kinetics and stoichiometry. Usually, modeling approaches assume steady-state behavior, but when time-dependent constraints can be established, nonsteady-state approaches can be applied. 

A mathematical model describing the processes of AC charging-discharge with account for EDL charging, intercalation of hydrogen into carbon, and its nonsteady-state solid-phase diffusion, electrode kinetics, ion transport over the electrode thickness, and characteristics of its porous structure was developed and confirmed experimentally. It is shown that the maximum path of diffusion of hydrogen atoms all other conditions being equal is inversely proportional to the hydrophilic specific surface area of the electrode. 

However, the study of the kinetics of polymerization under nonsteady-state conditions can lead to the separate determination of the mean lifetime of the active centers. This can be done when the polymerization is initiated by light of intensity / such that... 

B. E. Conway, H. A. Kozlowska, and E. Gileadi [1965] Significance of Nonsteady State a.c. and d.c. Measurements in Electrochemical Adsorption Kinetics. Applications to Galvanostatic and Voltage Sweep Methods, J. Electrochem. Soc. 112, 341-349. 

In Chapter 3, the theoretical minimum minimarum is explained. This toolbox serves the primary analysis of data of chemical transformations and the constraction of typical kinetic models, both steady-state and nonsteady-state. In the course of the primary analysis, the reaction rate is extracted from the observed net rate of production of a chemical component. Although the procedure is rather simple, its understanding is not trivial. 

In Chapter 6, we explained the concept of time invariances obtained in dual (or reciprocal) kinetic experiments. Such experiments can be performed in batch reactors or TAP reactors and start from special, reciprocal, initial conditions. This results in certain mixed quotientlike functions of selected nonsteady-state normalized concentrations from both experiments always being equal to the equilibrium coefficient of the reaction, during the whole course of the temporal experiment, that is, not only at the end, when equilibrium conditions are reached. Examples of such quotientlike functions are presented in Chapter 6. In this section, two new patterns will be presented, namely intersections and coincidences. 

More recently, specific reactors are used for nonsteady-state studies of catalytic kinetics which also allow observation of the intermediate steps in a reaction mechanism [22]. Transient studies of catalytic reaction schemes and kinetics are discussed in Chapter 10. 

Model-free kinetic analysis of nonsteady-state reactions is a recent development that began with the thin zone microreactor configuration [82, 88, 89]. A model-free kinetic method known as the Y-procedure has been used to extract the nonsteady-state rate of chemical transformation from reaction-diffusion data with no assumptions regarding the kinetic model the reader is referred to [90] for more details describing this procedure. 

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