Stationary state kinetics

If we apply stationary-state kinetics to AB and to AM, we find for the rate of formation of AB (by mechanism I) ... 

We have now included three possible termination steps among the two chain centers R and X for the most general ease. If we apply stationary-state kinetics to this system, we find, assuming long chains, ... 

Both of these methods are subject to uncertainty that arises from the manner of termination, i.e., by recombination or by disproportionation. This latter question, if not resolved, can introduce an error of 2 into the estimates of / . Assuming that all of these problems have been satisfactorily resolved, the study of photo-stationary-state kinetic systems gives the ratio of rate constants kp ki>. Within these limitations the results of various laboratories, obtained by the use of different techniques, show reasonably good agreement, usually within 10 to 20 per cent for a given monomer-solvent system. ... 

This mechanism illustrated by linoleic acid autoxidation is, however, typical for virtually all autoxidation processes. The general rate law based on stationary state kinetics is... 

Rates of radiation induced polymerizations are normally determined by dilatometric [85] or gravimetric [84] experiments. Some of the first quantitative results from cyclopentadiene [86] and a-methylstyrene [87] were obtained by competitive kinetic methods, based on the retarding effect of ammonia and amines. This approach tends to yield maximum values for Rp. More recently, however, a procedure combining stationary state kinetic and conductance measurements has been described [88, 89], and further refined [85]. Because the ions generated by 7-ray irradiation have a transient existence, the kinetic treatment leads to expressions which are very similar to those derived for homogeneous free radical polymerizations [90]. A simplified version of the kinetic scheme is as follows ... 

Assuming stationary-state kinetics are applicable, we have for primary radicals and propagating chain radicals the expressions ... 

Stationary state kinetics can also be used to derive a relation for the polydispersity index of the form... 

Cliirkov (52) has compared values of C obtained using non-stationary state kinetics with those obtained by Tanaka Morikava (49) Schnecko Kern (45) have discussed values of C obtained by kinetic and molecular wei t techniques and by... 

Contrarily to the styrene/TEMPO system, the Pp of the styrene/ DEPN system was observed to increase with an increase of Iq (=[P-X]o). ° This nonsteadiness is due to the large K of the DEPN system, as noted above. The crossover time tcross from the power-law to stationary-state kinetics is given by eqn [28]. For the PS-TEMPO/styrene/120 °C system, for example, we may roughly set (Pi/fet) = 2 x 10" M, fet=10 M s ... 

In the temperature range from 0 to 30 °C, on the other hand, the perchloric acid-initiated polymerization of styrene follows a stationary-state kinetics, giving a linear first-order plot for monomer consumption Based on the absence of UV absorptions... 

If some of the reactions of (A3.4.138) are neglected in (A3.4.139). the system is called open. This generally complicates the solution of (A3.4.141). In particular, the system no longer has a well defined equilibrium. However, as long as the eigenvalues of K remain positive, the kinetics at long times will be dominated by the smallest eigenvalue. This corresponds to a stationary state solution. 

Volumetric heat generation increases with temperature as a single or multiple S-shaped curves, whereas surface heat removal increases linearly. The shapes of these heat-generation curves and the slopes of the heat-removal lines depend on reaction kinetics, activation energies, reactant concentrations, flow rates, and the initial temperatures of reactants and coolants (70). The intersections of the heat-generation curves and heat-removal lines represent possible steady-state operations called stationary states (Fig. 15). Multiple stationary states are possible. Control is introduced to estabHsh the desired steady-state operation, produce products at targeted rates, and provide safe start-up and shutdown. Control methods can affect overall performance by their way of adjusting temperature and concentration variations and upsets, and by the closeness to which critical variables are operated near their limits. 

In the derivation above, we have included the kinetic energy of the nuclei in the Hamiltonian and considered a stationary state. In Eq. II.3, this term has been neglected, and we have instead assumed that the nuclei have given fixed positions. It has been pointed out by Slater34 that, if the nuclei are not situated in the proper equilibrium positions, the virial theorem will appear in a slightly different form. (A variational derivation has been given by Hirschfelder and Kincaid.11)... 

Fig. 17. Comparison of the variation of the time-average S02 conversion and the maximum bed temperature predicted for stationary cycling condition by an unsteady-state and a steady-state kinetic model for a packed-bed S02 converter operating with periodic flow reversal...

The derivation of a rate law from a postulated mechanism is a useful application of reaction mechanisms. It shows how the kinetics of the elementary reaction steps are reflected in the kinetics of the overall reaction. The following example illustrates this for a simple, gas-phase reaction involving an open sequence. The derivations typically employ the stationary-state hypothesis (SSH) to eliminate unknown concentrations of reactive intermediates. 

An unusual feature of a CSTR is the possibility of multiple stationary states for a reaction with certain nonlinear kinetics (rate law) in operation at a specified T, or for an exothermic reaction which produces a difference in temperature between the inlet and outlet of the reactor, including adiabatic operation. We treat these in turn in the next two sections. 

Procaccia, I. Ross, J. (1978). Stability and relative stability in reactive systems far from equilibrium. II. Kinetic analysis of relative stability of multiple stationary states. J. Chem. Phys., 67, 5565-71. 

This reaction and the synthesis of HBr have also received much less attention than the corresponding reactions of the HI system. The problem of the mechanism of the H2+Br2 reaction which Bodenstein and Lind33 found to be complex was later solved independently by Christiansen34, Herzfeld35 and Polanyi36. The well-known mechanism and the kinetic equation resulting from the stationary-state solution are given below... 

If we apply the term stationary state to those systems in which the concentration of growing chains does not vary throughout the reaction, then it is evident that none of the systems comprised under Case 1 can involve a stationary state. The population of growing chains is declining throughout most of the reaction and falls to zero before all the monomer has been consumed. The method of dealing with this type of kinetic behaviour, devised by Pepper et al., [26] has been used effectively by others [40b, 109, 110]. 

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