Stationary state assumption

The concentrations of the chain carriers R1 and R2 may be determined by the use of the stationary state assumption for each species. 

A simple kinetic analysis of reactions 15.5 and 15.6 leads to equations 15.7 (where the stationary state assumption was applied to the acyl radical concentration) and 15.8. 

This decomposition is usually first-order and only a fraction, f, of the radicals are utilised in step (62). Chain termination via step (64) and/or (65) is now an inherent feature of the reaction scheme. Often, the stationary state assumption can be applied to the A (as a group) so that... 

Now the stationary state assumption is applied to the Equation 8 to obtain ... 

This simple two-step mechanism, when combined with the stationary-state assumption for the presumably highly reactive positive ion (see Section 2.5, p. 93), leads to the prediction given in Equation 5.4 for the rate of product formation. (See Problem 1.) The term in parentheses in Equation 5.4 will... 

Reaction (13) is followed inevitably by reaction (2). Since ku was found to be much larger than fcuM,83 the stationary state assumption for 0( 7)) leads to... 

When the concentracion of active centres does not change during propagation, we speak of stationary polymerization. All previous and the majority of the presently valid analyses of radical polymerization kinetics assume stationary (or, better, pseudo-stationary [20, 21]) states. The conditions under which the stationary state assumption is a valid and useful approximation, can be expressed as follows [22] ... 

With the stationary state assumption, they derived and equation for copolymer composition (r, = km/km, r2 = k222Jkm, r = k2n/k2l2, and r2 =... 

An exact solution of scheme (114) would require the knowledge of various active centre concentrations, [ —M, — M2, and — M3], These data are unknown even for the simplest systems. Therefore use must again be made of the stationary state assumption+ yielding... 

The appropriateness of the stationary-state assumption is open to question in such a system when the amount of enzyme-substrate complex is large compared to free enzyme and (Eo) is not very much smaller than (S) (Sec. IIL9). When, however, (S) (Eo) or (E-S) (Eo) then the stationary-state hypothesis is valid for amounts of reacti (So) - (S) > (E-S). 

Instead of the quasi-stationary state assumption of Kramers, he assumed only that the density of particles in the vicinity of the top of the barrier was essentially constant. Visscher included in the Foldcer-Planck equation a source term to accoimt for the injection of particles so as to compensate those escaping and evaluated the rate constant in the extreme low-friction limit. Blomberg considered a symmetric, piecewise parabolic bistable potratial and obtained a partial solution of the Fokker-Hanck equation in terms of tabulated functions by requiring this piecewise analytical solution to be continuous, the rate constant is obtained. The result differs from that of Kramers only when the potential has a sharp, nonharmonic barrier. 

Table V shows examples of the gains obtained using the new computational scheme. The typical real-time/computer-time ratio was increased from 36/1 to 180/1. Perhaps, more significant is the fact that the Fade method has allowed us to obtain acceptable solutions in situations where Runge-Kutta either failed to converge or produced spurious solutions. One such instance is the integration of full differential equations for the free-radical species. The Fade method successfully computed solutions without algebraic substitution of stationary-state assumptions whereas Runge-Kutta failed to produce any solution.

The stationary state assumption In a given orbit, the total energy (kinetic + potential) will be a constant The kinetic energy KE is equal to (l/2)mv, while the potential energy V can be obtained by integrating Coulomb s law with respect to distance. While this assumption was in opposition to the classical prediction that the electron will spiral into the nucleus, it was necessary in order to explain the experimental observations. 

This approximation is known as stationary state approximation and holds good for all the intermediate compounds that are formed in multistep reactions. Applying this stationary state assumption, we get... 

Whole order irreversible or equilibrated reactions, stationary state assumptions, selected boundary conditions and geometries make simplifications possible. 

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