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THE EVALUATION OF SPECIFIC REACTION RATES IN CHAIN REACTIONS1 G. K. ROLLEFSON Department of Chemistry, University of California, Berkeley, California Received May 25, 1988 One task involving photochemical chain reactions which has been under- taken by many investigators is that of evaluating the specific reaction rates (or rate constants) for the various steps which occur. The problems which arise with each reaction studied are so similar that the discussion will be simplified if we consider the three main types of processes: (~) chain starting, (2) chain terminating, (~) chain continuing. I. CHAIN-STARTING REACTIONS A consideration of the mechanisms which have appeared in the literature leads to the conclusion that chains are started by some kind of an odd molecule, either an atom or a free radical., and are propagated by the alter- nate formation and disappearance of such molecules. Therefore, in this discussion, we are concerned with the rate of formation of odd molecules as a result of light absorption. In the simple cases, such as the absorption of light by diatomic molecules in the gaseous state in a continuous absorp- tion band, it has been quite well established that dissociation occurs for every light quantum absorbed. The same may be said for the truly con- tinuous absorption by more complex molecules, although, in such cases, it may be difficult to decide whether the spectrum is really continuous or only appears to be so because of inadequate resolution. Very often, how- ever, we must consider a competition between the dissociation process and otter processes such as fluorescence, deactivation by collision, or some reaction of the photoactivated molecule, either by itself or with other molecules, which does not involve the formation of odd molecules. For example, the photodecomposition of acetaldehyde at high temperatures is a chain reaction probably involving methyl radicals (15~. From the work on this reaction at lower temperatures we know that, in the first stage, we have competition between fluorescence, a polymerization reac- tion, and probably a direct decomposition into methane and carbon monox- ide as well as the dissociation to give methyl radicals (16, 27~. It is appar- ~ Contribution No. 7 to the Third Report of the Committee on Photochemistry, National Research Council. 75
76 G. K. ROLLEFSON ent, therefore, that in this case it would be definitely inaccurate to set the number of chains started equal to the number of quanta of light absorbed. This uncertainty is involved in every reaction in which the radicals are produced by a predissociation process. A similar uncertainty is intro- duced if we have two overlapping absorption bands corresponding to transitions to two difl Brent excited states. Even with such a simple mole- cule as chlorine, Aickin and Bayliss (1) have shown that the continuous absorption is complex and there is a continuum underlying the sharp line bands. In this case Bayliss (3) has shown that the observed facts con- cerning the combination of hydrogen and chlorine caused by light absorbed in the banded region of the spectrum can be accounted for by assuming that only the continuous absorption starts the reaction chains. Another source of uncertainty as to the efficiency of the dissociation process was suggested by Franck and Rabinowitch (11~. They expressed the view that the quantum yield of a primary dissociation process in solu- tion must be very low, as the surrounding molecules will prevent the sepa- ration of the parts. Rollefson and Libby (28) pointed out that such an eject should be observed if the speed of the separating parts is small, but most experiments have been performed with such energies that the parts would be separating with relatively large kinetic energies and thus be able to break through the surrounding cordon of solvent molecules, making the dissociation practically as efficient as in the gas phase. Dickinson (8) has discussed a number of reactions in solution and shown that they could be most readily explained by assuming that the first step was a dissociation of the light-absorbing molecule. Rabinowitch (22) has objected to such arguments on the grounds that the rate considerations used by Dickinson and others depend on the steady state concentrations of radicals or atoms rather than on their rates of formation, and these steady state values could be unchanged if the rates of dissociation and recombination were affected equally by the solvent. Rollefson and Libby pointed out that such a symmetrical modification was unlikely. Furthermore, it must be said that it is difficult to conceive of chains which are unaffected by dilution having a length such that the quantum yield of the overall reaction would be 2; yet such an assumption is necessary for C102 in carbon tetrachlo- ride solution if we do not assume a high efficiency for the primary step. II. CHAIN-TERMINATING REACTIONS The best known of the chain-terminating reactions are those involving the recombination of atoms. Many studies with atomic hydrogen have shown that the recombination occurs at every collision during which a third body is present to remove some of the energy (2, 10, 32~. Similarly, a comparison of the thermal and photochemical rates of formation of hydro- gen bromide has led to the conclusion that bromine atoms recombine by a
REACTION RATES IN CHAIN REACTIONS 77 triple collision mechanism (5, 14~. Ritchie (23) and Hilferding and Steiner (13) have studied the relative efficiencies of different molecules as the third body. Some idea of the range encountered is given by rate con- stants for the reaction Br + Br + M = Br2 + M + k.e. taken from the paper by Hilferding and Steiner (see table 1~. The varia- tion is very similar to that found for the quenching of fluorescence by these gases. On the basis of these observations it seems reasonable to assume that any atom recombination process occurs at approximately every triple collision if the reaction is homogeneous. The heterogeneous recombination of atoms depends quite markedly on the nature of the surface involved. It was found in the very first experi- ments with hydrogen atoms that dry glass or quartz surfaces are very much more effective in causing the recombination than ones which had not been dried (33~. Metallic surfaces were also found to be very eff ective in causing recombination. Experiments with the hydrogen-bromine TABLE ~ Rate constants for Br + Br ~ M M............... k X 10-15......... . . . | H2 | He | A | N2 | Br2 | HBr | HC1 | CO ... 1 1.25 1 0.47 1 0.11 1 0.82 1 2.6 2.1 1 4.7 1 6.3 reaction at such pressures that the bromine atoms recombine on the wall show that the rate depends on the previous treatment of the walls (13~. Another example is found in the hydrogen-chlorine reaction in which Bodenstein and Winter (6) calculated that only one collision in six thousand on a silver chloride surface resulted in removal of the chlorine atom. A comparison of their quantum yields with values obtained in the presence of glass surfaces indicates that glass is about ten times as effective. The failure of the atoms to react at every collision with the surface does not seem to be due to the requirement of any heat of activation, but rather to a low value of the accommodation coefficient. This idea is supported by a comparison of the photochemical temperature coefficients for the hydro- gen-chlorine reaction as obtained by Hertel (12) and by Potts and Rollef- son (21) with the value obtained by Rodebush and Klingelhoefer (24) for the reaction of chlorine atoms with hydrogen molecules, which shows that the latter reaction is capable of accounting for the entire temperature co- efficient of the former. The principal difficulties in the way of securing exact values for the rate constant of such a chain-terminating reaction are the determination of the accommodation coefficient and the rate of ap- proach to the wall. The latter rate is complicated by the fact that, usually, the heat of reaction sets up convection currents which make it
78 G. }I. ROLLEFSON virtually impossible to decide on an average path length. On the whole, the error in the estimation of the rate of a chain-terminating process in- volving atoms is probably not greater than a factor of one hundred, whether it is a homogeneous or a heterogeneous reaction. If the chain-terminating reaction involves more complex groups, more varieties of reaction are introduced. The surface reactions and the triple collision mechanism for association reactions are possible here as well as with atoms. However, it cannot be said with certainty that the associa- tion reactions do not involve heats of activation. Furthermore, it is possible for two radicals to combine to form a single molecule by a process which is the reverse of predissociation without having a third body present. In such a case, the quasi-molecule formed would have a sufficiently long life to lose some of its energy in a collision and become a stable molecule. Finally we have the possibility of two radicals reacting to form normal molecules. Some examples of these types of reaction which have been assumed are CO Cl + Cl ~ CO + C12 CH3 + CH3 > C2H6 C2H5 + C2H5 ~ C2H4 + C2H6* (1) (2) (3) * As the concentrations of these radicals are always very low, bimolecular proc- esses involving them must be considered improbable. The evidence for these reactions is not very conclusive, as the experiments have been such that other possibilities have been present. In support of reaction 2 we may cite the formation of ethane in the photolyses of lead tetramethyl (17), acetone (18), or methyl ethyl ketone (19~. Reaction 3 has been assumed frequently, but probably the best evidence for it is the formation of ethane and ethylene in the photolysis of ethyl iodide (9~. The magnitudes of the activation energies for these reactions are at present unknown. They are probably not large, but even a small activation energy would introduce a rather large uncertainty into the specific reaction rate. Until further data are available, we must conclude that the constants for reactions between two radicals are not sufficiently well known to be used in calculating rates of photochemical chain reactions. IlI. CHATN-CONTINUING REACTIONS The direct measurement of the rate constants for reactions of the type involved in the propagation of chains has been limited to a very few cases involving atoms. Most of these reactions involve hydrogen or one of the halogen atoms. The specific rate of the reaction Br + H2~HBr + H
REACTION RATES IN CHAIN REACTIONS 79 can be obtained from the thermal rate of formation of hydrogen bromide if we assume that bromine atoms and molecules are in equilibrium in the reaction mixture. On the basis of data obtained from the experimental study of both the thermal and photochemical formation of hydrogen bromide, Bodenstein and Lutkemeyer (5) give for the rate constant log k = - 38T9 + 13.862 which corresponds to an activation energy of 17,640 cal. The reaction is endothermic to the extent of 14,500 car., so the activation energy is only slightly greater than the energy required to offset the endothermicity of the reaction. The study of the rate of formation of hydrogen bromide also tells us that the ratio of the rate constants for the reactions H + Bra >HBr + Br and H + HBr~H2 + Br is 8.6 over a very wide temperature range. This fact suggests that the heats of activation for these reactions are zero, the only difference being in the so-called "steric factor". The rate of the reaction Cl + H2 > HCl + H was measured directly by Rodebush and Klingelhoefer (24), who used a known concentration of.chlorine atoms produced by an electric discharge in chlorine gas. They found that the rate was given essentially by the number of collisions multiplied by e-6000/RT with the uncertainty in the numerator of the exponent being approximately 1000 cal. A similar value was obtained from the measurements by Hertel (12) and by Potts and Rollefson (21) of the temperature coefficient of the photochemical reaction of oxygen-free mixtures of hydrogen and chlorine, if it was assumed that the chain-terminating step involved no heat of activation. A number of reactions of hydrogen atoms have been studied by preparing a measurable concentration of the atoms by means of an electrical dis- charge and passing these atoms into some other gas. The most important reaction studied by this method is the transformation of pare-hydrogen into or/ho-hydrogen. The fraction of the collisions between hydrogen atoms and hydrogen molecules which result in reaction is 2 X 10-6 and the heat of activation is approximately 7000 cal. Other reactions which have been tried include those with oxygen, water, the halogens, the hydro- gen halides, some hydrocarbons, hydrogen sulfide, and methyl halides. Under these conditions most of these reactions proceed too fast to obtain accurate measurements, so the only conclusion which can be drawn is that the hydrogen atoms are destroyed in less than one hundredth of a second,
80 G. K. ROLLEFSON which means that at least one collision in 104 is effective. energy is therefore either zero or very small. The activation Other methods which have been used to obtain estimates of the rates of the steps in a chain process are (1) studies of the overall rate with appro- priate assumptions concerning the chain-starting and chain-terminating reactions, and (2) studies of systems in which two reactions compete for the same reactive intermediate. The validity of both of these methods depends on the assumption that the mechanism used in the calculation is correct. It is essential therefore that any other mechanisms be excluded on the basis of the experimental evidence before rate constants obtained by these methods may be considered valid. We may illustrate the first method by referring to the formation of hydrogen chloride from the ele- ments. The mechanism which seems to explain the behavior of oxygen- free mixtures of hydrogen and chlorine is expressed by the following equations: k3 Cl2+hY >2C1 (1) k2 Cl + H2 > HCl + H k3 H + Cl2 ~ HC1 + Cl k4 C1 > 1/2 C12 (on the walls of the reaction vessel) This leads to the rate expression (2) (3) (4) d(HCl) kik2 dt k Iabs.(H2)' If the light absorbed corresponds to the continuum in the chlorine spec- trum, there is plenty of evidence to support the assumption that ki = 2. The constant k4 is subject to much greater uncertainty. Usually, it is assumed that every collision of an atom with a surface is effective in causing recombination, but recently Bodenstein and Winter (6) have presented data which indicate that only one collision in six thousand of chlorine atoms with a silver chloride surface leads to the formation of molecules. This efl ect does not seem to be due to a heat of activation but rather is an accommodation coefficient analogous to the steric factor for bimolecular reactions. Collisions with glass or quartz surfaces are much more effec- tive, but it is probable that the erratic rates of formation of hydrogen chloride that have been reported are at least partially due to the variation of this k4 in different experiments. If we assume that reaction 4 has no heat of activation, then the entire heat of activation for the overall rea¢- tion is due to reaction 2. The value of k2 may be calculated approximately by multiplying the collision number by e-Q/T, where Q is the heat of activation. The efficiency factor for collisions between molecules possess- ing the necessary energy cannot be determined any more exactly than we
REACTION RATES IN CHAIN REACTIONS 81 know k4. Usually in systems of this type the uncertainty in the heat of activation is sufficient to mask any uncertainty in the collision number or efficiency factor. This type of calculation may be applied, with the same degree of approxi- mation, to any other chain reaction involving atoms in the initial and final steps and having the rate determined by one step in the chain process. If radicals are involved in the initial or final steps, both of the uncertainties become much greater particularly on account of the activation energy of the chain-term~nat~ng step. Thus a reaction such as 2CCl3 + Cl2- > 2CCl4 which Schumacher and Wolf (31) assumed to be the chain-term~nat~g step in the chlorination of Chloroform probably requires some heat of activation, but the magnitude of this heat is not even approximately known. The same statement can be made about the other processes which we have discussed in the section on chain-terminating processes. A further complication arises frequently owing to the complexity of the assumed mechanism. Thus, instead of having the overall rate constant expressed in terms of the constants for initial and final steps and one step of the chain (k2 in the case of the formation of hydrogen chlorides we find that the constants of two or more steps of the chain appear in the rate equation. Such constants are indeterminate from rate measurements alone, and up to the present time no one has determined the constants for any system of this type from experimental data.2 The second method for evaluating the constants of steps in a chain, the use of competitive reactions, has not been used very extensively as yet. We have already cited the competition between hydrogen bromide and bromine for hydrogen atoms in the formation of hydrogen bromide. Other examples which have been studied quantitatively include the competition of carbon monoxide and hydrogen for chlorine atoms (4, 26), of oxygen and chlorine for COCl (25, 29), and of ozone and oxygen for oxygen atoms (30~. The experiments determine only the ratio of the two rate constants but if one rate constant, as, for example, that for the reaction between atomic chlorine and hydrogen, is known from other studies, then the other can be calculated. We have already seen that the rate constant for the reaction Cl + He > HCl + H 2 Bodenstein and his students claim to have made such determinations for the formation of phosgene, but the details of their calculations have not been published. It may be remarked here that in their published work they have neither proven their mechanism to the exclusion of others nor listed enough independent equations based on experimental data to determine all of the constants involved in their rate equa- tions. See, however, contribution No. 9 to this Report.
82 G. E. ROLLEFSO~- ;s pretty well known, therefore we may use this reaction as a means of measuring chlorine-atom concentrations in reaction mixtures and thus the specific rates of other reactions. Naturally this method is limited to those systems in which the chlorine atoms react at a rate comparable with that with hydrogen. Thus, the observation that in mixtures of ethylene, hydrogen, and chlorine the halogen adds to the ethylene with no appreci- able formation of hydrogen chloride tells us that the first step in the addi- tion reaction is very fast but does not permit an exact calculation of its specific rate. Many reactions of hydrogen atoms have been studied by determining the concentration of the atoms by the rate of the conversion of para-hydro- gen to or/ho-hydrogen. This is essentially the method of competing rates, except that the atoms are not destroyed by the test reaction. The rate constant for this conversion has been given as 2 X 109Te-7°°°/RT, which indicates the order of magnitude of the rates which may be studied by this method. One point which has been overlooked in some investigations is that if the hydrogen atoms react very rapidly with other substances in the reaction mixture the concentration of the atoms may not be great enough to cause appreciable conversion of pare-hydrogen to or/ho-hydrogen. This method has been applied by Cremer, Curry, and Polanyi (7) to the study of reactions of atomic hydrogen with alkyl halides. Their results were only semiquantitative and their experimental method limited them to reactions for which the activation energy was in the range 2800 < Q < 7200 cal. Farkas and Sachsse (10) studied H + 02 > HOe and H + CO > HCO. They concluded that the former occurred once in seven hundred fifty triple collisions and the latter once in ten triple collisions. Patat (20) has also used this method to determine the concentrations of hydrogen atoms present during the decompositions of a number of organic compounds. On account of the complexity of such systems only qualita- tive results concerning rate constants were obtained. In conclusion, it must be stressed that the great need at present is the determination, by methods free from assumptions, of a few rate constants for reactions of the type we have discussed. No matter how reasonable assumptions may seem, any rate constants based on them are little better than guesses. This is especially true in complex systems as, under such conditions, usually several mechanisms are capable of describing the facts, and hence there is no certainty that we are dealing with the right set of reactions. The constants which have been calculated for such systems in the literature must be looked upon as reasonable interpretations rather than as established facts.
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