The Steady-State Principle

Two free radicals can only be terminated by mutual deactivation when a sufficiently high free radical concentration is reached. Since new free radicals are formed continuously and others are destroyed, a steady state is attained. In the steady state of the first kind, the total free radical concentration is constant. In the steady state of the second kind, the individual free radical concentrations are constant. However, a steady state can only be attained when there are sufficient initiator molecules present (see Section 20.2.6.1). 

This steady state is reached within a relatively short time. Before the occurrence of the steady state, the production of polymer free radicals P (ignoring termination by initiator free radicals) is 

According to Bodenstein, equation (20-48), and d [P ]/dt = 0, the following holds in the steady state  

On achieving the steady state, [P ]f/[P ]stat be unity, thus, with equations (20-51) and (20-52), 

Equation (20-53) must be fulfilled experimentally to 1 %, If we put y = tanh X and x = tanh y, then we obtain 

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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... 

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),... 

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. 

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

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

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. 

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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