Steady state principle

This scheme requires a rate-determining (second) proton-transfer, against which there is considerable experimental evidence in the form of specific-acid catalysis, the solvent isotope effect and the hg dependence discussed earlier. Further, application of the steady-state principle to the 7i-complex mechanism results in a rate equation of the form... 

Biogeochemical Model Profile for Calculation of Critical Loads of Acidity The biogeochemical model PROFILE has been developed as a tool for calculation of critical loads on the basis of steady-state principles. The steady-state approach implies the following assumptions ... 

Polymer Chemistry Considering steady-state principle,... 

According to the stationary or steady state principle, whenever a short-lived reaction intermediate occurs in a system, its rate of formation can be taken as equal to its rate of disappearance. Applying this principle, we have... 

As the reaction proceeds, the intermediate complex formed in accordance with the suggested mechanism, decomposes instantaneously according to the same mechanism. On applying the steady state principle, we have... 

The following equation applies according to the steady-state principle as applied to... 

Applying the steady-state principle to the primary radical R-,... 

Applying the steady-state principle to the growing chains, i.e., equating the rates of reactions (d) and (f),... 

In order to solve Equation (8.3), expressions for [E] and [ES] are required that are obtained by applying Briggs-Haldane steady state principles. According to these principles, biocatalysis rapidly attains a condition of stasis under which all biocatalyst species are at a constant equilibrium concentration. In other words [E] and [ES] are constant with time. Stasis is reflected by... 

To review A steady stream of neutral molecules is drawn into the source from an inlet system and ionized where the stream of molecules intercepts the electron beam. The positive ions created are constantly drawn out with the ion gun, while the remaining neutral molecules are steadily pumped away. Thus, the source operates on a steady-state principle constant input of neutral molecules and output of ions (and leftover neutral molecules). 

The concentration of the primary radical cr is characterized by the fast establishment of a steady-state value. Applying Bodenstein s quasi-steady-state principle to this concentration yields the following description of the initiating reaction ... 

The polymer radical concentration cp , which is part of the propagation rate, can be obtained by applying the Bodenstein quasi-steady state principle. 

The second approach starts with an idea of possible mechanism, leading to a theoretical kinetic equation formulated in terms of concenhations of adsorbed reactants and intermediate species use of the steady-state principle then leads to an expression for the rate of product formation. Concentrations of adsorbed reactants are related to the gas-phase pressures by adsorption equations of the Langmuir type, in a way to be developed shortly the final equation, the form of which depends on the location of the slowest step, is then compared to the Power Rate Law expression, which if a possibly correct mechanism has been selected, will be an approximation to it. A further test is to try to fit the results to the theoretical equation by adjusting the variable parameters, mainly the adsorption coefficients (see below). If this does not work another mechanism has to be tried. 

Chain reactions belong categorically to the class of complex reactions, in the sense that cyclically regenerated intermediate chemical species are central to the reaction mechanism. When such reactions are characterized by low overall rates and intermediate species concentrations which remain small throughout, the analysis of kinetics is greatly facilitated by the applicability of the quasi-steady state principle. As we shall see, it is characteristic of the fast chain reactions studied in the shock tube that conventional quasi-steady state conditions do not prevail, and that macroscopic chain centre concentrations develop. 

In the foregoing we have introduced some of the underlying principles of chain reaction Wnetics in physical chemistry, and indicated the nature of their extension to describe rapid reactions at elevated temperatures. Clearly, these extensions could have been made entirely theoretically and the description of the processes worked out but for some quantitative parameters. However the actual development has occurred largely in a surge of activity that began in the middle 1950 s, in conjunction with fast reaction experiments which reveal the phenomenology and provide access to the quantitative parameters. In place of the conventional quasi-steady state principle, other comparably useful and mathematically approximate aids to the evaluation and comprehension of data have arisen in conjunction with the nonsteady ignition behaviour and the subsequent removal of the residual super-equilibrium population of chain centres. The detailed consideration of these experiments and their interpretation form sections 2.2 and 2.3 of this chapter. 

The concentrations of polymer free radicals P and initiator free radicals RJ should be constant, i.e., the steady-state principle is valid ... 

With the usual assumptions [steady-state principle, no penultimate chain end effect (see Section 22.4.2), no influence of chain length on the polymerization equilibrium], the following equation is derived for the case of one reversible (monomer 1) and one irreversible homopolymerization step with two irreversible heteropropagation steps ... 

Introducing the steady-state principle, d[R ]/df = 0, yields an recursive expression for the concentration of macroradicals with chain length i. 

By assuming the Bodenstein steady state principles for all intermediates and that all rate constants are independent of the degree of polymerization, Bohm has deduced the following equation. 

Let us carry out a check of the steady-state principle. For this purpose, let us calculate the time dependence of the end product formation rate from the relationships obtained by accurate solving the direct kinetic problem (see Table 2.1). Next, let us compare the result with the calculations from obtained formula (2.9). The corresponding plots represented in Fig. 2.17 show that the behaviour of the both curves coincide after less than 0.5 s at given values of the rate constants satisfying the condition ki > k. This indicates applicability of the steady-state concentration method to the considered model of the consecutive reaction. 

For each reacting species, a differential equation of the concentration change may be written. The Bodenstein-Semenov steady-state principle can be applied only for short-lived active species therefore, either the obtaining system is analytically nonsolved or its complex solution is impracticable. 

Remark 2 The approximation given by the steady state principle holds very well in most cases encountered exceptions are so rare that in general it can be used to describe actual reactions. Thus, for the open sequence... 

In general, unbranched chain reactions can be described quantitatively, if the mechanism of bond rupture is simple, by application of the steady-state principle to active centres. 

The concentration of active centres can be calculated from the steady state principle ... 

The different steps of these reactions always proceed in the same order, without repetition of a propagation-step. The steady-state principle is applicable and allows us to give a satisfactory description of them. 

The quantitative treatment by the steady-state principle proves most often convenient, which implies that the formation of the complex is rate-determining in neglecting for this reason the terms Col +/ i Sol, the rate equation has the form (Section 2). 

Kinetically, the steady-state principle is applicable to each radical, and gives the following, if we call the rate of the initiation reaction i, and make the plausible hypothesis that all termination reactions have the same rate coefficients and all propagation reactions the same coefficients kp... 

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