372 chain reaction

Chain mechanisms involve reactive intermediates that are called chain carriers. A chain mechanism usually contains an initiation step, in which chain carriers are formed one or more chain propagation steps in which products are formed and in which chain carriers are produced as well as being consumed and a chain termination step in which chain carriers are consumed without being replaced. Since chain carriers are produced as well as being consumed, the reaction can continue without further initiation steps. The following gas-phase reaction has been identified as a chain reaction  

The empirical rate law for the forward reaction in the presence of some HBr is 

Step 1 is the initiation step. The forward reactions of steps 2 and 3 are chain propagation steps, producing the two chain carriers, Br and H. The reverse reaction of step 1 is the termination step. The reverse reactions of steps 2 and 3 regenerate chain carriers but consume the product. They are called inhibition processes. 

To obtain the rate law we apply the steady-state approximation. For a three-step mechanism we must write three differential equations. We choose the time derivatives of [H2] and the concentrations of the two chain carriers [H] and [Br]. We choose [H2] instead of [Br2] or [HBr] because H2 occurs in only one step of the mechanism and will give a simpler differential equation. The simultaneous differential equations are 

We have applied the steady-state approximation and set the time derivatives of the concentration of the chain carriers H and Br equal to zero. To solve the algebraic versions of Eqs. (12.5-4b) and (12.5-4c), we addEqs. (12.5-4b) and (12.5-4c) to give 

Chain reactions are a t3q)e of overall reactions, which require two or more steps to accomplish. They are also known as consecutive reactions or sequential reactions. Examples of chain reactions include nuclear hydrogen burning, nuclear decay chains, ozone production, and ozone decomposition. Some steps of a chain reaction may be rapid and some may be slow. The slowest step is the ratedetermining step. During a chain reaction, some intermediate and unstable species may be produced and consumed continuously. 

Chain reactions may lead to either steady state (in which concentration of intermediate species reaches a constant) or explosion (in which the reaction rate increases exponentially with time). If the concentrations of intermediate catalyst species reach a constant quickly, there would be a steady state. If the concentrations of intermediate catalyst species grow exponentially, there would be explosion. 

Propagation or chain transfer interaction of an active intermediate with the reactant or product to produce another active intermediate 

Termination deactivation of the active intermediate to form products 

An example comparing the application of the PSSH with the Polymath solution to the full set of equations is given on the DVD-ROM for the cracking of ethane. Also included is a discussion of Reaction Pathways and the chemistry of smog LivingExsfnpie Problem formation. 

The mechanism of the basic feature of a chain reaction may be illustrated by that of a free-radical reaction, particularly the polymerization of styrene, which has been extensively investigated for years by many investigators. Here we describe the mechanism proposed by Mayo and co-workers (1951, 1959). 

A free-radical reaction may go through all or some of the following steps  

Biradical from thermal initiation Nonradical products Monoradical Chain radical 

Monoradicals and chain radicals are the same. The chain has not propagated as yet hence, both equations are labeled (2.4). BZ2O2 is benzoyl peroxide. The term kd is the rate constant of the decomposition of peroxide, ki is the rate constant of the thermal initiation of the biradical, k is the rate constant of the first-order recombination of radicals from the peroxide, and k is the rate constant of the reaction of these radicals with monomer  

The value off ranges between 0.5 and 1 and is usually 0.70. The rate of termination 

An understanding of the properties and behavior of nuclear chain reactors is achieved through a study of the neutron population which supports the chain. Information about the neutron population is conveniently expressed in terms of the neutron-density-distribution function. 

The detailed features of the chain reaction are determined by the various nuclear processes which can occur between the free neutrons and the materials of the reactor system. As in chemical chain reactions, the rates of the reactions involved in the chain are directly dependent upon the density of the chain carrier, in the present case the neutrons. Thus in order to determine the various properties of a reactor, such as the power-production rate and the radiation-shielding requirements, it is necessary to obtain the fission reaction rate throughout the system and, therefore, the neutron-density distribution. In fact, all the basic nuclear and engineering features of a reactor may be traced back ultimately to a knowledge of these distribution functions. 

The subject of reactor analysis is the study of the analytical methods and models used to obtain neutron-density-distribution functions. Since these functions are intimately related to various neutron-induced nuclear reactions, a knowledge of at least the basic concepts of nuclear physics is essential to a thorough understanding of reactor analysis. 

The first section of this chapter is a brief discussion of those aspects of nuclear reactions which are of principal interest to reactor physics. This presentation assumes that the reader is equipped with an introductory course in nuclear physics. The second section is an outline of the basic nuclear components of reactors and of the various types of reactors, and the last section is a summary of the principal problems of reactor physics and the analytical methods of attack. It is intended that the last section be used primarily for purposes of review and to aid the reader in orienting the various topics with regard to the over-all structure and scope of the subject. 

This information is included here primarily as a convenient reference for explaining the physical features of neutron phenomena relevant to reactors. 

To illustrate the nature of chain reactions, we consider a classic example the formation of hydrogen bromide from molecular hydrogen and bromine in a homogeneous gas-phase reaction  

This stoichiometric equation does not give any indication of the complex nature of the mechanism by which this reaction proceeds. 

In certain highly energetic collisions with any molecule M in the system, a bromine molecule may be dissociated in a homolytic split of the bond joining two bromine atoms  

The collision must be sufficiently energetic that enough energy is available to break the chemical bond linking the two bromine atoms. This type of reaction is called an initiation reaction because it geueiates a species that can serve as 

Equations (4.2.3) and (4.2.4) are the elementary reactions responsible for product formation. Each involves the formation of a chain carrying species [H- for (4.2.3) and Br-for (4.2.4)] that propagates the reaction. Addition of these two relations gives the stoichiometric equation for this reaction. These two relations constitute a single closed sequence in the cycle of events making up the chain reaction. They are referred to as propagation reactions because they generate product species that maintain the continuity of the chain. 

Gas-phase radiolysis can sometimes result in chain reactions involving H atoms or other radicals. As in other cases with chain reactions, termination is due to either recombination or reaction with other radicals. Typical chain length is -1000 or more. Some specific examples will be considered in Sect. 5.2. 

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 

Rice—Herzfeld mechanism for alkane pyrolysis is an example which, in its 

B and E are the chain carriers, D and F are the most abundant stable products. Other termination and propagation reactions, in which C may participate, are also possible. If only steps (44)—(47) occur and molecu-larity corresponds to reaction order, application of the stationary state approximation to B and E leads to the prediction that 

Equation (48) e ees with experimental results in some circumstances. This does not mean the mechanism is necessarily correct. Other mechanisms may be compatible with the experimental data and this mechanism may not be compatible with experiment if the physical conditions (temperature and pressure etc.) are changed. Edelson and Allara [15] discuss this point with reference to the application of the steady state approximation to propane pyrolysis. It must be remembered that a laboratory study is often confined to a narrow range of conditions, whereas an industrial reactor often has to accommodate large changes in concentrations, temperature and pressure. Thus, a successful kinetic model must allow for these conditions even if the chemistry it portrays is not strictly correct. One major problem with any kinetic model, whatever its degree of reality, is the evaluation of the rate cofficients (or model parameters). This requires careful numerical analysis of experimental data it is particularly important to identify those parameters to which the model predictions are most sensitive. 

In many cases it is difficult or impossible to isolate a particular step in a reaction scheme and evaluate the rate coefficient. If a simplified kinetic 

A discussion of luminescence is continued on the CD-ROM Web Module. Glow Sticks. Here, the PSSH is applied to glow slicks. First, a mechanism for the reactions and luminescence is developed. Next, mole balance equations are written on each species and coupled with rate law obtained using the PSSH and the resulting equations are solved and compared with experimental data. 

let us proceed lo some slightly more complex examples involving chain reactions. A chain reaction consists of the following sequence  

Term illation deactivation of the active imermediate to form products. 

The thermal decomposition of ethane to ethylene, methane, butane, and hydrogen is believed to proceed in the following sequence  

Certain constituents when added to the reaction mixture, slow down the rate of reaction. This phenomena is called inhibition and constituent called inhibitor. Such an effect is similar to the negative catalysis. But the constituent usually undergoes chemical change, inhibition is the preferred term. Inhibition may occur in chain reactions, enzyme catalysed reactions, surface reactions or many reversible or irreversible reactions. A trace amount of an inhibitor may cause a marked decrease in the rate of reaction. The inhibitor sometimes combines with a catalyst and prevents it from catalyzing the reaction. 

In chain reactions, the inhibitor 7n interacts with free radical P and makes it unavailable for the propagation of chain, thus providing an extra termination step 

The total rate of termination will be rate of termination due to inhibitor. Let us consider a general scheme 

With the help of steady-state approximation, [F°] can be obtained as 

Rate law (6.55) clearly indicates the retardation of rate on increasing the concentration of inhibitor. 

If we compare the energies needed to form HCl from the radicals with the total energy required for the formation of HCl from CI2 and H2 molecules, we found that the first is much lower due to the high reactivity of radicals. The radicals are 

This observation was accounted for by the assumption that the reaction occurs in a series of steps. The first is the dissociation of a bromine molecule 

This is followed by a sequence of reactions, called a chain  

A mixture of hydrogen and chlorine explodes when ignited. The overall reaction H2 + CI2 — 2HC1, which takes place by the chain mechanism analogous to that for H2 + Br2, liberates so much heat that the gas may begin to increase in temperature, instead of dissipating the heat to the environment. The reaction then proceeds more rapidly, the temperature increases more rapidly, and a very rapid reaction, called a thermal explosion, results. 

Open shell atoms and other molecular species that act as free radicals play a special role as intermediates in reaction mechanisms. Because they have incomplete electron shells they are usually highly reactive, even with stable molecules at ordinary temperatures. The concentrations of such free radicals in these reaction systems are usually low and the application of the steady-state hypothesis to them is valid. 

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The effect of a slow or "thermar neutron on a nucleus of is to split it into one or more neutrons and into large fragments of approximately equal mass. There is a liberation of energy equal to the loss in total mass, If the neutrons produced effect further fissions, a chain-reaction of successive fissions may be set up. Am and... 

Am undergo fission with thermal neutrons of these isotopes and Pu are the most important as they are most readily obtainable. Other heavy nuclei require fast neutrons to induce fission such neutrons are much more difficult to control into a self-sustaining chain-reaction. 

In nuclear chemistry, a fission reaction (see atomic energy) may be initiated by a neutron and may also result in the production of one or more neutrons, which if they reacted in like manner could start a chain reaction. Normally, moderators such as cadmium rods which absorb neutrons are placed In the reactor to control the rate of fission. 

After the primary step in a photochemical reaction, the secondary processes may be quite complicated, e.g. when atoms and free radicals are fcrnied. Consequently the quantum yield, i.e. the number of molecules which are caused to react for a single quantum of light absorbed, is only exceptionally equal to exactly unity. E.g. the quantum yield of the decomposition of methyl iodide by u.v. light is only about 10" because some of the free radicals formed re-combine. The quantum yield of the reaction of H2 -f- CI2 is 10 to 10 (and the mixture may explode) because this is a chain reaction. 

Additives function by reacting with hydrocarbon partial oxjdation products by stoppihg the oxidation chain reaction that would otherwise driye the combustion. 

The second category of polymerization reactions does not involve a chain reaction and is divided into two groups poly addition and poly condensation [4]. In botli reactions, tire growth of a polymer chains proceeds by reactions between molecules of all degrees of polymerization. In polycondensations a low-molecular-weight product L is eliminated, while polyadditions occur witliout elimination ... 

Mixtures of chlorine and hydrogen reaa only slowly in the dark but the reaction proceeds with explosive violence in light. A suggested mechanism for the photochemical chain reaction is ... 

Bromine, like chlorine, also undergoes a photochemical chain reaction with hydrogen. The reaction with bromine, however, evolves less energy and is not explosive. 

This equilibrium has been extensively studied by Bodenstein. Unlike the other halogen-hydrogen reactions, it is not a chain reaction but a second order, bimolecular, combination. 

CR Polymerase Chain Reaction. Widely used method for amplifying a DNA base sequence... 

W. B. Motherwell, D. Crich Free Radical Chain Reactions in Organic Synthesis (Academic Press 1992)... 

Uranium-235 is of even greater importance because it is the key to utilizing uranium. 23su while occuring in natural uranium to the extent of only 0.71%, is so fissionable with slow neutrons that a self-sustaining fission chain reaction can be made in a reactor constructed from natural uranium and a suitable moderator, such as heavy water or graphite, alone. 

There were two schools of thought concerning attempts to extend Hammett s treatment of substituent effects to electrophilic substitutions. It was felt by some that the effects of substituents in electrophilic aromatic substitutions were particularly susceptible to the specific demands of the reagent, and that the variability of the polarizibility effects, or direct resonance interactions, would render impossible any attempted correlation using a two-parameter equation. - o This view was not universally accepted, for Pearson, Baxter and Martin suggested that, by choosing a different model reaction, in which the direct resonance effects of substituents participated, an equation, formally similar to Hammett s equation, might be devised to correlate the rates of electrophilic aromatic and electrophilic side chain reactions. We shall now consider attempts which have been made to do this. 

The suitability of the model reaction chosen by Brown has been criticised. There are many side-chain reactions in which, during reaction, electron deficiencies arise at the site of reaction. The values of the substituent constants obtainable from these reactions would not agree with the values chosen for cr+. At worst, if the solvolysis of substituted benzyl chlorides in 50% aq. acetone had been chosen as the model reaction, crJ-Me would have been —0-82 instead of the adopted value of —0-28. It is difficult to see how the choice of reaction was defended, save by pointing out that the variation in the values of the substituent constants, derivable from different reactions, were not systematically related to the values of the reaction constants such a relationship would have been expected if the importance of the stabilization of the transition-state by direct resonance increased with increasing values of the reaction constant. 

The hydrogenolyaia of cyclopropane rings (C—C bond cleavage) has been described on p, 105. In syntheses of complex molecules reductive cleavage of alcohols, epoxides, and enol ethers of 5-keto esters are the most important examples, and some selectivity rules will be given. Primary alcohols are converted into tosylates much faster than secondary alcohols. The tosylate group is substituted by hydrogen upon treatment with LiAlH (W. Zorbach, 1961). Epoxides are also easily opened by LiAlH. The hydride ion attacks the less hindered carbon atom of the epoxide (H.B. Henhest, 1956). The reduction of sterically hindered enol ethers of 9-keto esters with lithium in ammonia leads to the a,/S-unsaturated ester and subsequently to the saturated ester in reasonable yields (R.M. Coates, 1970). Tributyltin hydride reduces halides to hydrocarbons stereoselectively in a free-radical chain reaction (L.W. Menapace, 1964) and reacts only slowly with C 0 and C—C double bonds (W.T. Brady, 1970 H.G. Kuivila, 1968). 

The second mechanism is the one followed when addition occurs opposite to Markovmkov s rule Unlike electrophilic addition via a carbocation intermediate this alternative mechanism is a chain reaction involving free radical intermediates It is pre sented m Figure 6 7... 

Section 11 10 Chemical reactions of arenes can take place on the ring itself or on a side chain Reactions that take place on the side chain are strongly influ enced by the stability of benzylic radicals and benzylic carbocations... 

FIGURE 28 14 The poly merase chain reaction (PCR) Three cycles are shown the target region appears after the third cycle Additional cycles lead to amplification of the target region... 

Cation (Section 1 2) Positively charged ion Cellobiose (Section 25 14) A disacchande in which two glu cose units are joined by a 3(1 4) linkage Cellobiose is oh tamed by the hydrolysis of cellulose Cellulose (Section 25 15) A polysaccharide in which thou sands of glucose units are joined by 3(1 4) linkages Center of symmetry (Section 7 3) A point in the center of a structure located so that a line drawn from it to any element of the structure when extended an equal distance in the op posite direction encounters an identical element Benzene for example has a center of symmetry Cham reaction (Section 4 17) Reaction mechanism m which a sequence of individual steps repeats itself many times usu ally because a reactive intermediate consumed m one step is regenerated m a subsequent step The halogenation of alkanes is a chain reaction proceeding via free radical intermediates... 

Polyethylene (Section 6 21) A polymer of ethylene Polymer (Section 6 21) Large molecule formed by the repeti tive combination of many smaller molecules (monomers) Polymerase chain reaction (Section 28 16) A laboratory method for making multiple copies of DNA Polymerization (Section 6 21) Process by which a polymer is prepared The principal processes include free radical cationic coordination and condensation polymerization Polypeptide (Section 27 1) A polymer made up of many (more than eight to ten) amino acid residues Polypropylene (Section 6 21) A polymer of propene Polysaccharide (Sections 25 1 and 25 15) A carbohydrate that yields many monosacchande units on hydrolysis Potential energy (Section 2 18) The energy a system has ex elusive of Its kinetic energy... 

Propagation steps (Section 4 17) Elementary steps that repeat over and over again in a chain reaction Almost all of the products in a chain reaction arise from the propagation steps... 

Termination steps (Section 4 17) Reactions that halt a chain reaction In a free radical chain reaction termination steps consume free radicals without generating new radicals to continue the chain... 

The mechanism of these reactions places addition polymerizations in the kinetic category of chain reactions, with either free radicals or ionic groups responsible for propagating the chain reaction. 

Chain reactions do not go on forever. The fog may clear and the improved visibility ends the succession of accidents. Neutron-scavenging control rods may be inserted to shut down a nuclear reactor. The chemical reactions which terminate polymer chain reactions are also an important part of the polymerization mechanism. Killing off the reactive intermediate that keeps the chain going is the essence of these termination reactions. Some unusual polymers can be formed without this termination these are called living polymers. 

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