Stationary Chain Reactions

The chlorine-hydrogen reaction mechanism, generally accepted and quantitatively supported by many experiments, is based on the concept that the primary active centers of the reaction are chlorine atoms which are generated through dissociation of Clg molecules the chlorine atoms subsequently react with molecular hydrogen [333] Cl -f- Hg 

HCl + H. The resultant hydrogen atoms react with chlorine molecules H + Clg 

HCl + Cl to regenerate chlorine atoms. Thus, the reaction of chlorine with hydrogen occurs via two active centers chlorine atoms and hydrogen atoms. 

The most important elementary steps of the radical chain mechanisms of photochemical and thermal reactions between chlorine and hydrogen (at moderate temperatures) include  

In the presence of impurities the active centers preferentially interact with the impurity molecules resulting in an immeasurably low reaction rate. Only over a certain time interval known as the induction period, when most impurities are consumed, is the chain start important, i.e. the reaction rate becomes measurable. 

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When the rate of initiation is very low the important moment of chain reaction becomes the kinetics of the establishment of the stationary concentration of free radicals. This time is comparable with the lifetime of the radical that reaction limits the chain propagation. The... 

A polymer-forming chain reaction requires at least one rate-constant, namely that for propagation, k, for its complete specification in this simplest case there is only one type of propagating centre, all the centres are formed in a time which is negligible compared to the duration of the reaction, their concentration remains constant throughout the reaction (Stationary State of the Second Kind), and there is no transfer. 

One common use of the stationary state approximation is with chain reactions. The simplest cases have three types of constituent chemical step, viz. chain initiation, chain propagation and termination. The... 

Under these conditions the increase in G(CH3COCH3) should be paralleled by an increase in G(H202), as observed. If Reactions 8 and 9 are rapid relative to Reaction 18, then it follows from kinetic analysis of the above reaction scheme, assuming a stationary concentration of H02 radicals, that the yield of acetone arising from the chain reaction is given by... 

Explosive reactions) l4)D.A Frank-Kamenetskii, ZhFizKhim 10, 365-70 (1939) 16 357-61 (1942) 20, 729-36 (1943) (Mathematical theory of thermal expin, called 1 stationary theory of thermal expin ) 15)K.K,Andreev, KhimReferatZhur 1940, No 10-11, p 129 CA 37, 1604 (1943) (Mechanism of expl reaction) 16)M.Mucahy A.Yoffe, Australian Chemlnst Jour Proc 11, 106-20 (1944) (chain reactions and gaseous expins) 17)Ibid 11, 134-46 fit 166-74 (1944) (Propagation of gaseous expins. 

Experimental evidence relating to stationary and non-stationary chain mechanisms in gas reactions. 

For ideal radical polymerization to occur, three prerequisites must be fulfilled for both macro- and primary radicals, a stationary state must exist primary radicals have to be for initiation only and termination of macroradicals only occur by their mutual combination or disproportionation. The rate equation for an ideal polymerization is simple (see Chap. 8, Sect. 1.2) it reflects the simple course of this chain reaction. When the primary radicals are deactivated either mutually or with macroradicals, kinetic complications arise. Deviations from ideality are logically expected to be larger the higher the concentration of initiator and the lower the concentration of monomer. Today termination by primary radicals is an exclusively kinetic problem. Almost nothing has been published on the mechanism of radical liberation from the aggregation of other initiator fragments and from the cage of the... 

DHPLC is usually performed on a styrene-divinylbenzene-based polymeric stationary phase, with a mobile phase that contains triethylammonium acetate as the IPR to provide adequate reversed phase (RP) retention for the negatively charged nucleic acid molecules. The samples are usually amplified according to polymerase chain reaction (PCR) protocols and then injected into the chromatographic system. 

It is an interesting result of such c alculations that we can show that the stationary-state concentration of H, (H)s8, is many powers of 10 higher than the equilibrium concentration (H)oq one would get from H2 2H. This, as we shall see, is quite common for chain reactions. 

The approach to the stationary state in a chain system is not instantaneous but takes a finite time which may be calculated from the kinetic mechanism if the individual rate constants for initiation and termination arc known. For the homogeneous chain reaction represented by case 1 Benson has shown how to calculate both the time ta required to reach any fraction a of the stationar37 -state concentration of X, and the fraction of reaction Fa occurring in that time. For thermal initiation ta is given by... 

The stationary-state treatment of this more complex chain reaction can be used to derive an expression that governs the total concentration of all radicals in the system, (H) + (CH3) + (C2IT6). Since the propagation reactions merely replace one radical by another but do not affect the total concentration of radicals, we can write for the stationary state for all radicals that the sum of all initiation reactions is equal to the sum of all termination reactions. This leads to the equation ... 

By measuring the dark period time which corresponds to the time required for the radicals to decay, one obtains the average radical concentration (from the rate of loss of monomer) and finally the ratio Together with the photostationary measurements of kt lkp, this allows kt and kp to be determined separately. This method is useful only in systems in which termination and propagation have different orders with respect to radicals. Although it studies the nonstationary period of the chain reaction, it studies it in a periodic or stationary way. 

As highly reactive species with very short life spans, the chain carriers remain at trace level (except in a detonation). The Bodenstein approximation of quasi-stationary behavior thus is applicable to them. In fact, it was first introduced by Bodenstein and his school in their study and mathematical analysis of chain reactions [1]. Its validity for chain carriers will be taken for granted throughout this chapter. 

This examination also illustrates another facet of chain reactions Granted quasi-stationary conditions and long chains, the concentrations of the chain carriers are coupled, in reactions like 9.5 through the requirement that the rates of the two propagation steps must be equal. Therefore, only one of the two chain carriers can be in dissociation equilibrium with its source, the other gets boosted by the propagation cycle to a higher than thermal concentration. 

The model also revealed the importance of non-linear interactions of chains, which determine the existence of two quasi-stationary oxidation modes. The difference in reaction rate between them is about four orders of magnitude. A critical transition between these modes is taking place due to small changes in reaction parameters, e.g., pressure. This critical transition leads to a very fast increase in the oxidation rate and an abrupt transition from the low-pressure slow oxidation to the fast high-pressure stationary chain-branching process. In the latter mode, radical generation is no more determined by a slow initiation reaction, but by very fast chain-branching reactions. It is the phenomenon... 

Chemical reactions may involve large numbers of steps and participants and thus many simultaneous rate equations, all with their temperature-dependent coefficients. The full set of rate equations is easily compiled as shown in Section 2.4, and to obtain solutions by numerical computation poses no serious problems. With a large number of equations, however, it may become too much of a task to verify the proposed network and obtain values for all its coefficients. Therefore, every available tool must be brought to bear to reduce the bulk of mathematics, and that without unacceptable sacrifice in accuracy. The present chapter critically reviews the principal tools for such a purpose stoichiometric constraints and the concepts of a rate-controlling step, quasi-equilibrium steps, and quasi-stationary states. Other tools useful in catalysis, chain reactions, and polymerization will be discussed in the context of those reactions (see Sections 8.5.1, 9.3, 10.3, and 11.4.1). 

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