Instantaneous steady-state approximation

It, therefore, appears that the equilibrium approximation is a special case of the steady-state approximation, namely, the case i > 2- This may be, but it is possible for the equilibrium approximation to be valid when the steady-state approximation is not. Consider the extreme but real example of an acid-base preequilibrium, which on the time scale of the following slow step is practically instantaneous. Suppose some kind of forcing function were to be applied to c, causing it to undergo large and sudden variations then Cb would follow Ca almost immediately, according to Eq. (3-153). The equilibrium description would be veiy accurate, but the wide variations in Cb would vitiate the steady-state description. There appear to be three classes of practical behavior, as defined by these conditions ... 

Consider a straight tubular runner of length L. A melt following the power-law model is injected at constant pressure into the runner. The melt front progresses along the runner until it reaches the gate located at its end. Calculate the melt front position, Z(f), and the instantaneous flow rate, Q t), as a function of time. Assume an incompressible fluid and an isothermal and fully developed flow, and make use of the pseudo-steady-state approximation. For a polymer melt with K = 2.18 x 10 N s"/m and n = 0.39, calculate Z(t) and Q(t)... 

In summary, Ogwada and Sparks (1986c) developed a model and assumed that the adsorption of ions from solution by soil particles occurs in a series rather than a parallel reaction mode. Thus, mass-transfer processes and CR occur consecutively. Under the steady-state approximation, the rate of mass transfer is approximately equal to the rate of the reaction, so that instantaneous change in the concentration of CA with time approaches... 

The steady-state approximation is a more general method for solving reaction mechanisms. The net rate of formation of any intermediate in the reaction mechanism is set equal to 0. An intermediate is assumed to attain its steady-state concentration instantaneously, decaying slowly as reactants are consumed. An expression is obtained for the steady-state concentration of each intermediate in terms of the rate constants of elementary reactions and the concentrations of reactants and products. The rate law for an elementary step that leads directly to product formation is usually chosen. The concentrations of all intermediates are removed from the chosen rate law, and a final rate law for the formation of product that reflects the concentrations of reactants and products is obtained. 

The quasi-steady-state approximation (QSSA) is commonly made for the moments of living polymer chains since, for most practical situations, an equilibrium is achieved instantaneously between chain initiation and chain transfer, fc,C [M] = ( cpR + daclMo- This equilibrium results from the fast dynamics of the initiation and transfer reactions compared to that of the overall polymerization rate. In this case, an even simpler system of equations is obtained than the one listed in Table 2.6. 

Because the quasi-steady-state approximation for radical concentration during autoacceleration is invalid [35], determination of individual rate coefficient becomes possible under the assumption of an instantaneous quasi-steady state at the point of the irradiation termination. It is assumed that the change in radical concentration during a single measurement at one reaction point is negligible, so the classical expression for the polymerization rate can be used ... 

Thus, it is commonly taken that there is no change in the instantaneous concentration of the carbocation intermediate—it is used as quickly as it is formed This assumption is called the steady-state approximation, that is. 

Now, if it is argued that there is no change in the instantaneous concentration of the carbanion (conjugate base) intermediate, that is, it is a reactive intermediate that, in its short lifetime, either reacts with a proton source and returns to starting material or goes on to product by loss of the leaving group, then its concentration remains small and constant and the steady-state approximation applies, so that... 

The data are from a free radical polymerization of butyl acrylate (BA) in butyl acetate. When fractional monomer conversion reached 0.4, an extra amount of azobisisobutyro-nitrile (AIBN) initiator was added (initiator boost). Its effect can be immediately seen in the rapid drop of as the quasi-steady state approximation (QSS A) predicts for kinetic chain length (e.g., see Chapters 1 and 5) which is the molecular weight of chains being produced at any instant in a free radical reactions. It is proportional to the concentration of monomer to that of initiator. Hence, the addition of initiator causes the instantaneous chain length, and hence to fall. 

This equation is analogous to Eq. (5.4.18) or (5.4.19) for the steady-state current density, although the instantaneous current depends on time. Thus, the results for a stationary polarization curve (Eqs (5.4.18) to (5.4.32)) can also be used as a satisfactory approximation even for electrolysis with the dropping mercury electrode, where the mean current must be considered... 

It is experimentally found that the concentration of radicals increases initially, but almost instantaneously reaches a constant, and that the number of growing chains is approximately constant over a large extent of reaction, that is, steady-state condition where d[Af ]/dt = 0 and... 

Puff models such as that in Reference 5 use Gaussian spread parameters, but by subdividing the effiuent into discrete contributions, they avoid the restrictions of steady-state assumptions that limit the plume models just described. A recently documented application of a puff model for urban diffusion was described by Roberts et al, (19). It is capable of accounting for transient conditions in wind, stability, and mixing height. Continuous emissions are approximated by a series of instantaneous releases to form the puffs. The model, which is able to describe multiple area sources, has been checked out for Chicago by comparison with over 10,000 hourly averages of sulfur dioxide concentration. 

The difference in the concentrations of the QSSA species calculated from differential equation (4.1), and algebraic equation (4.90), while the concentrations of the non-steady-state species are the true concentrations calculated from equation (4.1), is the instantaneous error of the quasisteady-state approximation. The instantaneous error induced by the application of the QSSA to a single species Ac/ [158] can be used to identify the possible steady-state species and is calculated by the following expression ... 

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