The Elucidation of Reaction Mechanisms by the Method of Intermediates in Quasi-Stationary Concentrations

The Elucidation of Reaction Mechanisms by the Method of Intermediates in Quasi-Stationary Concentrations J. A. CHRISTIANSEN Institute of Physico-Chemistry, University oj Copenhagen, Denmark Page I. Introduction: The Correspondence between Kinetics and Mechanism . . . 311 11. Gibbs' Fundamental Rule of Stoichiometry. . . . . . . . . . . . . . . 314 111. Intermediate Products and Sequences. . . . . . . . . . . . . . . . . 315 1. The Intermediates . . . . . . . . . . . . . . . . . . . . . . . . 318 2. The Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . 318 3. Qualitative Properties of Open and Closed Sequences. The Definition of Catalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 IV. Calculation of Stationary Velocities and Concentrations. . . . . . . . 325 1. The Partition Matrix. . . . . . . . . . . . . . . . . . . . . . 325 2. Applications. Cancelling of Matrix Elements . . . . . . . . . . . 329 3. The Analogy between Chemical Reaction and Diffusion. . . . . . . . 330 4. The Closed Linear Sequence: Catalysis and Enzymatic Reactions. . . . 332 5. Orientation of Diagrams. . . . . . . . . . . . . . . . . . . . . . 334 6. Branched Sequences. Cyclic Sequences. . . . . . . . . . . . . . . 337 7. The Stationary Velocities of a Branched Sequence: hegative Catalysis . 338 V. Integration of the Velocity Expressions and Comparison with Experiments 343 1. Determination of the Best Possible Values of the Constants . . . . . . 346 2. The Dependence of the Velocity Constants on Temperature. . . . . . 348 VI. Conclusive and Historical Remarks . . . . . . . . . . . . . . . . . . 350 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

I. INTRODUCTION: THE CORRESPONDENCE BETWEEN KINETICS AND MECHANISM

Since the early days of chemical kinetics it has been known that the stoichiometrical equation of a chemical reaction does not necessarily represent its mechanism. Very often the mechanism must be represented by a sequence of partial or component reactions which on addition give the overall reaction. This is evidently the case with catalytic reactions, but an immense number of noncatalytic reactions are known for which the same is true. It is one of the purposes of chemistry to set up the proper mechanisms of the chemical reactions and thus in case of complicated mechanisms to give information on the number and nature of intermediate steps. T o accomplish this we must study experimentally the kinetics of the reactions, and from the information thus obtained, try to construct a mecha311

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nism which will agree with the kinetics and with the overall reaction. It is recognized that this often represents great problems, but still many cases are known, especially among gaseous reactions where such results have been achieved. This also applies to a number of reactions in solution. Concerning the reliability of a proposed mechanism the experimental data and numerical results must be considered critically. T o be able to assert with certainty that a proposed mechanism is the only one which conforms with the experimentally observed kinetics demands sufficiently accurate and complete experiments. If the experimental conditions have been such that the experimental results may be regarded a s being incomplete or not sufficiently arcurate, it is impossible t o claim the proposed mechanism to be the only one which will conform with the experimentally observed kinetics. The latter remark refers t o the works in which experimental errors are overestimated to make the results fit a chosen mechanism. An overestimation of the accuracy of experimental results is out of place, but an underestimation may have more serious consequences in the said respect. If the differences between found and calculated values are systematic it is nearly certair: that the assumed sequence is erroneous. I n a not too complicated case, e.g., in the case of the hydrogen-bromine reaction investigated so thoroughly by the Bodenstein school, it can be shown with certainty that its well known mechanism is really the only one which conforms with the (very accurate) experiments by Bodenstein and Lind and later experimenters. In this arid often in more complicated cases, too, the calculations permit expressing the time from the beginning of the experiment by means of the sum of a number of known functions of the degree of advancement z, each multiplied by a constant. If the constants can be chosen so that if the measured values of 2 are inserted the sum becomes equal to the corresponding measured times, this is a strong evidence that the proposed mechanism is the right one, provided of course that the partial reactions in the proposed mechanism add up t o the actual total reaction. The relation between time and the degree of advancement should, however, hold over a very wide range from the beginning of the reaction nearly t o completion, and should also hold when the original concentrations (but not temperature or pressure) are varied. Concerning the possibility of the existence of another mechanism which should simultaneously conform with the measured kinetics, i t must be noted that if the assumptions concerning the mechanism are altered the relation between time and degree of advancement will be altered qualitatively, i.e., the number, the nature, and the dependence on the original concentrations of the functions will be altered simultaneously or

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separately. This will be due to the discontinuity of matter, and the finite number of valency states in which an atom or molecule can be found, thus permitting only a finite number of alterations in the reaction sequences. If, therefore, agreement is arrived a t when one mechanism is assumed, it will in general be impossible t o get agreement with another. I n this connection there may be an objection that if a n expression is given, e.g., for time as a function of the degree of advancement, and this contains a number of constants, one can always choose the constants so as to make the theoretically predicted curve fit the experimental curve. This would be possible if we could choose at will certain families of functions and if we were free to dispose of many constants. B u t the functions appearing in the equations cannot be chosen arbitrarily since they follow as logical consequences of the assumed mechanism, and the number of constants is restricted by the finite number of partial reactions in the sequence. Only if the range of the degree of advancement is small and if the precaution to make experiments with different initial concentrations is not taken, may i t be possible t o obtain agreement between the same set of experiments and different mechanisms, but not otherwise. To sum up the foregoing considerations we may compare the problem of the elucidation of a complicated reaction mechanism on the basis of kinetic experiments with the problem of solving a crossword puzzle: The solution must be found by a method of hit-and-miss, but when once it has been arrived at, there is no doubt that the solution found is the only possible one. As a by-product a t the elucidation of the mechanism, we get numerical values of certain constants, and if experiments a t several temperatures are performed, they may be represented as temperature functions. But there is a clear distinction between the latter kind of information, which is quantitative in nature, and the information concerning the reaction mechanism which of course is purely qualitative. Even when the constants for experimental reasons can be determined with only moderate accuracy, the mechanism may be unambiguously determined from the experiments. Generally i t may be said th at the primary aim of a reaction-kinetical investigation is the clarification of the mechanism of the reaction in question. It is a commonplace to add that nowadays the study of reaction kinetics is not the only means of getting information concerning mechanisms, owing t o the experimental possibilities of the tracer methods. The relationship between these two main methods is such th a t the preference of one to another is not justifiable. Just as there are types of problems (e.g., the classical problem in ester hydrolysis) which can be solved only by tracer methods, so there are others which require kinetical

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investigations for their solution and some in which an application of both methods is in order, the situation thus being one of cooperation rather than one of competition between the two lines of research. Finally it may be added that in the earlier days of organic chemistry the use of kinetic investigations was not so widely applied as might have been desirable. The ways of setting up reaction mechanisms were merely based upon the enormous knowledge of known reactions compiled in the literature. To us the only natural way is to corroborate such suggestions by kinetic experiments.

11. GIBBS’ FUNDAMENTAL RULEOF STOICHIOMETRY The rule which we are going to state was given by J. W. Gibbs (1) in his famous paper of 1878, but as he gave the statement and its proof somewhat implicitly, we may repeat it here in a more explicit form as an introduction to the following paragraphs. We ask the following question: Given a mixture of m different species of molecules, composed of e different elements in a mass-tight container, how many independent reactions can take place in this system? Let the total, constant amount of element i in the mixture be Ei gram atoms (i = 1, 2, 3 * . e). Similarly let the amount of the molecular speciesj be Sj gram mols ( j = 1, 2, 3 * * m). The law of the conservation of the elements then gives one equation per element in the mixture that is a total of e equations

-

C

j=m

E; =

XijS,

j=1

where the X’s are integers known from the composition of the molecules. We thus have e equations between the m variable quantities Sj. Therefore if m - e of the quantities Sj are chosen, the amount of all the other quantities is determined. But this implies that m - e degrees of advancement may be chosen at will, or expressed in another way: m - e chemical reactions may occur. In this form, however, the statement is false. This is immediately apparent from such an equation as 2KClOa = 2KC1

+ 302

The obvious reason is that in this system a relation apart from the equations expressing the conservation of the elements exists, viz., that the number of atoms of two of the elements are the same in all compounds occurring in the mixture. This causes two of the equations of conservation to merge into one, so that the number of independent reactions must be increased by one. If generally there are T such relations, of which the

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condition of electroneutrality is often one, the number of possible independent reactions is n = m - e r. As an example let us ask: How many independent reactions are possible between the substances KCIOI, KC103 KC1, and Oz? As m = 4, e = 3, and r = 1, the answer is 2. One of these may be the well-known reaction

+

2KC103 = 2KC1

+ 302

The other may then be either or

Of these three equations only two are independent as addition of the first two obviously gives the last one. Only experiment can tell us which one actually takes place (2), e.g., that only the first one takes place when the chlorate is heated with manganese dioxide which acts catalytically. 111. INTERMEDIATE PRODUCTS AND

SEQUENCES

Most reactions studied kinetically are stoichiometrically pure, i.e., their advancement can be described by one number. Referring to the foregoing paragraph this is tantamount to saying that n = m - e r = 1. However, as indicated in the foregoing paragraph, also other cases than the stoichiometrically pure reactions may be investigated. If n is greater than 1, i.e., if two or more reactions may occur simultaneously, the process is said to be mixed in the stoichiometrical sense of the word. To quote another example: Methanol vapor reacts with finely divided copper as a catalyst according to the two equations (3)

+

and

2CHaOH = 2Hz CHiOH = 2Hz

+ HCOOCHj + CO

where m = 4, e = 3, r = 1. That r = 1 comes from the fact that all the molecules contain oxygen and carbon in the same proportion. But it is also an experimental fact found by chemical analysis that in a mixture of methanol and sufficient water vapor only the reaction (4,4a) CHiOH

+ HzO = COz + 3Hz

takes place in the presence of the same catalyst although generally in a mixture of methanol and water vapor n = 3, as in this mixture m = 6 , e = 3, r = 0. That r = 0 comes from the fact that there is in this case no common relation between the oxygen and carbon content in the molecules appearing in the equations.

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As stoichiometrically pure reactions are much easier to investigate kinetically than the mixed ones, we shall in the following mainly treat such reactions without forgetting that more complicated types exist. Let us assume that it has been shown analytically that the reaction is pure and let us further assume that it has been shown by kinetical experiments that the reaction does not follow the kinetics derived from its stoichiometrical equation in the well-known way. Obviously then we have to split up the overall reaction in a number of steps, earh represented by a chemical equation, and thus we must assume n > 1. If for the time being we exclude the addition of foreign substances acting as catalysts, the number of elements is obviously constant. We must therefore increase the number of molecules occurring in the system by 1 for each added step and are thus compelled to destroy the stoichiometrical simplicity. To justify the addition of such reactions which are not evident from the stoichiometrical scheme a new theory was introduced in 1913. This may be called the theory of intermediates in stationary concentrations (or even better: the theory of intermediates in quasi-stationary concentrations) and has since then shown to be of the greatest importance in reaction kinetics. The first who used this principle was Chapman (B), and half a year later Bodenstein in his paper on the hydrogen-chlorine reaction ( 5 ) also used it. Since the latter defended its use so ardently, it is not unjustly often connected with his name. The molecules which do not occur in the overall reaction, the “intermediates, ” occur in the mixture in such minute quantities th a t they normally cannot be detected by ordinary chemical (or physical) methods of analysis, or, likewise, the half-life of the intermediates in the reaction mixture, is exceedingly short. They must therefore be assumed t o disappear at nearly the same rate as the one with which they are formed, the respective concentrations therefore being stationary (or nearly so) in time. As the intermediates occur in such minute concentrations, a reaction between two intermediates will be a very rare event as compared to a reaction between an intermediate and ordinary molecules. With few except ions therefore the partial reaction equations contain a t most one intermediate on either side, and the intermediates can therefore be numbered according to their appearance in the sequence, e.g., 1, 2 in the first equation, 2, 3 in the second, and so on. We may therefore speak of a net inflow s12 from 1 to the intermediate 2 and a net outflow sZ3from 2 to 3. When the difference between s12and s Z J is exceedingly small, as compared to one or the other, we are justified in saying that the

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317

concentration of intermediate 2 is quasi-stationary in time. Strictly speaking we cannot say that the concentration of 2 is stationary, as the calculations will show that it depends on the concentrations of ordinary molecules, which in their turn depend on time. Nevertheless we shall often leave out the prefix quasi and define a stationary state of the whole system as a state in which the concentrations of all the intermediates are stationary, i.e., they depend only implicitly on time through the concentrations of ordinary molecules. It may be mentioned here that in certain types of reaction sequences the principle of intermediates in stationary concentrations cannot be fulfilled, with the result th at instead of a steady reaction with measurable velocity, a n explosion occurs. (See p. 320.) As the question concerning the validity of the principle is of some importance, we may perhaps consider the situation from another viewpoint. Let us assume a certain sequence of reactions containing both ordinary molecules and intermediates. We may then use the elementary principles of reaction kinetics to set up a system of differential equations which in principle (not always in actual practice) can be solved. The solution of these equations tells us what happens to the whole system, i.e., the amounts of all the molecules and intermediates t o any given time, provided that we know their amounts a t a certain time, e.g., a t time zero. Now i t is easy to ascertain the amounts of the ordinary relatively stable molecules a t the start of the experiment, but for the intermediates, because of their reactivity, such a knowledge is impossible to acquire. On the other hand, it is in order in this case to use the method of intermediates in stationary concentrations, which leads not only to a n expression of the reaction velocity by means of the concentrations of the ordinary molecules but also to values of the concentrations of the intermediates as functions of the concentrations of the ordinary molecules. However, just at the start the expressions are invalid, since it takes some time for the amounts of intermediates t o become stationary. This “time of relaxation” may of course vary from case to case within wide limits, but so long as i t remains small as compared to the overall reaction time, these limits are irrelevant. Usually the time of relaxation can be roughly estimated by qualitative considerations, or, if necessary, by more quantitative calculations (7a,b). Usually the sufficient smallness of the relaxation time will follow empirically from the fact that the advancement of the reaction follows some law predicted with sufficient accuracy by means of the principle of intermediates in stationary concentrations from the beginning of the reaction. Further, the conditions are often rather ill defined at the beginning of the experiment.

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J. A. CHRIBTIANSEN

To avoid misunderstandings it must be added that many cases are known, e.g., from the chemistry of carbohydrates, where the intermediates are so stable that the above-mentioned method is inapplicable. As is also seen, when and only when the steady state has been arrived at, the evolution of the system can be described by only one number, i.e., one degree of advancement, provided only one overall reaction is possible. 1 . The Intermediates

The nature of the intermediates is of course dependent on the nature of the reaction. In gas reactions free atoms and radicals often occur as intermediates. In reactions in solutions similar intermediates occur, but ordinary ions and molecules also may, if they are reactive enough, play the role of intermediates. This means that their half-life in the system is short compared to the reaction time. Since the fact has distinct and sometimes far-reaching kinetic consequences, it is well to remember that there are two distinct classes of radicals, those with an odd number and those with an even number of electrons. It is well known that nearly all ordinary molecules have an even number of electrons. From such molecules an odd-numbered radical cannot be formed alone but must always appear together with another oddnumbered radical, and vice versa: An odd-numbered radical can disappear only from the mixture to form an ordinary (even-numbered) molecule by reaction with another odd-numbered radical. Of course an oddnumbered radical may very well react with an even-numbered molecule, but in that case it will be replaced by another odd-numbered radical, which again is reactive. The situation is quite different in the case of even-numbered radicals. Such radicals can obviously be formed without especially reactive partners from ordinary molecules and vice versa: They may disappear from the mixture by reaction with ordinary molecules without being displaced by another reactive partner. As well-known examples of odd-numbered radicals we may mention H, HO, HOz, NH2, CH, and as examples of even-numbered radicals 0, NH, CHz. It may be added, that certain quite stable molecules are as a matter of fact odd numbered (NO, NOn, and Cu++). 2. The Sequences

Let us consider sequences of the kind where one intermediate appears on both sides of the equations for the partial reactions. Other types of sequences are possible and may be treated on similar lines as those used below, but the treatment is made more difficult by the fact that in such

319

ELUCIDATION OF REACTION MECHANISMS

cases quadratic (or higher) equations have to be solved, instead of the simpler linear ones. This causes certain practical difficulties without adding much to our knowledge of the principles. Another and more essential reason for emphasizing sequences of the first type is that a reaction between an intermediate and an ordinary molecule is usually much more probable than a reaction between two intermediates, simply because the probability of collision between an intermediate and an ordinary molecule is much larger than the probability for collision of two intermediates which always are present in low concentrations. As indicated below two types, open and closed, sequences are known. I

Open Sequence A1 = Xz Bz Az Xz = Xa B3 A3 Xt = X4 B4 A4 X I = BLl

+ + +

+

+ +

Indices of Steps (+I) (52) (53) (54)

I1

Closed Sequence A1 YI = Yz Bz Az Ya = Y3 B3 A3 Ya = Y4 B4 A4 Y4 = YI Bo

+ + + +

+ + + +

These two schemes represents two types (I and 11) of so-called linear sequences of reactions. I n the equations the A’s and B’s symbolize

FIG.I

ordinary molecules, or ions. This implies that if an A or B is zero the corresponding intermediate (X or Y) reacts spontaneously (unimoleculady). If the A’s or B’s are not zero, the reaction between these molecular species and their corresponding intermediates will be bimolecular. Sometimes an A or B may represent two or several ordinary molecules (or ions), in which case the reaction in question is of an order higher than two. The two types of linear sequences of reactions are so chosen that they will result in the same Fro. 2 overall reaction. From Figs. 1 and 2 it is evident why these sequences are called linear: the intermediates in each of them are connected by t,he partial reactions as pearls on a string. For reasons evident from the figure we call the two types of sequences respectively open and closed sequences. Since 1921 the latter type of sequences has been called chain reactions (11,12), a name which should be reserved for sequences of the closed type. Sometimes it has been used also for open sequences, but as the characteristics of the two types are very different this use may cause confusion and should be abandoned.

111

320

3. A . CHRISTIANSEN

Geomet,rical representations of sequences are very helpful in the study of the kinetics of certain scheme of reactions and will be used in the following presentation. 3. Qualitative Properties of Open and Closed Sequences. T h e Definition of Catalysis

From the equations of the open sequence (p. 319),it is evident that the first intermediate X2 (in the open sequence) must be formed from a reactant (A,) in the overall reaction. If now we introduced this intermediate from the outside it would disappear either by the reaction ( - 1): X Z B B + A2 or by formation of resultants of the overall reaction. (The time taken for the disappearance of Xz mould be the time it takes for the system to reestablish the steady state which is supposed t o be very short.) It is therefore obvious that the said introduction can cause a n increase of the reaction products in an amount which is a t most stoichiometrically equivalent to the amount of Xz introduced. For the closed sequence (p. 319) the situation is totally different. Here the first intermediate Y, must be formed by a reaction which is different from those constituting the sequence, and is not used up during the reaction. On the contrary when the sequence has been gone through once, or as we may say, when one revolution of the closed sequence has been performed, it reappears and the reactions take place again in the same order as before, and so on. I n other words, if in this case we introduce the intermediate Y I and there is no mechanism by mhirh it or one of the other intermediates is removed, the addition of one molerule Y1 would cause an infinite number of revolutions i.e., of overall reactions, and if there is some mechanism which removes an intermediate, the number of revolutions caused by the introduction of one Y, may nevertheless be large. If it is possible to fulfil the conditions for attainment of the steady state this (large) number will be finite However as mentioned in a paper hy Christiansen and Kramers (13) and shown experimentally, notably by Hinshelwood (14) and Semenoff (15), cases are known in which the number of intermediates is increased by one revolution of the cIosed sequence. I n th a t case it is impossible to fulfil the stationarity conditions, which means that the reaction goes on with ever increasing speed, i.e., we get an explosion. Only by adding inhibitors which remove one or more of the intermediates can the reaction be turned into an ordinary smooth reaction. It is obvious that when the intermediates are odd-numbered radicals the sequence must be of the closed type. On the other hand, when they

+

ELUCIDATION O F REACTION MECHANISMS

321

are even numbered, the sequence is usually of the open type, but i t may also be of the closed type. The intermediates may be formed from the reactants in the overall reaction, but they may also be quite foreign to the substances occurring in the equation for that reaction. As calculations will show, the velocity of a closed sequence becomes proportional t o the amount of the intermediate Y1. This means that we can increase the velocity of the overall reaction by addition of a substance which does not occur in its equation. As this may be taken as the definition of a catalyst for the overall reaction, we might call Y1 a catalyst irrespective of its origin either from the reacting system itself or from foreign sources. It is more common usage t o reserve the name catalyst for the latter alternative, but even then ambiguous cases remain, since the catalyst may sometimes be formed by a reaction between the foreign substance and the reacting system itself, the substrate. I n the well-known mechanism for the hydrogen-bromine reaction one may for instance not be inclined to consider the bromine atoms as a catalyst. As a matter of fact their concentration can be increased at will within limits by irradiation of the mixture (16) which causes a definite increase of the velocity of the overall reaction, a n effect which may very well be called catalytic. For such reasons the author would prefer the wider definition of catalysis, but the question of a suitable definition is in this case, as in many others, not only a question of principles but also one of convenience. I n any case it is easily seen by reversing the argument th a t the mechanism of a catalytic reaction can always be represented by a closed sequence. From the fact that the overall reaction velocity turns out to be proportional t o the amount of one of the intermediates it appears a t once that the velocity is known only when information concerning this amount is available. I n an ordinary catalytic (or enzymatic) reaction it is comparatively easy to get the required information, at least in the cases where the catalyst is not used up during the course of the reaction by some side reaction. The case is different when the catalyst is formed by the system itself. I n these cases one extra condition is required to determine the velocity. I n the hydrogen-bromine reaction the bromine atoms are formed by the dissociation of bromine molecules and disappear by the reversed reaction. Steady course of the reaction thus requires that chemical equilibrium in regard to that reaction is reached. As two intermediates (bromine atoms) appear on one side in one of the chemical equations we get in this case a quadratic kinetical equation. It may be mentioned in passing that this reaction was one of the first, if

322

J. A. CHRISTIANSEN

not the first, reaction not obeying one of the classical laws (unimolecular, bimolecular) whose mechanism was completely elucidated by means of its kinetics (8,9,10). The mechanism of the reaction leading to the formation of hydrogen bromide may be pictured as in Fig. 3. This represents not only the closed sequence Br + H z e H B r + H Brz+ HBr Br H

+

+

but includes also the equilibrium Brz

2Br

which defines the concentration of bromine atoms. The figure represents therefore not simply a linear closed sequence, but a linear closed sequence

Brz

FIG.3

FIG.4

and a branch. We call it a branched sequence. The reason for this branch is evident when Fig. 4 is considered. I n this there is nothing which shows that the reaction has a well-defined velocity. This is first accomplished when a reaction leading to the formation and disappearance of bromine atoms is assumed. We may insert here the proof that the mechanism of thereaction between hydrogen and bromine must be the one mentioned above. If we do not assume that molecules other than Hz, Brz, and HBr are in the mixture, the number of possible reactions is 1, according to the stoichiometric rule, p. 315. I n that case the overall reaction, as well as the mechanism, would be represented by the equation Hz

+ Brz = 2HBr

which leads to a simple bimolecular reaction in contradiction to the experimental facts. If we next introduce bromine atoms as intermediates, for stoichiometric reasons it is also necessary to assume hydrogen atoms as intermediates. In that case the number of possible independent reactions becomes 3. One possible set of such reactions is the sequence given above, A second one would be

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ELUCIDATION O F REACTION MECHANISMS

Hz = 2H

H Br

and a third one

+ Brz = HBr + Br + HZ= HBr + H H

Hz = 2H Brl = 2Br Br = HBr

+

In all there are 10 possible combinations of the above 5 equations in sets of 3 each. Of the 10 sets only 5 can be combined to result in the overall reaction, and the only one of these which can be reconciled with the kinetic experiments is the first one above, which proves our case; cf. Skrabal’s somewhat different considerations (35b). I n the hydrogenbromine reaction we may use the expression that reaction chains are started by the formation of bromine atoms from molecules and broken by their disappearance by the reverse reaction.

FIG.5

It is not necessarily true in all instances that the starting and the breaking of chains are caused by reciprocal reactions. I t must, however, be remembered that if one of two reciprocal reactions are known to take place under certain conditions then in principle the reverse of that reaction must also take place, but the effect of this may be vanishingly small as compared to that of other chain-breaking (or chain-starting) reactions. If the chains are started by the formation of one intermediate and broken by the disappearance of another, the case becomes somewhat more complicated as in Fig. 5 showing a closed (linear) sequence with two open branches. We may characterize the closed sequence by the symbol (23452). This means that the component reactions are (23)) (34), (45)) (52) in that not irrelevant order. For readers not familiar with this sort of reasoning we may write down the sequences in question, from now on giving up the distinction by symbols between intermediates in open and closed sequences. Main Reaction

Az A3 A4 As

++ Xa = Xa + B3 Xa = Xr + B4 ++ Xs X4 = Xs + Bs = Xz + Bz

Az -t Aa

Side Reaction

(23) (34) Bz (45) OL (52)

€33

A5

- _ _ A1 Bz

+ + As = + B4 + Bs + Bz (R) A4

AI = Xz + P + Xz = X5 + + Xa = Bo

(12) (25) (50)

+ + a = 6 + As + Bo (S)

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J. A. CHRISTIANSEN

The sequences should be compared with Fig. 5 . We have two stoichiometrically but not kinetically independent reactions, their velocities being respectively r and s. To illustrate the relation between the different flows and the two reaction velocities, we remark that the flows 23, 34, and 45 are obviously the velocity of the main reaction, r, while the flows 12 and 50 equals the velocity s of the side reaction. This is shown in Fig. 5 b y means of letters and arrows. The diagram also shows the symbolic analogy between our flows and real physical flows. Thus we may speak of sources and sinks, 1 being a source and 0 a sink, and of translational and rotational flows; for example, we may say that the flow s52 is a superposition of a translational flow (- s) and a rotational flow ( r ) . s may be assumed to be always positive. The case s = 0 is in principle the same as the one treated above (p. 322), where we may speak of catalysis with X2 as a catalyst. As the chain (23452) is broken in this case only by the reaction 21, the chain length then has its maximum, but its numerical value cannot be defined unless we know the kinetics of the reactions (12) and (21), which may be unknown; compare the discussion in the literature of the hydrogen-bromine reaction (see also p. 334). T o arrive a t this limiting case the reaction (50) must be prevented, This obviously can be done by making the concentration of a zero, while any increase of that concentration will increase s, the velocity of the side Xs, and reaction. This causes the amount of XSand therefore also of Xz, X4 to decrease, which means that, the veIocity of the overall reaction (R) decreases. We may say that the side reaction induces the main reaction and t ha t the induction factor is r / s . If the reaction (50) is completely irreversible, i.e., if the reaction (05) does not occur a t all, and if, furthermore, the velocity of the reaction (21) is vanishingly small a s compared to that of (50), the induction factor becomes identical with the chain length, defined as the number of overall reactions (R) occurring for every intermediate Xz formed according t o reaction (12) (cf. p. 341). One can see that the substance a acts as a negative catalyst (inhibitor) and t ha t it disappears during its action. Such anticatalytic effects were studied in the renowned works by Moureau and Dufraisse on “antioxygenes” from about 1920 (17). They also found th a t the inhibiting effect on the autoxidation of certain aldehydes by certain substances, e.g., quinols, disappeared after a while. Some years later Backstrom (18) and co-workers showed by chemical analysis that the inhibitors were in such cases actually used u p (oxidized) during their action. Indications of similar effects are known from catalytic reactions, especially in the case of enzymes, where it is sometimes found that the enzyme gradually loses its activity during its action. Each case must, of

ELUCIDATION OF REACTION MECHANISMS

325

course, be investigated to determine whether this degradation is really due t o the action itself or simply to a reaction which has nothing t o do with the enzymatic (catalytic) reaction studied. From the field of inorganic heterogeneous catalysis cases of degradations are well known, but in these the explanation often may be, for example, a recrystallization of a finally divided catalyst, which would also occur a t the same temperature when the catalyst is not acting, or a poisoning of the surfa,ce by impurities in the reacting system, of which the same may be true.

IV. CALCULATION OF STATIONARY VELOCITIES AND CONCENTRATION 1. The Partition Matrix We start with the case of a linear open sequence with a definite number of steps, e.g., four. We do this for the simple reason that the general expressions arrived a t are often difficult to write and to read, whereas an example with four steps in the sequence is neither too simple nor too complicated t o show the general form of the results obtained. Let the sequence (cf. p. 319) be

and the corresponding diagram:

We now define a certain number of reaction probabilities, meaning the probability per unit of time that an intermediate or in some cases a n ordinary molecule will react according to some prescribed partial reaction. I n all such calculations it is essential to use a system of symbols which are well suited to the purpose. Although in principle the choice of symbols is a matter of pure convention and thus arbitrary, one may make the expressions so complicated that their meaning becomes veiled by choosing the symbols in an irrational way. No one who has even the slightest knowledge of the elementary theory of the solutions of systems of linear equations with several unknowns would dream of not choosing the symbols for the constants in some systematic way. I mention this because even today it happens that an author in reaction kinetics chooses, for example, the symbols for two opposing reactions in a way which gives

326

J. A. CHRISTIANSEN

no indication whatever of the fact that the two reactions are reciprocal relatively t o each other. I n reaction kinetics the order of the partial reactions is relevant, and it is therefore natural to number them 1, 2, 3 * * . The corresponding For a reversed reactions may then be numbered -1, -2, - 3 linear open or closed sequence this numbering is satisfactory, b u t when we come t o cases of branched sequences the principle fails. I n th a t case it becomes necessary t o number substances (states), usually the intermediates, instead of reactions, which then must be characterized by a pair of numbers whose order defines the direction of the reaction. For example, if the states are 1 and 2, the transition from 1 to 2 is symbolized by 12 and the opposite by 21. We shall use the two systems alternatively, preferring the former (numbering the reactions) when possible because it is slightly more convenient than the latter, which may be used in nearly all cases. It must be remembered that a t times the same pair of intermediates occurs in two different partial reactions (cf. the hydrogen-bromine reaction). I n such cases the two pairs of symbols must be chosen so that they are visually different, either by numbering the partial reactions or, if numbering of states is preferred, by using a dash to distinguish one reaction from the other. It is often convenient to number the “head” of the series of substances (intermediates) with 1 and the “tail” with zero. We shall confine ourselves to the case of homogeneous mixtures. An extension t o heterogeneous mixtures along the same lines is easy in principle, but such cases are complicated by the peculiar properties of surface layers and by the processes of interaction between surface layers and the bulk of the solid. It may be mentioned here th at the first example in which the author tried to apply the principle contained in this paper was on the hydrolysis of methanol (19) (cf. p. 315) with a copper-magnesia catalyst. This reaction is, of course, heterogeneous. I n homogeneous reactions the concentrations (not amounts) of different substances, denoted by lower-case letters corresponding t o the capitals used for designating the substance in question, will be used as variables. If w1denotes the probability per unit of time of reaction i, the velocities of the different reactions are:

-

9

.

ELUCIDATION O F REACTION MECHANISMS

327

The reaction probabilities w are constants if the corresponding A’s or B’s are zero, and proportional to the corresponding values of the concentration a or b if an A or a B stands for one molecule. I n the rare cases in which they stand for a sum of two or more molecules the w’s are proportional to the Corresponding concentration products. The symbol s is preferred to u partly because of the possibility of confusion with v for volume, partly because a reaction velocity has much more in common with a “stream” (current) than with the velocity of a particle in mechanics. The conditions of stationarity are obviously SI

=

s2

=

s3

=

sq

=s

and we thus get four equations with the concentrations of three intermediates and one reaction velocity as unknowns. If we choose l/s, x z / s , x3/s, x4/s as the unknowns, the last (right-hand) members of the four equations become 1. If we further replace a l w l by a1and bow-., by I ~ Lthe ~ , determinant of the equations becomes

a1

0 0 -@-4

-w-1 ~2

0 0

0 -w-Z

W3

0

0 0 -w-3

W4

The methods of solution may be found in any standard textbook on linear equations, and they may be written in a form which is easy t o remember: Let L = alw2w3w4- I . ~ ~ ) _ ~ W - ~ W _ ~ W - ~ = alw1w2w3w4 - b O ~ - 4 ~ - 3 ~ - 2 ~ - - 1 Now let { M ]be the following matrix

This matrix can be written down a t once since it is necessary only to remember t o start with wz in the first row of the first column and t o know the rules for the arrangement of the rest of the indices, which are obvious from the above. The number of rows and columns in the matrix is equal to the number of partial reactions in the sequence, while the number of factors in each matrix element is one less. Let further M1 be the sum of the elements in the first row of { M I ,M 2 the corresponding sum for the second row, and so on. The solutions are

328

J. A. CHRISTIANSEN

then l/s = MI/& XZ/S = M 2 / L ,xB/s = M3/L, x4/s = d!f4/L. T h a t this is true can easily be verified. Obviously we may write 1 = AM1, x2 = AMP, 2 3 = AM,, 2 4 = AM4, where A is the common factor s/L. These inserted in the original equations (pp. 326-327) give in the four cases the same identity:

-

=

A ( t D i l ~ 2 ~ 3 ~ 46-4w-yw-zw-1)

s

If bo = 0 or if the reaction (4) is irrcversibIe i.e., if w - ~= 0, the elements hside the frame in the matrix disappear and L simplifies into alw 1wzw3w4. Therefore in this case a1 - 1 w-1w-2 - - + w-1 1 + _ _ -1 + w-1w-2w-3 ____- 1

s

- _-

w1

w1 w2

-1+

x2

S

w1

w2

W2 w3

+ 1 -+

_ -1 w-2

WlW2WB

-S

1

w2w3

w4

. -

w2 w3

2 3

w4

w-2w-3

-_1

w-3

w3 w4

w3

1 -

--

2 4

w4

S

The reader will find no difficulty in writing down if necessary similar expressions for the case, where bow-4 is not zero. I n the special but very frequent case considered above it is seen that if the last reaction (+4) in the sequence is somehow prevented from taking place, the ratios between the different concentrations converge as they must against the equilibrium values. If, namely, w4 -+ 0 we get alwl

=

XPOW-1;

xzow2 = zr0w-z;

23OW,

=

x4°w-3

It is furthermore seen that the expression for al/s may be written l / S = l/UlWl

+

l/XzoW2

f

1/23OW3

+

l/X4OW4

where xZo, z3', 2 4 ' are virtual equilibrium concentrations i.e., purely thermodynamic quantities, while the w's are purely kinetic. This shows that in the sequence of reactions of the type considered we get the reciprocal of the resulting velocity b y adding a number of reciprocal velocities, one for each partial reaction. Finally it appears from the above th at al, 22, 23, x4 are proportional to MI, M a , Ma, M4, respectively. For this reason the matrix may appropriately be called a partition matrix. The sum of all the members in the Bzi) E, we thus partition matrix we shall call M . Calling the sum (u1

+

get

al/E

=

M J M ; xZ/E

=

M2/M;

*

.

*

ELUCIDATION OF REACTION MECHANISMS

329

I n these expressions the M’s are functions of the concentrations of the different ordinary molecules in the mixture and therefore vary slowly with time, but they are not explicit functions of the time. This is what we mean by saying that the concentrations are quasi-stationary. 2. Applications.

Cancelling of Matrix Elements

T o apply the above expressions to practical cases we must discuss them in more detail. I n the first instance we are interested only in (overall) reactions which are sufficiently slow to be accessible to kinetic measurements. Therefore at least one of the members in the expression for l/s must be large, i.e., its denominator must be small. Let us repeat our diagram (Fig. 6) in a slightly different form. This is supposed t o mean th at the reactions (+1) or (12) and (-4) or (04) are endothermic. From a kinetic aspect this means that w l ( w 1 2 ) and

FIG.6

contain an exceedingly small exponential factor, which is neither contained in w-l(wsl) and W,(W40) nor in the other reaction probabilities occurring in our system. Now as the different w’s apart from the exponential factors are not very different, factors of several powers of ten being admitted, we may with more than sufficient accuracy consider products containing exponential factors as vanishingly small as compared t o products which do not contain such factors. The diagram shows at a glance that the factors in question are w1and w - 4 . Inspection of the partition matrix then shows, th a t x2/a11 xa/al, x4/a1are disappearingly small or that we shall commit a very small error indeed by neglecting the difference between MI and M . From this it follows th at al (or b,)may be determined not only by physical, but also by chemical analysis. This remark refers to the fact, that if we determine the concentration of A1 by some reaction in which it is used up, we should get not the actual concentration of All but the w-d(WO4)

+ 2 xi as the reactions ( - 1), ( -2), (- 3) would occur during the 4

sum a1

2

analytical reaction.

If on the other hand we choose t o determine bo, we

should get the sum bo

+ 1xi. 4

2

It need hardly be said th a t the sum

330 al

+

J. A. CHRISTIANSEN 4

xi + bo for purely stoichiometric reasons must be constant during

2

a kinetic experimeiit.

1xi into 4

Although it would not be impossible to take also the sum

2

consideration, it is of course much more convenient to cancel it completely. In a similar way a case such as shown in Fig. 7 may be treated, where al and x2 are practically in mutual equilibrium and we should get for the irreversible reaction (L = ulw1w2w3w4)

where

(a1

+ x2) is measured as one substance by chemical analysis.

FIG.7

FIQ.8

As a third example we have a diagram such as Fig. 8, showing at once that z2 and x4 are disappearingly small as compared t o al and z3if stationarity has been attained. If however W-Z(w32) and w3(w34) are so small that they are comparable in magnitude with the reciprocal “time of reaction,” which is of the same order of magnitude as wl,the method of stationarity fails and we are compelled to use the much more difficult general method. Here we may determine x3 by chemical or physical analysis so that, in some cases at least, the reactions 1 -+ 3 and 3 --+ 0 may be investigated separately. 3. The Analogy between Chemical Reaction and Diffusion

Many years ago W. Nernst in his textbook “Theoretische Chemie” pointed out that the velocity of a chemical reaction may be represented in analogy to Ohm’s law by i = e/r where i represents the velocity, e the “chemical force,” and r the chemical resistance. A t that time the law of Arrhenius governing chemical reaction velocities and its underlying assumptions was so well established that no need was felt t o follow up the idea of Nernst. The Arrhenius picture can

ELUCIDATION OF REACTION MECHANISMS

33 1

be expressed by a two-step diagram. If, however, we insert in the diagram any number of unimolecular steps, this will have no experimental consequences so far as the isothermal reaction is concerned, and it is thus seen that i t is possible to replace the discontinuous Arrhenius picture with a continuous one by dividing the one large step in a great number of small steps. But this is tantamount to considering the elementary reaction as a diffusion of the different particles in the reacting molecule, or if the reaction considered is not kinetically unimolecular in the reacting complex of molecules. From this point of view the Arrhenius picture and the Nernst picture merge into one. It is also seen th a t the picture is very nearly the same as that proposed by Eyring and Polanyi (20) in 1931, a picture which has since been further developed by Eyring with co-workers and others and has got the name of the transition state theory. This differs so far a s can be seen from the statistical mechanical theory of R. Marcelin of 1915 (21) only by its use of quantum mechanics instead of classical mechanicg. Marcelin’s theory, the theory of critical complexes, again is at least strongly influenced by the paper by Arrhenius of 1889 (22), containing his theory of “active molecules.” By means of the expression from p. 327 for a n open sequence

l/s

or

=

MIL

s = L/M1

it can be shown that (‘the driving force” is not the affinity of the reaction, b u t the activity difference alfl - bofo where f l and f o are activity coefficients, while the resistance is represented by the sum M 1 divided b y the product w1w2w3 . (cf. p. 328). Exactly the same consideration can be applied t o ordinary diffusion, easiest in the unidimensional case. As was shown in 1936 by Christiansen (23), this leads t o the following expression for s the diffusion current in a force field. -sp = D(cp)’ = Da(cp)/az where p is an “activity factor” defined by RT In cp = V , in which V is the potential energy of the diffusing particles in the field of force a t the place z. The expression is only formally different from the Einstein expression s =

- D [ ( ~ c / ~ x-) cK,/RT]

where K , is the force acting on 1 mol. If the diffusion is stationary, i.e., if s is independent of x we get by integration

332

J. A . CHRISTIANSEN

where clcpl and cop0 are the activities a t the two ends of the diffusion column, or in the source and in the sink respectively. If the potential energy is zero a t both ends we get

Is

= c1

- co

where I represents the integral. This integral may be used if we consider diffusion in a column with interposed diaphragms each with their own value of cp/D, as has been done recently by Eyring and co-workers (24). We may in such cases measure a n effective diffusion coefficient D by means of the equation (c1

- co)/s =

z/D

where 1 is the length of the diffusion column, and all the quantities on the left-hand side are measurable. It thus follows th a t

I = l/D

It is obvious that if V is large a t certain places I will get contributions practically only from these places. If the solution is ideal and no field of force is applied to the column, we may say that cp is everywhere 1, and we thus get D = D. We may, however, consider the matter from a different point of view. The dissolved molecules diffuse between the solvent molecules, which cause them to move in a force field of very variable intensity. If this were not so we should expect the diffusion constant t o be the same everywhere. Denoting by D this constant of diffusion, we get cpdx = l D / D

/01

Again those ranges for which the potential is large give the main contribution t o the integral. As cp = exp ( V / R T ) it is seen that D, which is what we actually measure as the diffusion constant, must depend on temperature. This is actually the case. Similar considerations are t o be found in a recent paper by Eyring and co-workers (24).

4. The Closed Linear Sequence: Catalysis and Enzymatic Reactions Let the overall reaction be

333

ELUCIDATION O F REACTION MECHANISMS

and let us assume that the sequence is

The diagram may then be illustrated by

or simply by

2 1

3 4

We thus get four equations expressing stationarity of the intermedi-

ates:

XIW1 x2w2 x3w3

Choosingsi/s (i = 1

- -

24204

- xzw-1

=s

- x3w-2 = s - x4w-3 = s - XLW-4 = s

4) as the variables, the determinant becomes

-w-1

w1

0 0

WZ

0 0

-w-4

0 -w-2 W3

0

0 0 -w-3 w4

Solution of the equations gives the partition matrix { M )

=

j

w2w3w4

W-lW3W4

w-1w-2w4

w-1w-2w-3

W3WqW1

w-2w4w1

w-zw-3w1

w-2w-3w-4

W4WIW2

W-3WIW.2

W-3W-qW2

W-3W-qW-1

W1WZW3

w-4w2w3

W-4W-lW3

w-4w-1w-2

The solutions may then be written X ~ / S=

Mi/L, (i = 1

.

*

’

4);L = WIW2W3W4

i

- w-1w-zw-3w-p

334

J. A. CHRISTIANSEN

This shows a t once that if we do not know the values of either one of the

ztls or of their sum, the velocity remains indeterminate. A well-known example of the first case is the hydrogen-bromine reac-

tion where the concentration of the intermediate Br is determined by its equilibrium value in the case of the reaction in the dark and in the photochemical case the stationary value resulting from the dissociation by light and the recombination (16), the latter being independent of the illumination. I n such cases the sequences are strictly speaking not linear but branched and shall not be discussed further in this section. I n a catalytic (enzymatic) reaction we know the total amount of the catalyst (the enzyme), or we may a t least have ascertained by experiment that its amount is constant during the reaction. From the stoichiometry of the component reactions in the sequence it follows that Zx, = El where E is the total constant concentration of the catalyst, the enzyme. Calling the sum of all the members in the distribution matrix ( M ) for M we thus get by addition

zx,/s = M , / L E/s = M / L

or

in accordance with the fact that the velocity of a catalytic (enzymatic) reaction is proportional t o the total amount of catalyst (enzyme) (7b). Also in this case i t is essential for the practical application of the result, to the study of the mechanism of a definite overall reaction that a number of the sixteen products in the sum M are so small that they can simply be omitted because they contain exceedingly small exponential factors. 5 . Orientation of Diagrams

To select the nondisappearing products we ascribe t o the orientation of the diagram a qualitative meaning so that a step upwards means that the partial reaction in question has a certain “energy of activation” while a reaction corresponding t o a nearly horizontal step or a step downwards has none or a relatively small one as the case may be. Let us look again a t a diagram corresponding t o the above closed sequence (Fig. 9). The reaction may obviously be symbolized by a rotational flow through a viscous medium or by a rotating electric current, the driving force having its origin in the FIG.9 affinity of the overall reaction. From the figure it may further be seen that the intensity of the flow is proportional to the difference L, which thus corresponds to the electro-

0

335

ELUCIDATION O F REACTION MECHANISMS

motive force, while M / E represents the resistance. I n cases where the overall reaction has no heat of reaction it is evident that the two members of L must contain the same exponential factors. The algebraic sum of the energy differences corresponding to the four steps 12, 23, 34, 41 must then be zero, and the diagram if drawn correctly will show the exact values of the different energy steps, i.e., the activation energies. If, however, the overall reaction has a heat of reaction, the diagram cannot represent the energy steps exactly, because in a closed diagram the differences in level must always give the algebraic sum zero. If we want a diagram which can give a true picture of what is going on we must use instead of the two-dimensional one the surface of a cylinder or a prism with a vertical axis. On this the four intermediates are represented by four lines parallel to the axis or by the vertical edges of the prism. To represent the chemical reaction we then use not a plane rotational flow but a flow in screw line around the cylinder which repeats itself at lower and lower levels. In this way we may free ourselves from the geometrical restrictions which are inherent in the two-dimensional diagrams (cf. the diagram p. 342, Fig. 10). If the exponents of the exponential factors are taken t o be as foIIows: Reaction (1) (2) (3)

Exponent --HIIT

(4)

0

-HdT 0

Reaction (-4)

(-3) (-2) (-1)

Exponent

-H a / T - H-r/T 0 0

where the H ’ s are taken as vertical distances in the diagram, it is obvious there is now no geometrical reason why HI H z should equal H--( H--a as it would in the case of the circular diagrams. The difference in height between two consecutive loops on the cylinder may be taken as a measure of the decrease in enthalpy of the overall reaction. It must be mentioned that we have hereignored the difference between -AH and -AG (the decrease in the Gibbs free energy). This is seen as follows. The condition of equilibrium is obviously

+

WlW2W3W4 =

+

w-1w-2w-3w-4

which for ideal dilute solutions takes the form ala2a3a4k~k2k3ks = blb2b3b4k-lk-2k-3k-4

where the k’s are a t constant pressure functions of the temperature alone. The a’s and b’s represent as usual the concentrations of the A’s and B’s. If we now place In ki = -G,/T Bi

+

336

J . A . CHRISTIANSEN

where the B's are constant while the G's vary slowly with temperature, we get for the equilibrium constant

K = ~lk2kIk4/k-~k_zk-3k_4 In K = -AG/T AB where AG = +(GI Gz GI G4) - (G-I G-z G-3 G-4) AB = B1 B2 Bt B4 - (B-1 B-2 B-3 B-a). and

+

+ + + + + +

+

+ + + +

+

It can easily be shown by differentiation of the expression for In K that AG must have the properties of a Gibbs free energy, and the same must then be true of the single members Gi. To avoid confusion it must be added that what we have called AG is - T In K TAB and not as usual - RT In K , and that energy values are in degrees Kelvin while ABisapure number. For the sake of illustration it is sufficient t o use the plane diagram. If, for instance, the orientation and the levels are as indicated by the numbers below 3 2 4 1

+

the reaction probabilities 201, wz,w-3, w-4 all contain vanishingly small exponential functions. A glance at the partition matrix p. 333 shows that for this reason only the second and the third members in M I remain of the whole sum M . If the orientation is 3 4 2 1

w 2and w - ~must be small, which means that only the three last members of MI and the first three members of M z are retained in the sum M . If the orientation is 4 2

3

1

w3 and w-4 are small and again six members remain, viz., the last two in MI, the middle two in M 2 , and the first two in M I . All the diagrams mentioned above are circular in type. However in a closed linear sequence of a t least four steps another type is possible, viz., one of a lemniscate type, e.g., 2 3

4 1

ELUCIDATION O F REACTION MECHANISMS

337

I n that case W I Z = wl, w14 = w-4, w34= w3,w32 = wP2 all contain the exponential function. To get a qualitative picture of the partition matrix for this case we replace the very small reaction probabilities by 0 a n d the others by 1. The matrix then appears as follows:

jB E :;; :::j 101 101 101 101

=

010 010 010 010

From this i t follows that all the members in the sum M are small but that in comparison to x1 and z3,xz and 2 4 are small, which also may be seen directly from the diagram as Xz and Xq are on “peaks” while X1 and X, are in ‘(valleys.” As therefore a passage from XI t o X3 can only take place across a peak, the question arises whether it is permissible to assume stationarity in such a case. The mathematical discussion is somewhat lengthy and has been given elsewhere (7b), but even without detailed calculations the qualitative result is rather obvious. If the amount of catalyst (enzyme) is small as compared to the amounts of reactants, the time of relaxation is always short a s compared t o the reaction time, i.e., in th a t case it is permissible t o assume stationarity. The case of a diagram of this form has been treated provisionally (7b) by the author and in much more detail by Darling (25)) who discusses a great number of possible distributions in level of four points in such a diagram. It must be added that for each of the great (but finite) number of possibilities of distribution on the levels in circular diagrams or diagrams of the lemniscate type a number of permutations between the different reactants and resultants appearing in a sequence of given overall reaction are possible. The number of possibilities thus may be bewilderingly large, and it may seem t o be a hopeless task t o find the correct mechanism. However the experiments will always show the investigator certain pecularities of the reaction which exclude a number of assumptions. Other assumptions may be so improbable from a chemical point of view that they also can be excluded. In short, a chemist who has provided the experimental material and who has some experience in kinetics will have a good chance of finding the right mechanism, i.e., the mechanism from which his experimental results can be derived. 6. Branched Sequences. Cyclic Sequences

A branched sequence usually corresponds t o more than one stoichiometric equation. If we count only one “head,” the rule is that every

338

J. A. CHRISTIANSEN

tail and every closed sequence corresponds to an independent stoichiometric equation. There are degenerate cases where some closed sequence gives n o overall reaction, which is the same as saying that its overall reaction equation is an identity. Such closed sequences, which may appropriately be called cyclic, may have kinetic consequences but they certainly have no chemical effects. Application of the steady state condition in such cases shows that any rotational flow is possible, i.e., the intensity of flow is indeterminate. This fact has puzzled chemists and physicists for many years. The first explicit statement concerning this puzzle known to the author is by Wegscheider in a paper of 1902 (26). The solution of the puzzle is according to modern views that in the case of a cyclic process the only possible stationary value of the rotational flow in a cyclic sequence is zero, which is known as the principle (or axiom) of microscopic reversibility. If we apply this principle to a cyclic process passing through the steps 1,2,3,4, it follows that wl~wz3w34w41= ~ 2 1 ~ 3 ~ ~ or 4 3 ~ 1 4

As in this case the same concentrations in the same powers obviously must appear on both sides of the equation, we get one relation between the eight velocity constants, vie.,

instead of the ordinary equilibrium condition which can of course be fulfilled with any values of the eight constants. 7. The Stationary Velocities of a Branched Sequence:

Negative Catalysis

For the sake of illustration let us consider a sequence like that in the following figure (cf. p. 323) :

s-r

ELUCIDATION O F REACTION MECHANISMS

corresponding t o the following sequences

Az

(a3

=Xz+B

Ai

+ + + A6 = B3 + B, + + BZ A3

339

B5

A4

(R)

We thus have two (one tail, one closed sequence) stoichiometrically independent reactions. The condition of stationarity may then be written s = 812 = 8 2 5 f s 2 3 = 8 2 5 s 3 A = 8 2 5 8 4 5 = $50

+

+

Introduction of the expression for sij 8..

13

. - 2twij -

Xjwji

gives five linear equations with the unknowns s and xz, x3, x4, x6, which may be solved as usual. In practice, however, this is a very lengthy procedure. It is much easier to proceed as follows. Denote the velocity of the overall reaction of the open sequence (S), the flow (50) by S, and the velocity corresponding to the closed sequence ( R )by r . We then find: S ~ = Z S ; S25 ~ 2 = 3

r ; s34

=s

- r ; S50

=

= S

r; sd5 = r

For the sake of simplicity we assume that the reaction (50) is practically irreversible. The system of equations for the sequence ( 8 ) or (1250) thus becomes

alwlZ - X ~ W Z I= s x2w25

-

x5w52

=

Xgw50

=s

8

-r

340

J. A . CHRISTIANSEN

or by appropriate division x2w25/(s

alw12/s - X ~ W ~ ~=/ S1 - r ) - z5w52/(s - r ) = 1 x5w50/s

=

1

If we consider a l l x2, and 2 5 as the unknowns and w12/s, wzl/s, wZ6/(s - r ) , etc., a s known, the equations are of the same form a s those solved before, and we may therefore write down the solution for al after the model given p. 328: s w d s - r> I W Z I . u)52. al=-+ w12

w12w26

w12

w25

w50

Similarly we get for x2 2 2

s

-r I

= __ w25

w52.- S w25

w50

Now in reality 2 2 , s, and r , but not all are unknown, and we thus have two equations between three unknowns. A third equation between the same three unknowns can be derived from the consideration of the linear sequence (23450). Applying a method which is the exact analogue of the above we get

Denoting the polynomial in the parenthesis by P we thus may write

xz = rP

+s

~ ~ ~ w ~ ~ w ~ ~ / w ~ ~ w ~ ~ w ~

Elimination of x 2 from the last two equations yields

r(P

+

1/w25)

=

s(1/w25

+

w52/w25w50

-w ~ ~ w ~ ~ w ~ ~ / w ~ ~ w

which in connection with the expression for al can be solved for r and s. The result depends largely on the properties of the sequence ( R ) or (23452). If this sequence is cyclic, we obtp.in the condition

w 54w 43w32w 2 5 = w 52W 23W 34w 4 5 If ( R ) is not cyclic but if equilibrium in respect of the overall reaction ( R ) has been established, the same condition must be fulfilled. I n those

cases the two last members in the parenthesis above cancel, which of course means a considerable simplification of the expressions for r and s. For our present purpose we shall discuss the opposite case, viz., th a t the reaction ( R ) is irreversible. We may assume, for example, th a t

ELUCIDATION O F REACTION MECHANISMS

I n that case P 3 is2 zero. sion, yields

~

=

r(wdw23

34 1

l/wza, which, inserted in the above expres-

+ 1)

=

+

s(l

w52/w50)

When this is introduced in the expression for al and r and s are separated we get alw12w5O(w23 w25)/s = w50(w21 w23 f w25) f w21w25 a l w 1 2 ~ 2 3 ( ~ 5 0 w d / r = w50(w21 w23 w25) w21w25

+ +

+ + +

+

If we diminish the concentration of a and thus let w50 approach zero, s must obviously also approach zero while r remains finite, and in the limit is determined by al/r

=

WZI/WI~W~~

As the number of primary reactions 12 per unit time is alw12, the maximum chain length n is thus w23/w?.1. This ratio is great if either the ratio between the concentrations of A, and p or the ratio between the velocity constants k 2 3 and kzl is large, or if both conditions are simultaneously fulfilled. I n the two latter cases this would mean t h a t the heat of activation should be larger for reaction 21 than for 23, so th a t the reaction 21 must have a certain activation energy. As we have assumed that w32 is effectively zero, i.e., th at the reaction 32 has an activation energy, the reverse reaction 23 has none. I n the other limiting case, where w50 is chosen sufficiently large, we get alw12w23/r

=

w23

+

w25

as w21 has already been supposed to be much smaller than w23. I n this w25) which cannot exceed 1. case the chain length becomes w23/(w23 It is thus seen that the substance a acts as an inhibitor for reaction ( R ) . The general expression for the chain length n is for the case considered (w32 0;w23 >> wZ1)

+

---f

n

= W2dw50

+ w ~ ~ ) / ( w w ( w+z ~ + 202.5)

WZlw25)

This expression shows th at the lower limit of n can be reached at small concentrations of the inhibitor only if the velocity constant k5o is large as compared t o k 5 2 , where it must be remembered th a t wK2= where aa is the concentration of As and that w50 = f f k 5 0 where CY is the concentration of the inhibitor. In cases of pronounced inhibition by traces of the inhibitor we must therefore assume th a t the reaction 52 has a n activation energy and consequently th at 25 has none. As the same is true of reaction 23, the ratio w23/w25 is of the order of magnitude 1. Furthermore the reaction 21 has been assumed above t o have a n activation energy.

342

J. A. CHRISTIANSEN

If therefore the concentration of the inhibitor is not exceedingly small we get for the chain length

which shows that under the said assumption n is a linear function of the reciprocal inhibitor concentration. This relation is useful in graphical

FIG.10

representations of the experimental results. Figure 10 shows a diagram which is qualitatively in accordance with this assumption. It is assumed t o have been drawn on a vertical rectangular prism, whose projection on a vertical plane is shown. The heights represent energy levels, strictly speaking, levels of free energy. The “modulus,” i.e., the distance between two consecutive points on the same vertical line represents the affinity of the overall reaction (23452). Of course branched sequences without closed sequences can be found. They may be treated according to principles similar to those used above.

ELUCIDATION O F REACTION MECHANISMS

343

Let us take as a very simple case a reaction with a diagram as shown in Fig. 11 and the two sets of partial equations

+

*Xi

+ B2

A1 *Xi B2 Xi-* B3 __---

X2-1 B4

AI-+ BP

A I + Bi f B4 (SI)

which gives a1

a1

A1

+ B3 (S3)

= =

($3 (83

+ +

Sd/W12

Sd/W12

+ +

--

s3w21/w12w23 s4w21/w12w24

This shows at once that s3/s4

=

w23/w24

Since wz3 and w24 according to the mechanisms are constants, we thus find that the isomeric substances BS and Bq are formed always in the same proportion, although there is no stoichiometric connection between them. Wegscheider (27) seems to have been the first to state (in 1899) this consequence of reaction kinetics which to us seems nearly self-evident. In the works of Bray (28) and Abel (29) on the FIG.11 reactions between hydrogen peroxide and the halogens and their compounds many examples of branched sequences can be found.

V. INTEGRATION OF

THE

VELOCITYEXPRESSIONS AND COMPARISON

WITH

EXPERIMENTS

Let us revert to the case of linear, open or closed sequences with only one overall reaction, and let us assume that the reaction has been followed nearly to completion. It will follow from the sequel that in kinetic experiments it is always advisable to plan the experiments so that the determination of the degree or the number of advancement takes place a t equidistant times even when the reaction comes near t o completion, where the change is slow. It is further advisable to change the technique if practicable when the degree of advancement cy has become about 0.7 to 0.8 so that the values of its complement (1 - cy) may still be determined with reasonable accuracy. The number of advancement we shall denote by x (not to be confused with the concentration of intermediate i, xi) according to a very old tradition in kinetics. The result of the experiments then gives x as a function of time t, or more conveniently for the following t as a function of x. On the other hand, if the steady state method can be used, the formulas in the preceding paragraphs can be expressed in the form

344

J. A. CHRIGTIANSEN

where the (a’sare known rational or in some cases irrational functions of x and of the concentrations of ordinary molecules (not intermediates) a t time zero, while the A’s are constants which may contain the initial concentrations of ordinary molecules. The number of different functions is equal t o the number of steps in the sequence. T o determine the mechanism we thus have t o find a (finite) number of functions which by linear superposition gives the reciprocal velocity. For a preliminary orientation it is often a good practice t o find dt/dx by numerical (graphical) differentiation for different values of x in different experiments. This method has been used extensively by Bodenstein and his school. It is in many cases just as easy and more accurate to integrate the equation which gives

and compare the results directly with the experiments. The integrations are easy to carry out and, especially in the case of open sequences, the integrals occurring are nearly always of one or more of the forms x, In [a/(a- s)], [l/(a - s) - l/a], [ l / ( a - s ) ~ l/a2],etc. I n closed sequences with odd radicals as intermediates, other types of integrals, e.g., (4;- 4=), arc tan x, and others appear. In the case of linear sequences it is of interest to note that the theory of resolution of fractions in connection with the fact that each member in the sequence of functions q l , ‘p2, - * . is the foregoing multiplied by a rational fraction, shows that each integral q,ds is a linear combination of the foregoing integrals and a new one, It is therefore convenient always t o try expressions (valid for the single experiment) of the form

Jd:

t = Bill

+ BJz

*

.

*

where I , , I2 . . * respectively are, for example, one of the simple functions mentioned above, while the B’s are constants. When agreement with the experiment has been arrived at, one may proceed t o study the dependence of the constants on the original composition of the reacting mixture, all of course a t one temperature. When next the acquired expression for t is differentiated in respect to x, the mechanism, the sequence, can in many cases be read off directly from the expression for d t / d x , but there is no straightforward procedure by which the mechanism can in all cases be determined. The disentangle-

ELUCIDATION O F REACTION MECHANISMS

345

ment of reaction mechanisms may therefore appropriately be called a n art. Rut on the other hand, when a complete mechanism, which agrees with the experiments, has been found, one may for reasons given in the introduction be pract>icallycertain that it is the right one. It is well known that the mechanism found by experiment is often incomplete, this expression meaning that the equations for the partial reactions which have been derived from the kinetics do not add together t o give the equation for the overall reaction. Let us consider then an open sequence of four steps with the orientated diagram

2

3 4

1

0

A glance a t the partition matrix and L shows a t once t h a t the reciprocal velocity becomes a sum of two members only:

ads

=

l/wl

+ w-1/wim

while the rest of the sequence remains unknown. As it may very well happen that w-l/w2is a constant or that one of the members is too small t o be detected, only one member may remain, i.e., the information concerning the mechanism is rudimentary. To take a n example, DrostHansen (30)recently investigated the reaction 2Sr,Oe--

+ 60H-

a

5S203--

+ 3Hz0

and found that its velocity was proportional to the product of the concentrations of the pentathionate ions and the hydroxyl ions. This shows with certainty that the left-hand side of the first partial reaction in the sequence is the one in the equation: S60e--

+ OH-

= S,Oa--

+ -0sS2SOH

It is probable that the right side is as shown, since Foss (31) has given good arguments for the view that the main featur,: of th e reactions of the polythionates are displacements, usually of thiosulfate groups. It cannot be proved, however, by the kinetics of the reaction a s no inhibition by thiosulfate could be detected and still less could the fate of the intermediate HOSS20s- be ascertained kinetically. Even in the days of van’t Hoff many such cases were well knownmultimolecular reactions which were kinetically unimolecular or bimolecular or (rarely) trimolecular. But it was relatively recently, mainly in the second decade of this century, that chemists began t o realize th a t some reactions may very well be kinetically of a mixed type. To quote a case where the mechanism was complete, mention may be made of the

346

J. A . CHRISTIANSEN

work of Andersen (32) on the decomposition of hydrogen peroxide with ferric ions as a catalyst. He found that in the interval 1

> x / a > 0.05

his single runs could be accurately described by means of the formula

t

= A 1 In

(a/.)

+

- l/a)

where x is the amount of remaining peroxide a t time t and a the same at time zero. Variation of the acidity and the concentration of ferric salt showed that A1 and A a were both proportional to the hydrogen ion concentration and further, that A l was proportional to the reciprocal ferric ion concentration, while A2 was (practically) independent of that variable. Furthermore the dependence of the two constants on temperature was investigated (33). It turned out (33a) that these results led t o the following sequence of reactions Fe+++

+ HO- +

H202 $

FeOOH++

+ HzO

+ H62Hbz- + HzOZ--)O2 + HzO + HOFeOOH++ $ Fe+++

*

where HOz- means some “active” form of the hydrogen peroxide anion. It is seen that the three equations add together to give the well-known overall reaction, and as the irreversible reaction is the last one, the sequence is complete. 1. Determination of the Best Possible Values of the Constants

In a kinetic experiment we aim at finding a linear combination of certain known functions 11,1 2 , . * of x , the number of advancement, which is equal to the time (cf. p. 344). The dependence of t on x may therefore be illustrated by a straight line passing through the origin a t an angle of 45’ in a diagram in which the abscissa is the time and the ordinate the function f(x) = ZBJi. This function may for obvious reasons be called a (chemical) clock or a chronometric integral (chronomal). It is therefore best t o plan the single run so that the points plotted are evenly distributed on that part of the line which is accessible to sufficiently accurate determinations of x, let us say from zero t o 95% of the maximum value of x. It may be emphasized again that it is often advantageous to refine the analytical technique toward the end of the reaction. The nature of such possible refinements varies from case to case and must be left t o the judgment of the investigator. It may in

ELUCIDATION OF REACTION MECHANISMS

347

some cases be possible t o get useful results even above 95% of the maximum value. Anyway by the simple device of extending the measurements in a single run so far as possible, a very pronounced variation in the concentration of a t least one of the reactants can be obtained with relatively little extra labor. It follows from the above that it is a good plan t o take readings at equidistant times, although one may be tempted to use larger time intervals toward completion of the reaction. The actual choice of the constants which is necessary during the process of ascertaining the mechanism may be made by a method of trial and error. However, when the mechanism has once been ascertained, or rather when the calculations have shown that the functions I are chosen so that some linear combination of the 1’s gives a sufficiently accurate measure of the time, one may want t o find the best possible values of the constants B for the experiment in question. The problem of determining the constants can be solved by the statistical method known as regression analysis. This method is not only the standard method but it has also been proved th a t when the functions are known and a number of corresponding values of time and degree of advancement have been ascertained by experiment the constants determined by this method are the most probable ones. Details of the way in which the calculations are performed may be found iil a textbook on statistics. The method resembles the well-known “method of least squares,” but it has the characteristic feature th a t a system of rationally determined weight factors is introduced. If the calculations are carried through to completion, the method gives not only the most probable values of the constants themselves but also the probable errors of the different constants. Although the work of calculation becomes easier when measurements have been taken t o equidistant times, the method is nevertheless time consuming, so much so th at investigators who have not a calculation department at their disposal may not care to use it. If regression analysis is used, a warning must be given. The method is purely mechanical and will therefore always yield a result. But if the functions chosen are not appropriate, the curve will show systematic deviations from the straight line of a wavelike character. The procedure must therefore be the following: A sufficient number of (z,t) pairs is selected from the experimental material. They are inserted in the chronomal and the resulting, usually linear, equations are solved for the constants. If insertion of these constants yields a fair agreement with the straight line, they may be improved by regression analysis, but if the deviations

348

J. A . CHRISTIANSEN

outside the selected points are large and systematic the work spent on a regression analysis would be wasted. After the choice of the constants has been made, e.g., b y regression analysis, it is absolutely necessary either t o draw a large-scale diagram of the relation between the real times and those shown by the clock or better t o prepare a table containing the real times and those calculated by means of the chronomal. By this procedure any systematic deiriations can a t once be detected. If the constants have been found by regression analysis and systematic deviations still exist, this means that the mechanism of the clock is incorrect, i.e., th a t the functions li have been incorrectly chosen. If on the other hand the constants have been chosen by a preliminary estimation, it may be possible t o adjust them so th a t the desired straightline graph is obtained i.e., so that only unsystematic deviations remain. Of course this adjustment of the constants may need repetition, but t o the average chemist such “graphical experiments” are usually easier t o perform than a regression analysis, and the method has the advantage th a t any gross error, i.e., a wrong choice of the functions, is disclosed. 2. The Dependence of the Velocitg Constants o n Temperature

When the mechanism of the reaction has been settled a t one temperature the knowledge of the different velocity constants must usually be completed by experiments a t other temperatures. For this purpose it is essential t o get the best possible values of the constants a t each temperature. It must be said that it is not always possible t o determine the values of the constants themselves, but it is always possible t o determine ratios of the form k l / k z or analogous ratios. As we have some theoretical knowledge of the absolute values of the constants, when the type of reaction and the dependence on temperature is known i t is often possible t o judge whether the values for the constants or ratios between pairs of constants have reasonable values. I n this way one may sometimes be able to corroborate or t o correct the assumed mechanism. We give the following example. When Sten Andersen had completed his experiments (32) at 25” he was able t o propose a mechanism containing oxygen atoms as intermediates whose results conformed with the experiments so far as the dependence on the different concentrations was concerned. However, when he extended his measurements (33) to other temperatures ranging from 10°C. t o 35”C., the choice of oxygen atoms as intermediates could not very well be maintained. The reason was th a t with this choice the ratio between two velocity constants belonging t o reactions of the bimolecular type became practically independent of temperature and of the

ELUCIDATION O F REACTIOEJ MECHANISMS

349

order of magnitude lo9. It is known from the general theory of bimolecular reactions that the frequency factors of the velocity constants of such reactions arp roughly of the same order of magnitude and this is quite incompatible with the above result. The difficulty was resolved (33a) by replacing oxygen atoms as intermediates by a n ‘‘active” form of hydrogen peroxide anion (cf. the sequence p. 346). Another case may be mentioned. In the investigation of the dependence on temperature of velocity constants it sometimes happens th a t the logarithm of the velocity constant plotted against the reciprocal Kelvin temperature (the Arrhenius diagram) appears not as one nearly straight line but shows a distinct bend connecting two lines with different slopes. Obviously this may be explained by the assumption th a t the constant is really composed of two each with its own heat of activation, but a closer examination shows that we have t o distinguish between two qualitatively different cases. The first one is well known and has been discussed by Hinshelwood (34). A simple example is the following. A certain substance A may decompose unimolecularly in two different ways, but for experimental reasons i t may be impossible to distinguish between the two ways. The velocity constant k thus is a sum of two, i.e., k = k l kz. As the single constants with sufficient approximation may be represented by expressions of the form

+

k

=f

mexp ( - A / T )

:a_

where f is the so-called frequency factor and A is the “ h e a t” of activation expressed in degrees Kelvin, it is obvious th a t if the heats of activation are :different the curve in the Arrhenius diagram must have a bend a t the temperature Tb Ink where f1exp ( - A Tb) = ft exp ( - A 2/Tb). I n Fig. 12 separate lines have been drawn representing In kl and In k2 respectively. It is obvious from the figure that a t temperatures higher than Tb, In k = In ( k l k,) must con1/T verge asymptotically against line 2, while at lower FIG.12 temperatures i t must converge against line 1. This shows t ha t if a velocity constant is a sum of two the curvature in the Arrhenius diagram must always be as shown. The second case is not so well known. Let the mechanism be represented by a n open sequence with two distinct steps which are both unimolecular. A=x~ x1+ Products

+

350

J . A. CHRISTIANSEN

I n that case the formula gives a t once a/s = l / k l

+

The reaction is thus kinetically unimolecular but the velocity constant is defined by

+

Ilk

= 1/~1

where K~ = kl and K Z = kllcz/k-l. I n this case we get In k

=

~ / K Z

+

- In ( 1 / ~ 1

l / ~ q )

Assuming that K~ and K 2 can be represented by formulas similar t o those used in the first case, the bend must be at the temperature Tb defined by f i exp ( - A i / T b ) = fz exp ( - A z / T b )

znkl \

just as in the former case. But the curvature must be just the opposite which appears from the diagram below. The two lines represent In K~ and In K~ respectively. It is evident that a t temperatures different from Tb the largest member in the sum 1 / ~ 1~/ ~ 2 represents asymptotically the whole sum. But the larger member must be the one which has the smaller In K and therefore the curvature must be I / T as shown. If therefore experiments have shown the existFIG.13 ence of a distinct bend in the Arrhenius line, the curvature shows whether the reaction is composed of two parallel reactions (case 1) or of two reactions in series (case 2). Of course the sequences may be more complicated, but in principle the result is always the same. If the reactions are in parallel the curve is concave upwards, and if they are in series it is concave downwards.

+

VI. CONCLUSIVE AND HISTORICAL REMARKS The methods described above have been developed during a period of many years. They came as natural consequences of efforts to clear up as easily as possible the mechanisms of reactions which had more or less unorthodox kinetics. Some of the ideas are therefore old while others, for example the representation of a closed sequence by means of a screw line, are of quite recent date. The same is true also of the construction and application of the partition matrix. Other methods in which the same principles are used do actually exist. This is true of the method used by Skrabal (35), who by his criticism

ELUCIDATION OF REACTION MECHANISMS

351

especially of the steady state method has contributed so much t o sharpen and refine the ideas of chemical kinetics. Other scientists, among them Hearon (36), have simply taken up the fundamental ideas, especially the expressions for the reciprocal velocity of linear (open or closed) sequences and used them as they stand for their special purposes or have developed them in several directions. In this connection it may be mentioned that Hammett (37) recommends theuse of such expressions. As a more recent example it may also be mentioned that Schginheyder (38) with the same method arrived at a rather unexpected mechanism for an enzymatic reaction, the saponification of racemic i-caprylyl glycerol, by means of a certain lipase. Two classical papers on the mechanism of enzymatic reactions employ also the same principles, although t,he method is different. The first one is a paper of 1903 by Henri (39) which nowadays is not so well known as it deserves. In this paper Henri develops the theory which ten years later was used with such success by Michaelis and Menten (40),but while the latter authors restrict themselves to a rather special case, Henri’s ideas comprise more general cases and his paper contains. many valuable observations which, even today, are well worth reading. It should be added that chemical kinetics, like any other science, can be developed only through the cooperation of the members of a team comprising all contemporary workers in that field and that any advance rests on the work of former and contemporary scientists. Such considerations form the background for the presentation of problems concerning the elucidation of the mechanisms of chemical reactions given in this paper. The author wishes to conclude with the remark that the determination of the mechanism of a chemical reaction is an experimental problem. The methods used for the interpretation of the experiments may vary, but the experiments themselves form the fixed basis from which we draw our conclusions, and when there is contradiction between the results arrived at by means of some theory and the experiments, the theory and not the experiments is to blame. The editors have added the following historical notes. The first paper by Christiansen treating the disentanglement of the mechanism of a chemical reaction was the one from 1919 (9). I n this the mechanism of the hydrogen bromide formation was elucidated on the basis of the experiments of 1907 by Bodenstein and Lind (16a). It was independent of but simultaneous with the papers by Herzfeld (8) and Polanyi (10). In 1921 in a book in Danish (11) the theory of a reaction proceeding in two steps through an “active” intermediate was set forth and used to

352

J. A. CHRISTIANSEN

explain the occurrence of reactions, especially gas reactions, which are of the first order, although their mechanism of activation is certainly bimolecular. .This explanation was independent of but simultaneous with Lindemann’s, which was announced a t the meeting of the Faraday Society in 1921. I n the same book the word chain reaction (Kcedereaklion, in Danish) was introduced in chemical literature. Bodenstein and Nernst of course knew the phenomenon but they did not use the word. I n connection with a discussion on the characteristics of chain reactions it was emphasized that “ negative catalysis” (inhibition) can in some cases be explained only by the assumption th a t the reaction in question is a chain reaction. The ideas from the dissertation were then further developed in a paper with Kramers whose intimate knowledge of the theories of Bohr concerning atomic structure contributed essentially t o the correct representation of the theory. I n this paper inter alia the possible occurrence of explosive chain reactions, i.e., chain reactions in which no steady state concentrations of the intermediates is obtainable, was mentioned. The surprising results by Moureau and Dufraisse (17) concerning antioxidants were explained as being due to autoxidation proceeding as a chain reaction, the chains being broken by the antioxidant. In 1924 in a note on negative catalysis (12a) Christiansen’s views were further substantiated. This paper induced Backstrom t o enter on his well-known investigations on inhibition of autoxidations (18). The idea that bimolecular reactions generally, in gas and in solution, should have about the same frcquency factors was mentioned in another paper (41) in 1924. For gas reactions it was not a t all new a t that time, and the step t o reactions in solution is not large. The expression for the reciprocal velocity of a reaction proceeding through a linear sequence of reactions with short-lived intermediates was first published in 1931 (19). In th at paper it was used to disentangle the mechanism of the thermal hydrolysis of methanol catalyzed by copper. I n 1935 in a paper (19a) on the theoretical aspects of the method of 1931 a number of definitions and notations used in the foregoing paragraphs of the present paper were introduced. A further development of the method, especially concerning its application to catalytic (enzymatic) reactions appeared in 1949 (7b). I n this the partition matrix and orientated diagrams are introduced. It contains also a discussion of the time necessary for the establishment of the steady state.

REFERENCES 1. Gibbs, 3. W., Collected Works. Vol. I, 1906,p. 140. 2. Otto, C. E., and Fry, H. S., J . Am. Chem. SOC.48, 269 (1924).

ELUCIDATION O F REACTION MECHANISMS

353

3. Christiansen, J. A,, J . Chent. SOC.1926, 413. 4. Christiansen, J. A., J . Am. Chem. Sac. 43, 1670 (1921); (a) Christiansen, J. A., and Huffmann, J. R., Z. physik. Chem. A161, 269 (1930). 5. Bodenstein, M.,Z. physik. Chem. 86, 347 (1913). 6. Chapman, D. L., and Underhill, L. K., J. Chem. SOC.103, 500 (1913). 7. Christiansen, J. A., (a) 2. physik. Chem. 128, 430 (1927); (b) Acta chem. Scand. 3, 493 (1949). 8. Herzfeld, K. F., Z. Elektrochem. 26, 301 (1919). 9. Christiansen, J. A,, Danske Videnskab. Mat.-fys. Medd. 1, No. 14 (1919). 10. Polanyi, M., Z. Elektrochem. 26, 50 (1920). 11. Christiansen, J. A,, Reaktionskinetiske Studier. Copenhagen, 1921, p. 58. 12. Christiansen, J. A., (a) J . Phys. Chem. 28, 145 (1924); (b) Trans. Faraday Sac. 24, 596 (1928). 13. Christiansen, J. A., and Kramers, H. A., Z. physik. Chem. 104,451 (1923). 14. Hinshelwood, C. N., Kinetics of Chemical Change. Editions from 1926 to 1947. 15. Semenoff, N., Chemical Kinetics and Chain Reactions. Oxford, 1935. 16. Bodenstein, M., and Liitkemeyer, H., Z. physik. Chem. 114, 208 (1925). 16a. Bodenstein, M., andLind, S. C., Z. physik. Chem. 67,168 (1907). 17. See, for example, the report b y C. Moureau and Dufraisse in Reports of the Solvay International Council on Chemistry, Brussels, 1925. 18. Backstrom, H., J. Am. Chem. SOC.49,1460 (1927); Medd. Nobel Inst., Stockholm 6, No. 15 and 16 (1927). 19. Christiansen, J. A., 2. physik. Chem. (Bodenstein-Festband), 69 (1931) ;(a) B28, 303 (1935). 20. Eyring, H., and Polanyi, M., Z. physik. Chem. B12, 279 (1931). 21. Marcelin, R., Ann. phys. 3, 120 (1915). 22. Arrhenius, S., 2. physik. Chem. 4, 226 (1889). 23. Christiansen, J. A., 2. physik. Chem. B33, 145 (1936). 24. Zwolinsky, B. J., Eyring, H., and Reese, C. E., J . Phys. & Colloid Chem. 63, 1426 (1949). 25. Darling, S., Studier over den enzymatiske transaminering. Dissertation, Aarhus, 1951. 26. Wegscheider, R., Z. physik. Chem. 39, 266 (1902). 27. Wegscheider, R., 2. physik. Chem. 30, 594 (1899). 28. Bray, W. C., and Livingstone, R. S., J . Am. Chem. Soc. 46, 1260 (1923); cf. several later papers by the same authors in the same journal. 29. Abel, E., 2. physik. Chem. 96, 1 (1920). 30. Christiansen, J. A., Drost-Hansen, W., and Nielsen, A. E., Acta Chem. Scand. 6, 333 (1952). 31. Foss, O., Kgl. Norske Videnskab. Selskabs Skrifter No. 2 (1945). 32. Andersen, V. S., Acta Chem. Scand. 2, 1 (1948). 33. Andersen, V. S., Acta Chem. Scand. 4, 914 (1950). 33a. Christiansen, J. A., and Andersen, V. S., Acta Chem. Scand. 4, 1538 (1950). 34. Hinshelwood, C. N., Kinetics of Chemical Change., 1942, p. 45. 35. Skrabal, A,, (a) Homogenkinetik. Leipzig, 1941; (b) Monatsh. 80, 21 (1949). 36. Hearon, J. Z., Bull. Math. Biophys. 11,29 (1949); also later continuations. 37. Hammett, L. P., Physical Organic Chemistry. New York and London, 1940. 38. SchGnheyder, F., Volquartz, K., Biochim. et Biophys. Acta 6, 147 (1950). 39. Henri, V., Lois g6n6rales de I’action des Diastases. Dissertation, Paris, 1903. 40. Michaelis, L., and Menten, M. L., Biochem. 2. 49, 333 (1913). 41. Christiansen, J. A., Z. physik. Chem. 113, 35 (1924).