Pseudo-steady-state-hypothesis

For ease of solution, it is assumed that the spherical shape of the pellet is maintained throughout reaction and that the densities of the solid product and solid reactant are equal. Adopting the pseudo-steady state hypothesis implies that the intrinsic chemical reaction rate is very much greater than diffusional processes in the product layer and consequently the reaction is confined to a gradually receding interface between reactant core and product ash. Under these circumstances, the problem can be formulated in terms of pseudo-steady state diffusion through the product layer. The conservation equation for this zone will simply reflect that (in the pseudo-steady state) there will be no net change in diffusive flux so... 

A simple example is the so-called Michaelis-Menten kinetics for enzymatic reactions A + E +C->B + E, which, when the pseudo-steady-state hypothesis is invoked, gives for the concentration of A, for instance, a,... 

However if c3 and c4 are constant then c2 = (k2 + k3)c4/k1c3 must be constant, and no reaction takes place. There is therefore a basic inconsistency in the attempt to make the mechanism SR account strictly for the reaction Si. In spite of this, such kinetic equations as (28) have been found to be extremely useful and quite accurate in kinetic studies. The chemical kineticist therefore claims that over an important part of the course of reaction c3 and c4 are approximately constant, or often that they are both small and slowly varying. This is called a pseudo-steady-state hypothesis and however pseudo it must appear to the mathematician it is sufficiently important to merit formalization. We shall therefore propound a formal definition and illustrate further how it may be used. 

Mobility, permeability, and the pseudo-steady-state hypothesis. Math. Biosci 13, 1-8 (1972). 

In this chapter we discuss four topics the pseudo-steady-state hypothesis, polymerization, enzymes, and bioreactors. The pseudo-steady-state hypothesis (PSSH) plays an important role in developing nonelementary rate laws. Consequently, we will first discuss the fundamentals of the PSSH, followed by its use of polymerization reactions and enzymatic reactions. Because enzymes are involved in all living organisms, we close the chapter with a discussion on bioreactions and reactors. 

Using the pseudo-steady-state hypothesis (PSSH), Equations (7-11) and (7-14) can be combined to obtain a rate law for solely in terms of the concentration of azomethane. First we solve for the concentration of the active intermediate AZO in terms of the concentration of azomethane, AZO. Flora the PSSH,... 

Rate Laws Derived from the Pseudo-Steady-State Hypothesis... 

This is a pair of nonlinear equations and no simple solution can be written down, but they can be reduced to a single equation by making a so-called pseudo-steady state hypothesis. This is the assumption that, when Cq is much smaller than Uo, the concentration c is never very large and varies very slowly. We can then set dc/dt = 0. This hypothesis appears to be very pseudo indeed at first sight, but it can in fact be justified by what is known as the singular perturbation theory of differential equations. Setting dcjdt = 0 in Eq. (4.6.7) we can solve for c in terms of a ... 

F. G. Heineken, H. M. Tsuchiya, and R. Aris, On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics, Math. Biosciences, 1, 95 (1967). 

These rate laws [Equations (7-3) through (7-5)] are pretty much us in the design of any reaction system because the concentration of the a intermediate AZO is not readily measurable. Consequently, we will ust Pseudo-Steady-State-Hypothesis (PSSH) to obtain a rate law in terms of i surable concentrations. 

SwnmaiT of the Cumene Decompo.sition 680 Reforming Catalysts 681 Rate Laws Derived from the Pseudo-Steady-State Hypothesis 684 Tempertnure Dependence of the Rate Law... 

Note that a/ will gradually increase during the course of the reaction and will reach its saturation value, dg/Ku, when B is depleted. Dropping the accumulation term for a (t) represents a form of the pseudo-steady-state hypothesis. 

Rate of ftsmanon of P a rj - pj by pseudo steady-state hypothesis ... 

The pseudo steady state hypothesis and the long chain approximation are encountered constantly in publications concerning chain reactions. Find in the recent literature a paper using this analysis. Describe the problem addressed by the authors, the reaction and chain mechanism proposed, and the analysis employed. Then see whether you agree with the result and the interpretation proposed in the paper. 

Enzyme kinetics. The kinetics of transformation processes catalysed by a single enzyme are often described using the Micha-elis-Menten equation (1). The derivation of this equation is, however, based on two assumptions. The pseudo steady state hypothesis ( ) with respect to the intermediary enzyme-siibstrate complex is valid and the reverse reaction from product to substrate can be... 

In Eq. 5.3.d-l it has again been assumed, in accordance with the pseudo-steady-state hypothesis, that the amount of A involved in the coking reaction is small. Also, the effective diffusivity is presumed to be unaffected by the coke formation. deactivation function, assumed to be described by... 

It is common to make a pseudo-steady-state hypothesis for the initial concentration in the secondary particle at t = 0, [MsJo, to obtain the initial condition for Equation 2.126. This is done by setting the right-hand side of Equation 2.126 to zero and solving the resulting ODE for the radial profile of [Mj]. This hypothesis is made because the choice of [Mslo = 0 generally leads to stiff differential equations that may be more challenging... 

A number of regulatory mechanisms at the level of energy generation and consumption operate with such small relaxation times that a pseudo-steady-state hypothesis with respect to these mechanisms is justified. Hence, the introduction of these details seems to be unnecessary. 

Fri>ni time to time a paper emerges in the literature aimed at emphasising the net essity of the foundation of the principle of quasistationarity, pseudo-steady state hypothesis, or Bodenstein(-Semenov) method. The essence of the method seems to be an absolutely crazy idea — from the mathematical point ofview. In a system of differential equations let us consider the variables that late on small values to be constant. So if a function is small, so is its derivative It turns out that among the conditions that occur in chemical reaction kinetics it does work well. 

There may be situations where the rates of the individual steps are all of the same order of magnitude and we could not invoke the pseudoequilibrium hypothesis. In such a case, we can use the pseudo-steady-state hypothesis, or Bodenstein (1927) approximation At some point during the reaction, the rate of formation and the rate of disappearance of an intermediate are identical causing a steady-state concentration, or surface coverage, of the reactive intermediate. 

To eliminate these inaccessible concentrations the pseudo steady state hypothesis is called upon. When that hypothesis is satisfied those two rates become zero. The set of equations can then be solved for the radical concentrations, yielding e.g., for (3- Me-1-... 

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