Monomolecular Reactions in Modern Chemical Kinetics

O. Z. Roginsky, Leningrad

The classical doctrine of the rates of chemical reactions, created by the works of physical chemists of the second half of the nineteenth century, did not single out monomolecular reactions from the general range of kinetic problems as a special object of investigation. Processes of this type attracted the attention of researchers only in the 1920s of the present century, in connection with the discussion that arose around the radiation theory of chemical processes; after that they no longer left the forefront, having by the present time become one of the main problems of kinetics. The serious advances achieved in this field in recent years are poorly known to broad circles of Soviet chemists and physicists, as are a number of new problems and tasks that have arisen in connection with them—which is quite natural, since review articles on this question are still entirely lacking, and the number of original Soviet works on monomolecular reactions is very small. Therefore, before setting forth the present state of the question, I shall give a brief introduction, depicting the problem in its successive development.

§ 1. Difficulties of the Formal Kinetics of Monomolecular Reactions

The fundamental foundations of classical chemical kinetics are the law of mass action, the kinetic and thermodynamic interpretation of which encounters no obstacles...

difficulties, and Arrhenius’s doctrine of the critical energy increment, or, as it is now customary to say, of the activation energy, which is an elegant interpretation of the exponential increase of reaction-rate constants with temperature and which for a long time had no rigorous justification. The usual physical explanation of the Arrhenius equation

\[ K = B \cdot e^{-E/kT} \tag{1} \]

is reduced, as is known, to the assumption of a direct connection between the temperature dependence of the rate constant and the Boltzmann distribution of energies, which also leads to an exponential increase in the probability of finding in the system molecules with an energy exceeding some critical value, large in comparison with \(kT\). These two notions naturally and simply lead to rate equations for reactions of the second and higher orders, but their application to monomolecular reactions encounters fundamental difficulties, the meaning of which is clear from a comparison of equation (2), giving an expression for the number of collisions in a gas \(Z\) as a function of concentration, with the equation for the constant of monomolecular reactions (3):

\[ Z = \sqrt{2\pi}\cdot \sigma^{2}\cdot u n^{2}\cdot e^{-E/kT}, \tag{2} \]

\[ -\frac{dn}{dt}=K(n_{\mathrm{нач}}-n_{\mathrm{пр}}), \tag{3} \]

where \(\sigma\) is the effective radius; \(u\) is the mean velocity of the molecules at the given temperature \(T\); \(n\) is the number of molecules in \(1\ \mathrm{cm}^{3}\); \(E\) is the heat of activation; \((n_{\mathrm{нач}}-n_{\mathrm{пр}})\) is the concentration of the decomposing substance at the moment \(t\). From comparison of these formulas it is evident that, whereas the reaction rate is proportional to the first power of the concentration, the number of collisions is proportional to the square of the concentration.

Therefore, within the framework of ordinary kinetics, monomolecular reactions always remained an entirely formal category of processes for which, for one reason or another, the influence of collisions of the molecules of the reacting substance [continues]

( substances) with one another or with molecules of the catalyst upon the rate is, for one reason or another, masked, as a result of which the observed kinetic picture is obtained. It is clear that from this point of view all monomolecular processes must be regarded as processes that, in their physical essence, are bimolecular. The experimental material then available on the rates of chemical reactions, collected chiefly in comparatively complex systems, mainly in solutions, agreed well with such a conception; and it is no accident that the classical school example of monomolecular reactions became the hydrolysis of sugar—a process for which, already by stoichiometry, besides the component determining the order of the reaction, the presence of water is necessary and, under ordinary conditions, also that of a catalyst. In view of the practical constancy of the concentrations of the last two components under the conditions of ordinary experiments, they may be included in the rate constant, which also explains the observed order. In other cases, for example in the decomposition of phosphorous hydrogen, monomolecularity was observed in an obviously heterogeneous autocatalytic reaction and was explained by the practical constancy of the magnitude of the catalyzing surface (under the conditions of the experiments the work is carried out on a portion of the adsorption isotherm obeying Henry’s law). All this led to an extremely cautious attitude toward monomolecular reactions, and, despite the fact that a number of processes in solutions whose monomolecularity is now beyond doubt had been known for a long time (table in the article by Christiansen and Kramers), the majority of kineticists until quite recently did not believe in the possibility of gas reactions that are substantially monomolecular, and treated the already known processes with justified distrust, without subjecting them to systematic study (the works of Trautz and Freundlich[^2] not accidentally failed to evoke sufficient response). As late as the 1920s it was necessary to prove the very possibility of the existence of true monoreactions[^3]. Therefore the first thoroughly studied homogeneous monomolecular process in the gas phase, discovered in 1921 by Daniels and John-

*

... in the decomposition of nitrogen pentoxide\(^4\), could not fail to attract attention, which was expressed at first chiefly in the persistent challenging of the correctness of the data obtained\(^5\) and which called forth a long series of further experimental works.\(^6\) The study of the decomposition of nitrogen pentoxide played, in the development of the problem that interests us, approximately the same role as the photolysis of a mixture of chlorine with hydrogen in photochemistry; therefore we shall dwell somewhat further on this process in order to become acquainted with the typical features of the group of reactions that interests us.

Already Daniels and Johnston investigated the decomposition of nitrogen pentoxide in the pressure interval from 600 to 5 mm and did not observe any appreciable changes in the rate constant \(K_{\mathrm{mol}}\) on passing to low pressures. In this case the reaction is completely homogeneous, since a very considerable change in the surface of the vessel or, for example, complete dust removal has no effect whatever on the rate.\(^7\) Later, in connection with the discussion of the mechanism of the process and the testing of Langmuir’s theory of monomolecular reactions (§ 3), the range of pressures studied was considerably extended toward low pressures, with determinations of \(K\) being continued down to pressures of 0.001 mm. It turned out that down to 0.06 mm no appreciable decrease of the constant is observed, but below 0.06 it begins to fall and eventually passes into a bimolecular one. At \(p = 0.003\) mm the value of \(K\) is already only 50% of the normal value. Thus we have two regions: a very broad interval, whose limiting conditions differ from one another by at least \(10^5\) * to unity and in which \(K\) does not change its magnitude, and a region of low pressures, in which \(K\) is a certain function of the pressure. Dissolution of pentoxide in most indifferent solvents has a negligible effect on the constant\(^8\), or addition to the gas-

* This ratio must be increased at least another 10 times if one takes into account the high concentration of \(N_2O_5\) in the solutions in the experiments of Joos, Ahrens, and Daniels.

the decomposition of nitrogen pentoxide with large amounts of various gases that do not react with it.^9 As is evident from the table given below, in eight different solvents the value of \(k\) for equimolecular solutions varies within narrow limits, although the measurements were made in solvents with very different physical constants. The principal data of the table are taken from the work of Eyring and Daniels.^8 The data of Giben,^8 which broaden the range of solvents studied, have not been included, being only semiquantitative.

A certain, but in general insignificant, influence on \(k\) is exerted by increasing the concentration of the solution. In passing from normal to saturated solutions the constant increases by barely 10%. Of all the solvents investigated, only in concentrated nitric acid is \(k\) appreciably, almost 25 times, smaller than normal,^10 but the heat of activation in this case is also considerably above the norm. An analogous observation is found for other monomolecular reactions as well (for example, for the dissociation of triphenylsulfobromide,^11 the isomerization of pinene,^12 etc.).

Such exceptional insensitivity to changes in the medium is in striking contradiction with the generally known strong influence of the solvent on the rates of most chemical reactions.* In particular, there is no indication whatever that the well-known empirical Walden relation is obeyed, which connects the rate constants of a reaction in different solvents with the dielectric constant of the solvent. A substantial increase in the stability of nitrogen pentoxide (and a sharp decrease in \(K\)) occurs only on passing to the solid phase, which, as has already been pointed out by some authors,^10 is apparently connected with the high heat of vaporization, which in one form or another must enter into the heat of activation. The temperature dependence of the rate constant of nitrogen pentoxide is completely normal, and the heat of activation is calculated to be \(24\,700\) cal. As is readily seen from Table I, for reactions in solutions the heat

* Quite recently Hinshelwood showed that certain reactions with rather complex kinetics, such as the decomposition of \(\mathrm{Cl_2O}\), are also very little sensitive to the solvent.

TABLE I

Rate constants and heats of activation for the decomposition of equimolecular solutions of nitrogen pentoxide in various solvents

Solvent	15°	20°	25°	35°	40°	45°	Heat of activation
Nitrogen tetroxide	0.159	—	—	—	—	—	25,000
Ethylidene chloride	—	0.322	—	2.54	(4.22)	(7.26)	24,900
Chloroform	—	0.274	0.554	—	(3.78)	(7.05)	24,800
Ethyl chloride	—	0.238	0.479	—	(3.70)	(6.21)	21,400
Carbon tetrachloride	—	0.235	0.469	—	(3.62)	(6.28)	24,200
Pentachloroethane	—	0.220	0.430	—	(3.26)	(6.02)	25,000
Bromine	0.114	0.215	—	—	—	—	24,000
Nitromethane	0.0747	—	—	—	(2.14)	(4.33)	24,500
Gaseous pentoxide	0.079	0.165	—	0.808	(2.52)	(4.73)	24,700
Carbon tetrachloride, saturated solution	0.183	—	—	—	—	—	—
Nitromethane, saturated solution	0.135	—	—	—	—	—	—
Propylene chloride	—	—	—	0.220	—	—	27,600
Nitric acid	—	0.0238	—	—	—	0.197	28,300
Gaseous pentoxide at low pressures	—	—	—	0.10	—	—	—

of the reaction is on average equal to 25,000 cal. Among other properties of the reaction, we should also note its low photochemical sensitivity over a very broad spectral interval and the absence of any indications of catalytic decomposition on the walls. All these features are expressed to a greater or lesser degree in other monomolecular reactions as well. Let us also note the discrepancy of the kinetics with the stoichiometric equation $2N_2O_5 = 2N_2O_4 + O_2$, which assumes bimolecularity; this discrepancy can be eliminated if it is assumed that the pentoxide initially decomposes into oxygen and a mixture of lower oxides, and that the tetroxide is already a secondary product of the reaction of the lower oxides with the pentoxide. If this is true, then the reaction is sharply endothermic ($-Q =$ about 23,000 cal), which represents a rather rare case for monomolecular processes.

§ 2. The Radiation Theory of Monoreactions

In the development of the theory of monomolecular reactions, a major role was played by the so-called radiation theory of chemical reactions, proposed in its time by Perrin and Lewis and playing an essential part in the formation of modern ideas on the mechanism of chemical reactions. In contrast to the usual conception, which regarded the moment of collision of two molecules as the basic stage of every chemical reaction, the radiation theory did not attach any significant importance to collisions, considering all chemical transformations to be explicit or disguised photochemical ones; moreover, according to this conception the difference between ordinary photochemical reactions and thermal reactions consists only in the active region of the spectrum: in the first case these are visible and ultraviolet rays, in the second—infrared radiations. The transfer to an individual molecule of the energy required for reaction (activation of the molecule) in both cases occurs by a radiation path, and the activation itself in thermal reactions is a consequence of excitation associated with the absorption of a quantum of radiation that is in thermal equilibrium with the walls of the vessel. The law of variation of the energy distribution in this radiation is given by Planck’s formula:

\[ U_\nu=\frac{8\pi h\nu^3}{c^3}\cdot \frac{1}{e^{\frac{h\nu}{kT}}-1}. \tag{4} \]

It is easy to see that the radiation density \(U_\nu\) in a certain spectral region \(\nu, \nu+\Delta\nu\), corresponding to the heat of activation \(E=h\nu N\), will increase exponentially with temperature, since for \(h\nu \gg kT\), equation (4) becomes

\[ \frac{8\pi h\nu^3}{c^3}\cdot e^{-\frac{h\nu}{kT}}, \]

i.e., it will give the very same exponential dependence that the rate of chemical reactions gives, and the independence of the constant …

rates on pressure, which at that time was considered valid at all pressures. By means of more or less speculative constructions one could also attempt to reduce to radiation schemes the change in reaction rates with a change of solvent (the radiation density increases with the refractive index) and a number of other phenomena. There was also no lack of attempts at a radiation interpretation of the catalytic action of solid surfaces, etc. Especially tempting was the application of the radiation scheme to cases of monomolecular reactions, since their most characteristic feature—the independence of the rate of decomposition from the number of collisions—received a very simple and elegant qualitative interpretation and acquired an absolute character (as was indicated above, in reality \(k\) is constant only in a certain pressure interval). As early as 1920 Lindemann pointed out the insensitivity, contradicting the radiation theory, of a number of chemical reactions to sunlight, where the intensity of infrared radiation of the corresponding frequencies is about \(10^{12}\) times greater than in the radiation of a black body at \(300^\circ\) K, which in a number of cases ought to be effective according to this theory. In some cases the decomposing compound has no absorption at all in the corresponding spectral region. A second difficulty was pointed out by Lewis\({}^{15}\), who noted that in many cases there is no basis for assuming the existence of radiation densities sufficient to explain the observed rates. This question was investigated in detail by a number of authors\({}^{16}\), chiefly on the example of the decomposition of nitrogen pentoxide. If, at a given temperature, the radiation density is

\[ h\nu=\frac{E}{N}=U_\nu, \]

where \(N\) is Avogadro’s number, then the \(k\) following from the radiation theory can be estimated approximately. Already the first estimates of this type, made by W. Mac Lewis from the number of collisions of molecules with quanta of a stream of radiant energy of the given density \(U_\nu\), gave very unsatisfactory results. The calculated rate constant

turns out to be approximately \(10^6\) times smaller than that observed experimentally.

There was no shortage of attempts to overcome this discrepancy within the framework of the radiation theory by means of various additional assumptions about the nature of absorption, about chains, or about the photochemical sensitivity of nitrogen pentoxide to a broader spectral region[^17]. However, all these constructions could not bring the theory into satisfactory agreement with experiment. The final and general rejection of it followed after experiments investigating the direct action of infrared radiation on the reaction. I shall confine myself here to mentioning the elegant works of Mayer and Rice, Urey, and Washburn[^18], who passed a well-collimated molecular beam of nitrogen pentoxide through diaphragms along the axis of a hot furnace. In this way intense irradiation was achieved without the possibility of activation by impact with the surface. Despite the fact that the radiation density under these conditions exceeds that at low temperatures by \(10^{10}—10\), no appreciable acceleration of the reaction could be detected. This and a number of similar works led to the final rejection of the radiation theory. The only positive result of the discussion, which lasted several years, was the definitive establishment of the proposition that activation occurs by collisions, and a certain expansion of the range of studied gaseous monomolecular reactions. Considerably more substantial results were brought by another scheme, proposed as early as 1922 by Lindemann, but developed and becoming generally accepted much later.

§ 3. Lindemann’s Conception

Lindemann was the first to point out[^19] that the contradiction between the quadratic dependence of the number of collisions on concentration and the linear dependence of the rate of a monoreaction disappears if one assumes that the percentage of chemically effective activating collisions is very small and that, consequently, despite the reaction, the system must contain...

S. Z. Roginskii

the concentration of active molecules corresponding to thermodynamic equilibrium must be maintained. This conception makes it possible not only to explain monomolecularity qualitatively, but also to predict certain new features of this type of reaction, in particular the obligatory transition to bimolecularity at sufficiently low pressures, when the basic premise of the theory can no longer be fulfilled. Statistical mechanics gives the following expression for the probability of finding in the system molecules with internal energy exceeding some critical value:

\[ W=\frac{1}{\Gamma\left(\frac{n}{2}\right)} \left(\frac{E_0}{kT}\right)^{\frac{n}{2}+1} \cdot e^{-\frac{E_0}{kT}}, \]

where \(\Gamma\) is the gamma-function symbol, and \(n\) is the number of quadratic terms of which \(E>E_0\) in the molecule is composed. For small molecules the first two factors have little significance, but for large \(n\) the magnitude of this product becomes quite substantial. In a state of equilibrium the number of activating collisions must be equal to the number of deactivating ones; it is obvious that the total number of collisions involving active molecules per unit time in unit volume is always equal to \(aN^2W\), where \(a\) is Boltzmann’s constant:

\[ a=4\sigma^2\sqrt{\frac{\pi kT}{m}} \]

and \(N\) is the concentration of molecules in the distribution. It is obvious that \(aNW\) is the rate of activation at low pressures and the rate of deactivation at high \(p\).

Deactivation in the volume is possible in two ways: the first way is deactivation upon collision with a new cold molecule; the second is a chemical transformation with the formation of a new chemical individual. If in the reacting system the concentration of active molecules is equal to \(y\), then the rate of deactivation will be \(a\cdot y\cdot N\). The number of chemically transforming molecules depends on the concentration of active

molecules and on some constant \(b\). Obviously, the equality always holds:

\[ aN^{2}W=ayN+yb,\qquad Z=Z_{1}+Z_{2}, \tag{6} \]

whence

\[ y=\frac{aN^{2}W}{aN+b}, \tag{7} \]

and the rate constant \(K_{m}\) in the general case will be written:

\[ K_{m}=b\cdot y/N=\frac{aN^{2}W}{aN+b}\,b. \tag{8} \]

For sufficiently large \(p\) (and hence \(a\) and \(n\)) we obtain the equation (9) for monomolecular reactions

\[ K_{m}=K_{\infty}=\frac{aN^{2}W}{aN}\,b=NWb. \tag{9} \]

As is easy to see from equations (8) and (9),

\[ K=\frac{K_{\infty}\cdot aN}{N+\dfrac{K_{\infty}}{aW}} = \frac{K_{\infty}p}{p+K_{\infty}\cdot\dfrac{kT}{aW}}, \tag{10} \]

where \(p\) is the pressure, and \(k\) is the gas constant \((p=kTn)\). This equation is more conveniently reduced to the following form:

\[ \frac{1}{K}=c+\frac{b}{p}, \]

where

\[ c=\frac{1}{K_{\infty}} \quad\text{and}\quad b=\frac{kT}{aW}. \tag{11} \]

Substituting the expression for \(W\) for a molecule with \(n\) quadratic terms, we obtain the expression for \(b\) in expanded form (12):

\[ b= \frac{ \sqrt{m}\,(kT)^{\frac{n-1}{2}}\cdot e^{\frac{E_{0}}{kT}}\,\Gamma\!\left(\frac{n}{2}\right) }{ 4\sqrt{\pi}\,\sigma^{3} I_{0}^{\frac{n-2}{2}} }. \tag{12} \]

Let us also note the assumption, made by us in hidden form in deriving equations (10)—(12), concerning the equal probability of chemical transformation of active molecules with equal \(E>E_{0}\). From equation (12) it is obvious that in the case of large mo-

the molecules of the heat of activation \(E_0\), calculated from thermal rate coefficients, give not the critical energy, but this quantity diminished by some multiple of \(kT\). In other words, kinetically we always measure not \(E_0\) itself, but only the excess of \(E_0\) over the mean energy content of the molecule for the given temperature:

\[ \frac{d \log K_\infty}{dT} = \frac{d \log W}{dT} = \frac{E_0-\frac{n'-2}{2}\,kT}{kT^2}. \]

If the number of quadratic terms is large, this correction may be very appreciable. From equations (10) and (11) it is obvious that monomolecularity can occur only at sufficiently high pressures, when the second term may be neglected and

\[ K=K_\infty=\frac{1}{c}=\mathrm{const}, \]

and that consequently for every reaction there must exist a certain minimum pressure, below which a deviation from monomolecularity begins, and that finally, at sufficiently low pressures, every monoreaction must pass over into a bimolecular one. This corresponds in equation (6) to the conditions when \(Z_2\) becomes larger than \(Z_1\); monomolecularity corresponds to the condition \(Z_1 \gg Z_2\), and the transition region to equality of both terms in order of magnitude. This fundamental conclusion of Lindemann’s theory has been confirmed for all, without exception, of the monomolecular reactions studied in any detail. For a long time the general regularity was violated by nitrogen pentoxide, for which the minimum pressure is especially small, but for it too Tolman, Ramsperger, and Schumacher were able to prove a decrease of the constant at pressures below one hundredth of a millimeter.

From Lindemann’s scheme there also naturally follows the possibility of restoring monomolecularity and the normal value of the rate constant by adding a sufficient concentration of indifferent gases, since the transition to bimolecularity at low pressures is determined—

...is explained only by an insufficiency in the number of deactivating collisions, which may be collisions not only with molecules of the same kind, but also with foreign molecules.

This phenomenon, predicted by the theory, was in fact first found by Hinshelwood[^23], who observed the restoration of the normal rate constant upon adding various foreign gases to the organic substances he studied, and was later confirmed by a number of other authors. By adding a sufficiently large amount of a foreign gas, it is possible, at small partial pressures of the decomposing substance lying considerably below the minimum pressure, to obtain rates corresponding to \(K_{\infty}\). At the same time, different gases act with unequal strength, for example on \(K\) for methyl ether, according to Hinshelwood’s data: hydrogen acts more strongly than other gases and almost as strongly as the corresponding increase in the concentration of the ether itself. The remaining gases He, \(N_2\), CO, \(CO_2\) do not exert a significant action on \(k\). These relations are connected with the conditions of energy exchange between the molecules of the decomposing substance and the molecules of the additive; therefore it is natural to expect considerable specificity. This is in fact the case: thus, if in the decomposition of ether hydrogen occupies first place, then in the decomposition of nitrosyl chloride hydrogen is not distinguished more than the other gases, and a mixture of chlorine and nitrogen dioxide acts most strongly; for nitrogen pentoxide the addition of helium has a strong effect, etc.

Fig. 1. Circles—values of \(t\) for ether without additives; crosses—the same, with additions of large amounts of hydrogen.

It is very essential that, in complete agreement with the theory, indifferent impurities can significantly influence \(K\) ...

TABLE II

Time of 25% decomposition of dimethyl ether at different pressures and with addition of hydrogen. \(T=504^\circ\text{C}\).

I Initial pressure of ether, mm	II Initial pressure of \(H_2\), mm	III Time of 25% consumption	I Initial pressure of ether, mm	II Initial pressure of \(H_2\), mm	III Time of 25% decomposition
28	0.0	23 min	422	0	8 min 23 sec
58	0.0	25 min	509	0	7 min 45 sec
91	0.0	19 min	586	0	8 min 04 sec
150	0.0	15 min	894	0	8 min 1 sec
241	0.0	11 min 07 sec	150	0	15 min — sec
312	0.0	11 min 05 sec	150	200	11 min 05 sec
394	0.0	9 min 50 sec	150	300	8 min 43 sec
			150	400	7 min 44 sec
			150	600	7 min 55 sec

only at small pressures; their influence is entirely absent in the monomolecular region, and, with the exception of examples of evident catalysis, an increase of the rate constant by additions above \(K_\infty\) has never been observed. In a number of cases for small molecules equation (11) was fully confirmed. Thus, for nitrous oxide, as Fowler and Nagasaki showed,^24

\[ \frac{1}{K}=10.5+\frac{3.55\cdot 10^3}{p}, \]

where \(p\) is the pressure of \(N_2O\) in mm Hg; for nitrosyl chloride at \(T=130^\circ\text{C}\), as is easily calculated from Schumacher’s data,

\[ \frac{1}{K}=19.5\cdot \frac{5160}{p}, \]

at \(T=100^\circ\text{C}\),

\[ \frac{1}{K}=50+\frac{87000}{p}. \]

The correctness of the proposed scheme can also be checked in another way.^25 Nagasaki made an attempt to use the same formula for taking account of the influence of indifferent gases on the rate constant by means of very simple transformations, allowing for the possibility of deactivation not only by means of

collisions with cold molecules of the decomposing substance, but also with molecules of the additives. Characterizing the effectiveness of each type of collision by a certain empirical coefficient \(\chi\), Nagasaki obtained the following formula for \(K\):

\[ \frac{1}{K}=c+\frac{b'}{p+\chi' p'+\chi''p''+\cdots}, \]

where \(\chi'\), \(\chi''\) and \(p'\), \(p''\) are, respectively, the specific coefficients and the partial pressures of the additives. Hence, knowing the value of the rate constant for different compositions of the gas from experiments with various additives, one can determine \(k\).

TABLE III

Values of \(\chi\) for various gases in the decomposition of nitrous oxide, according to Nagasaki

Added gas	\(\chi\)
Oxygen	0.21	\(\displaystyle \chi_1=\frac{A_1}{A}=\frac{D'}{D},\) where \(A\) and \(A'\), \(D\) and \(D'\) are coefficients characterizing the probability of activation \((A)\) and deactivation \((D)\) for the decomposing gas and the impurity. Equation (15) is derived from the obvious equality for the activation equilibrium: \(\displaystyle Ac^2+A_1c_1c+\cdots+A_nc_nc=\) \(\displaystyle =Dc c'+D_1c_1c'+\cdots+D_nc_nc'+\frac{C'}{t}\)
Nitrogen	0.26
Air	0.25
Carbon dioxide	1.2
Helium	1.0

The correctness of the scheme proposed by Lindemann can also be checked in another way, by comparing the rate of activation, calculated from the number of collisions of special molecules, with the observed rate of reaction. As we have already said, the check gives quite satisfactory results only in the case where it is assumed that the total energy \(E\) of the active molecule is composed of \(n\) parts, where \(n\) is the number of quadratic terms entering the expressions for the internal energy of the molecules, which, generally speaking, may not be [[unclear: word broken after “непо-”]]

directly related to its heat capacity*. If this assumption is not made, and the original Arrhenius formula, which was used in calculating the number of activating molecules for simple bimolecular reactions, is employed, the maximum possible rate proves to be considerably—by \(10^4\)—\(10^6\) times smaller than the reaction rate. Therefore the assumption that the energy of a large number of bonds is used constitutes an indispensable part of the Lindemann concept in its modern form. It is curious to note that this very simple, but essential correction to Lindemann’s original scheme was found very late**. Above we gave expressions for the probability of activation of a molecule. They can be applied to calculating the number of the quadratic terms to be taken into account, but for this, besides the basic formula, which contains at least two unknowns, \(Z\) and \(n\) (which may be regarded, though with little accuracy, as known), additional data are needed. Usually, in a calculation proceeding from the fundamental postulate of the Lindemann scheme, it is assumed that when \(p\) is slightly below the minimum \(Z\), \(Z_1\), and \(Z_2\) are of the same order. Then, in the expression for \(Z_1\), one chooses \(n\) that leads to the correct value of the absolute magnitude of the rate constant. Despite all the simplification of such a calculation, it nevertheless gives approximate values for \(n\) that are equal for reactions.

The results of several calculations by this formula are used in Table IV.

As we have already indicated above, equation (11) gives good results only for relatively small molecules; on passing to large ones, instead of the expected linear relation of \(K\) to the reciprocal pressure \(\frac{1}{p}\), more or less complicated curves are obtained (Fig. 2). Since the basic premises of the Lindemann concept are experimentally splendidly substantiated, it remains to suppose that the particular form of the derivation used above is insufficiently successful,

* Since what matters here is essentially not simply the total content of vibrational and rotational energy, but the part capable of being used in activation.

TABLE IV

Constants of some gas monomolecular reactions

Substance	\(p\), cm	\(T^\circ\) C	\(B\)	\(\lg_{10} B_2\)	\(E\)	\(E'\)	\(n\)
Azo compounds
\(\mathrm{CH_3N_2CH_3}\)	0,026—70,8	570° K 278—327	\(10^{16}\)	10,12	51 200	64 300	25
\(\mathrm{CH_3N_2C_3H_7}\)	0,0058—13,1	570° K 250—332	\(2{,}8\cdot 10^{15}\)	8,88	47 480	65 150	35
\(\mathrm{C_3H_7N_2C_3H_7}\)	0,025—4,60	540° K 250—290	\(5{,}6\cdot 10^{13}\)	7,20	40 900	62 000?	42?
\(\mathrm{CH_3N_2(NCH_3)}\)	0,019—8,0	490° K 200—230	\(4\cdot 10^{11}\)	8,08	33 800	40 660	15?
Ethers
\(\mathrm{CH_3OCH_3}\)	3,0—90,0	800° K 422—552	\(1{,}5\cdot 10^{13}\)	9,4	58 500		12
\(\mathrm{C_2H_5OC_2H_5}\)	2,5—50	800° K 426—588	\(3{,}1\cdot 10^{11}\)	9,2	53 000		7
\(\mathrm{CH_3OC_2H_5}\)	2,6—54,0	700° K 386—460	\(9{,}2\cdot 10^{11}\)	8,7	47 000		9
\(\mathrm{CH_3OC_3H_7}\)	2,2—33,6	700° K 400—450	\(2{,}2\cdot 10^{12}\)	8,3	49 000		12
\(\mathrm{C_2H_5CHO}\)	2,0—40,0	450—600°	\(1{,}4\cdot 10^{13}\)		54 000		11
\(\mathrm{C_{10}H_{16}}\) pinene	17,0—116,0	184—237	\(5{,}4\cdot 10^{14}\)		43 700		\(>20\)
\(\mathrm{CH_3OCH_3}\)	2,4—90,5	378—445	\(10^{12}\)		52 000		\(>14\)
\(\mathrm{N_2O_5}\)	0,0002—70,0	0—65	\(4{,}5\cdot 10^{13}\)		24 700		30
\(\mathrm{N_2O}\)	8,1—800	560—667	\(4\cdot 10^9\)		53 000		2
\(\mathrm{NO_2Cl}\)		100—150	\(10^{11}\;(2\cdot 10^5)\)		20 500 (13 400)
\(\mathrm{CH_3COCH_3}\)			\(1{,}5\cdot 10^{15}\)		68 500		--

and try to replace it by another. Indeed, these difficulties can be eliminated if, while preserving the basic ideas of Lindemann’s conception, one assumes that the probability of transformation of an active molecule is different for different \(E\), and that for the reaction a concentration of all the energy \(E\) in a small number of degrees of freedom is required.

Fig. 2. \(1/k\) as a function of \(1/p\) for: propionaldehyde \((I)\) \(849^\circ\), diethyl ether \((II)\) \(788^\circ\), propionaldehyde \((III)\) \(796^\circ\), dimethyl ether \((IV)\), azomethane \((V)\) \(603^\circ\), azomethane \((VI)\) \(563^\circ\) (according to Kassel).

Such calculations, under various special assumptions, were carried out almost simultaneously and in a very similar way independently of one another by Rice and Ramsperger\(^{24}\) and by Kassel\(^{25}\). The essential difference from the calculation given above, for the simpler case, consists in the fact that molecules with different energy contents must be considered separately, and instead of the simple expression for the probability of finding molecules with \(E\) exceeding the critical \(E_0\), one must take:

\[ W=\int_{E_0}^{\infty} W_E\,dE_0; \tag{13} \]

in exactly the same way, for each \(E\) we obtain our own expression for

\[ y=\frac{aN'W_{E_0}}{aN+b_{E_0}}; \tag{14} \]

then \(k_{E_0}dE_0\)—the total number of molecules with energy within the limits between \(E\) and \(E+dE\), decomposing per unit time, divided by the total number of all molecules per unit volume—will be equal to:

\[ k_{E_0}dE_0=\frac{aNW_{E_0}b_{E_0}\,dE}{(aN+b_E)} \]

and

\[ k=\int_{E_0}^{\infty}K_n\,dE=\int_{E_0}^{\infty}\frac{W_{E_0}b_n\,dE_0}{1+b_n\frac{kT}{ap}}. \tag{15} \]

It is clear that at high pressures the denominator of the subintegral expression becomes equal to unity, \(k\) does not depend on \(p\), and the expression for \(k\) coincides with that expected from the simple scheme. Continuing the analysis along the same lines, one can find an expression for \(W_{\bar v}\,dE_0\) and find the final expression for \(k\) that interests us. Since it is not possible here to give the complete derivation because of its unwieldiness, we shall confine ourselves only to the final, rather complicated expression for \(k\):

\[ K=\frac{K_{\infty}}{kT\,\Gamma\!\left(\frac{n+1}{2}\right)} \int_{E_0}^{\infty} \frac{ e^{-\frac{E-E_0}{kT}} \left(\frac{E-E_0}{kT}\right)^{\frac{n-1}{2}} }{ 1+\frac{\beta_2}{p}\cdot \frac{(E-E_0)^{\frac{n-1}{2}}}{\varepsilon^{\frac{n-2}{2}}} } \,d(E-E_0), \tag{15} \]

in which

\[ \beta_2=k_{\infty}\cdot \frac{\Gamma\!\left(\frac{n}{2}\right)} {\Gamma\!\left(\frac{n+1}{2}\right)} \sqrt{\frac{\pi}{m}}\, \frac{1}{4\sigma^2}\, e^{\frac{u_0}{kT}} . \tag{16} \]

Its rigorous integration is not possible and is of no substantial interest; therefore, in order to check it, one is satisfied with comparing the dependence on pressure obtained by expansion in a series with that observed experimentally. It is clear that, as in the case of the check by equation (12), \(n\) and \(\sigma\) remain undetermined, and their values are selected from experiment. Usually a definite value is postulated for \(\sigma\) by analogy with one or another molecule for which the gas-kinetic radius is known, which, generally speaking, is incorrect. A large part of the values of \(n\) given in Table VI was calculated in this way. Of course these calculations are very inaccurate, since it is necessary to select two quantities that are not amenable to independent determination; but a more rigorous solution of the problem is not yet possible, since, on the one hand, we as yet know nothing about the probabilities and laws of transfer of vibrational energy in collisions and within the molecule, and, on the other hand, experimen-

...the material in the part concerning the kinetics and the elementary experiments is extremely scanty and does not provide a sufficient number of points of reference, and therefore it is obvious that the simple statistical considerations used by Rice and Kassel in the derivation are a very crude approximation.

§ 4. The quantitative effect of the decomposition products

The restoration of the limiting decomposition rate by foreign gases explains one peculiar feature of most monomolecular reactions. Even when mono-reactions are carried out at initial pressures lying above the minimum, from a certain point \(p\) becomes smaller than \(p_{\min}\), and one might expect a corresponding decrease of the constant and a change in the order of the reaction with the further course of the transformation. In reality this is usually not observed, and in most cases the order of the reaction and the magnitude \(K\) remain constant over the entire length of the decomposition curve of each individual experiment. The sole indication of the validity of the Lindemann kinetic scheme in this case is the difference of the constants at different initial pressures. It can be shown by direct experiment that such constancy of \(K\) is connected with the restoring, activation-equilibrium action of the reaction product. Indeed, in all these cases the addition of reaction products at \(p < p_{\min}\) easily restores \(K = K_{\infty}\). This was observed by Ramsperger for azomethane, and the restoration of \(K\) proved to be connected with ethane while the second product of the azomethane reaction was inactive. Likewise, for nitrogen pentoxide, nitrogen tetroxide, etc., has a noticeable influence on the rate of decomposition in solutions. A fact of the same order, described by Schumacher for the decomposition of nitrosyl chloride, is still quite incomprehensible and has not yet been verified. Of the two final reaction products, nitrogen dioxide and chlorine, each separately has a very slight effect on \(K\), but when mixed in stoichiometric proportions they act...

... act very strongly and almost exactly compensate for the fall in rate that would have occurred as a result of the decrease in the partial pressure of nitrosyl chloride in the course of decomposition. All these facts as yet have no rational explanation, and their clarification will be possible only after a detailed investigation of the question of energy exchange in molecular collisions.

§ 5. Heat of Activation

The kinetic theories of monoreactions did not go beyond the question of the number of activating collisions and the probability of deactivation. The question of the mechanism of reactions and of the physical meaning of the activation energy did not play any substantially important role in them. Hence the formal character of these theories, which did not create special difficulties in solving the question of the absolute magnitude of \(K\) and of its changes with pressure, but has now become a serious obstacle to the further development of the field. Therefore the last two or three years have been marked by a number of attempts to clarify the mechanism of monomolecular reactions, which to a considerable extent reduces to the question of the physical meaning of the heat of activation in these processes. All the work on this question is grouped around two basic concepts. The first, in the most consistent form developed in Germany by Polanyi and Wigner\(^{26}\), regards the elementary basic act as the process of rupture of one of the bonds in the molecule, the work of rupture of this bond \(D\) being at the same time the heat of activation of the corresponding reaction. The free radicals (atoms) formed initially may then enter into one or other secondary reactions, but the primary process is the one that determines the rate observed experimentally. This concept was developed in very general form by Polanyi and Wigner for homogeneous and heterogeneous reactions and, independently of this, was advanced in various particular forms to explain definite groups or types of reactions; such, for example, are the works of Rice\(^{27}\), Berke\(^{28}\), Frost and Nimtz on the cracking of large molecules, the works of Kondrat’ev\(^{30}\) and Ko-

Bose^31 on monomolecular reactions, etc. The physical meaning of this scheme is clear from Fig. 3, which depicts a schematized potential curve for a certain bond. In the normal state the atoms in the bond are at a distance \(r_0\), corresponding to the minimum of the potential energy; when vibrational energy is imparted to the bond, the atoms will vibrate between certain distances \(r_1\) and \(r'_1\), corresponding to some new energy level.

Fig. 3.

Activation corresponds to the vibrational energy sufficient to separate the atoms to the distance \(r_2\), at which \(V\) reaches the limiting value required for rupture. With further increase of \(r_1\), \(V\) will no longer increase, and consequently the atoms (radicals) will be able to move adiabatically apart to any distances (bond rupture). The probability of rupture of a bond possessing energy \(D\) is very great, and therefore such a physical scheme contains no arbitrary or incorrect assumptions; the question lies not in its fundamental possibility, but in its applicability to the group of reactions that interests us and in the possibility of obtaining correct quantitative conclusions in this way. This question may now be regarded as essentially settled in the negative sense, but for a proper assessment it is necessary to dwell on the concrete conclusions drawn from this theory. Like all the new work on monoreactions, the rupture scheme with the formation of free radicals takes as proven the basic premises of the original theory of Lindemann, but concentrates attention on the region in which \(K = K_{\infty}\), where, consequently, the influence of collisions does not affect the constant and it is possible to try to determine the rate of the elementary processes that underlie it. The calculation of the absolute value \(k\) according to the theory of Polanyi and Wigner is not difficult to carry out under the condition of a sufficient schematization of the process. Polanyi and Wigner, in particular, used for the calculation a model of a system

large number of identical oscillators lying at some average distance from one another and connected by central quasi-elastic forces. Activation in such a system, possessing energy \(E\), corresponding, as it were, to an enormous uniformity in all parts of the molecule, may be regarded as a local fluctuation (running away) of the energy flow, propagating through the system with the speed of sound. Analysis of this case leads to very simple equations for the reaction probability and the rate constant. If two bound atoms are likened to a linear oscillator, the constant will be equal to:

\[ K = \nu e^{-\frac{E}{kT}}, \tag{22} \]

where \(\nu\) is the frequency of oscillations corresponding to atomic bonds (of the order of \(5 \cdot 10^{12}\)—\(5 \cdot 10^{13}\)). If the same calculation is carried out for a three-dimensional oscillator, the equation obtained is

\[ k = \nu \frac{2E}{kT} \cdot e^{-\frac{E}{kT}} \tag{23} \]

Since \(\frac{E}{RT}\) is usually of the order of 40, values of \(B\) from \(2 \cdot 10^{14}\) to \(2 \cdot 10^{15}\) are obtained.

Equations (22) and (23) are entirely analogous to the expression for \(k\) derived in its time by Deshman\(^ {32}\) from radiation theory on the assumption that \(E = h\nu N\), where \(N\) is Avogadro’s number, but which previously had no serious justification.

As confirmation of the correctness of their theory, Polanyi and Wigner cite the following statistical curve, compiled from the data of the summary table of Christiansen and Kramers (1923), taking into account the known monomolecular reactions as of the beginning of 1929 that were not included in that table.

In Fig. 4, decimal logarithms of \(B\) are plotted on the abscissa axis, and on the ordinate axis—the number of monomolecular reactions with the corresponding \(B\). Polanyi and Wigner consider the very sharp maximum at \(\log_{10} B = 13—14\) a sufficiently weighty confirmation of their scheme.

S. Z. ROGINSKY

Similar ideas in a less general form were developed independently of Polanyi and Wigner by V. Kondrat'ev, who attempted to relate the heats of activation of a number of reactions to the work of breaking certain bonds in the molecule, calculated from optical data. Another Soviet investigator, N. Kobozev, went considerably further in a work entitled “Activation as a Chemical Process.” For several monomolecular reactions (the decomposition of acetone, the isomerization of pinene, etc.) and a number of bimolecular ones, Kobozev attempted to show the coincidence of \(D\) with \(E\); moreover, in contrast to Kondrat'ev, who analyzed reactions of comparatively simple molecules, for which \(D\) is determined with sufficient accuracy, Kobozev had to use rather unreliable data for \(D\) for large molecules.

Fig. 4. Order of magnitude of \(\lg K\) for the equation
\[ K = B \cdot e^{-A\kappa}. \]

Fig. 5.
\(n\) — number of reactions in the given interval.
Most probable values for \(E\) and \(D\), in cal.
Indicated reactions include: \(NO_2 = NO + O\); \(SO_2 \rightarrow SO + O\); \(CH_3CHO \rightarrow CH_3CO + H\); \(H_2CO \rightarrow HCO + H\); \(C_2H_5OH \rightarrow C_2H_5O + H\); \(C-H\) bond, etc.

The works of Kondrat'ev and Kobozev represent an attempt to verify the assumption of the equality of \(E\) and \(D\), which is the point of departure of the theory of activation by dissociation into atoms and radicals.

It is easy to prove, however, that this equality in reality does not exist and that the overwhelming majority

monomolecular reactions proceeds at values of \(E\) amounting to no more than \(1/4\)—\(1/2\) of the value \(D\) of the least strong bonds in the corresponding molecules. In order not to encumber the article with an analysis of individual cases, we shall confine ourselves here to the statistical graph of Fig. 5 for \(E\), showing that for the overwhelming majority of monomolecular reactions \(E\) lies within the limits between 25 and 55 Kcal. As is evident from the values of \(D\) for various bonds given here in the graph, the latter, as a rule, for bonds of kinetic interest have a magnitude exceeding 100 Kcal (Table V). The coincidence of \(E\) and \(D\), obtained for individual cases by earlier authors, is due to the incorrectness of the \(D\) values used, which were corrected by later publications, and to the choice of reactions atypical in the magnitude of \(E\). Thus the first and basic assumption of the theory of Polanyi and Wigner is not justified for the majority of monomolecular reactions known at the present time1,2. This makes, incidentally, very doubtful, in the case where there are no long chains, the various “radical” theories of kinetics, which seek to explain the composition of the products in the direction of particular reactions by various interactions3,4 of the radicals formed upon decomposition.

TABLE V

Most plausible values of \(D\) for certain bonds of interest for kinetics

Bond	\(D\) in Kcal	Method of determination	Author
\(H_2\) bond \(H-H\)	\(100 \pm 2.3\)	Optical	Frank (1931)
\(O-O(O_2)\)	\(117.4 \pm 0.2\)	Optical	Frank (1931)
\(Cl-Cl(Cl_2)\)	\(56.9 \pm 0.2\)	Optical	Frank (1931)
\(N-N(N_2)\)	\(208—\)	Optical	Frank (1931)
\(C-H\) al	\(115\)	Indirect	Mecke (1931)
\(C-C\) al	\(110—115\)	Indirect	Mecke (1931)
\(H-OH(H_2O)\)	\(112\)	Optical	Mecke (1931)
\(NO-O(NO_2)\)	\(75—77\)	Optical	Mecke (1931)
\(C=O(H_2CO)\)	\(>150—\)	Optical	Mecke (1931)
\(N-H\) (ammonia)	\(>110\)	Optical	Mecke (1931)
\(C-O\) (ether)	\(>100—\)	[[unclear: method not visible]]	[[unclear: author not visible]]
\(C-H(CH_2OH\cdot CO+H)\)	\(107\)	[[unclear: method not visible]]	[[unclear: author not visible]]

In view of the great popularity of the works of Polanyi and Wigner and of Kobozhev among chemists, we shall indicate a number of further objections to their schemes, not connected with the inequality \(D \gg E\).

The most essential is the contradiction of equations (21) and (22) with the conclusions of the kinetic school concerning the entry into \(B\) of the distribution multiplier (in the simplest case

\[ \frac{\left(\dfrac{E}{RT}\right)^{\frac{1}{2}n-1}}{\Gamma\!\left(\dfrac{n}{2}\right)} \]
).

It is quite obvious that the rate constant of reactions for which the position of the critical pressure and the absolute value of \(K\), according to kinetic schemes, require the introduction of the distribution multiplier, must contain it also according to the Polanyi and Wigner scheme when the latter is carried out more rigorously; and its absence in Polanyi and Wigner is only a consequence of an extreme schematization with complete neglect of collisions. But for \(\dfrac{E}{RT}\) of the order \(20\)—\(40\) and \(n\) of the order \(8\)—\(30\), this additional multiplier may reach \(10^{13}\), and for part \(B\) of the following Polanyi and Wigner scheme there will remain

\[ \frac{B \Gamma\!\left(\dfrac{n}{2}\right)}{\left(\dfrac{E}{RT}\right)^{\frac{1}{2}n-1}}, \]

i.e., a quantity not at all coinciding in order of magnitude with \(\nu^{\ddagger}\) (Table IV). Among other contradictions of the Polanyi and Wigner scheme we shall note the arbitrariness and artificiality of the assumption of the chain character of all monoreactions with \(B > 10^{18}\), and the existence of a regular relation between \(B\) and \(E\) (see below), which is plainly in contradiction with this scheme. All these considerations undoubtedly prove the inapplicability of the rupture scheme as a general theory of monomolecular reactions. Given the enormous strength of ordinary, even the most labile, chemical bonds, processes proceeding with their rupture can play an essential role only at very high temperatures, or in the presence of strong chain development in the same temperature region in which the majority of kinetic material has been collected; as a rule, only reactions proceeding more economically are possible.

Thus, without directly overcoming such a high energy barrier. This essential result will be examined and substantiated in more detail in § 6. Here we shall confine ourselves to pointing out the origin of the sharp maximum on the Polanyi and Wigner curve.

A more detailed examination of the experimental material shows that the indicated form of the curve is accidental in character and is to a considerable extent due to three or four reactions with \(B\), lying within these limits, each studied in a large number of solvents, which gives a corresponding raising of the curve; if, however, each reaction is counted only once, and the data from works appearing in 1929–1931 are also plotted, the graph assumes an essentially different form and becomes fully explicable by the statistical graph \(E\).

Fig. 6.

§ 6. Mechanism of monomolecular reactions

The failure of numerous attempts to construct a complete theory of monoreactions is not accidental. It is connected with the obvious inadequacy of classical mechanics and statistics for elucidating the nature of chemical forces and the mechanism of the elementary processes underlying stoichiometric transitions. Having at one’s disposal only the laws and concepts of macroscopic mechanics and electrostatics, on the one hand, and, on the other, the qualitative material of descriptive chemistry, one cannot go beyond crude mechanistic or speculative schemes, and one has to be content with a formal analysis of phenomena that does not touch the essence of the processes. This explains the striking lag that existed in theoretical chemistry for almost 40 years in kinetics, in which elementary pro-

processes from statics and the statistics of macroscopic processes. Electronics and the old quantum theory were the first to introduce a fresh current into this field, placing photochemistry and, in part, electrochemistry on a scientific footing. But only in quantum mechanics has chemistry found a theory that provides the key to its central microprocesses and is capable of quantitatively characterizing the forces acting here. It would be no exaggeration to say that, in the seven years of its existence, quantum mechanics has managed to provide incomparably more for the understanding of the fundamental chemical categories than classical physical chemistry was able to do in the whole half-century of its existence. The enormous successes achieved thanks to quantum mechanics in the field of the theory of the homeopolar bond38, of activation energy39, etc., have prepared the ground for a more profound study of chemical kinetics and of the elementary processes underlying it.

One of the serious difficulties here is the multistage character of the overwhelming majority of reactions and the difficulty of distinguishing macroeffects connected with elementary processes occurring in a separate atom or molecule from molecular-kinetic effects. In this respect, homogeneous monomolecular reactions (or reactions of zero order on surfaces) present a serious simplification; in them, as we have seen above, the processes of these two types are separated in time and in the macroscopic picture, and where, at sufficiently low pressures, the rate does not depend on collisions, there is observed, as it were directly, the spontaneous decomposition—conditioned by the internal forces of the molecule—of active molecules contained in the statistical distribution. It is therefore natural that attempts have appeared to construct a quantum-mechanical theory of monomolecular reactions40 in order to eliminate obstacles insurmountable for classical theories by the methods of wave mechanics.

The foundations of the quantum-mechanical theory of chemical affinity are already known to the readers of Uspekhi; therefore I shall not touch upon them here and shall proceed directly to the exposition of quantum-mechanical work on kinetics.

MONOMOLECULAR REACTIONS IN CHEMICAL KINETICS

In the preceding paragraph the case of rupture of the isolated bond under consideration in a large molecule has already been analyzed. In this case, too, the quantum-mechanical interpretation of transitions does not open up any great new possibilities, since the transition from the state \(V_{r_1}\) to \(V_{r_2}\) will require the full dissociation energy, which corresponds to very large quantum numbers, where the distinction between the classical and quantum interpretation of the question is in general smoothed out. In reality, as is easy to show, in monoreactions we are dealing with considerably more complex processes, which in the final analysis reduce to the rupture of some chemical bonds and the simultaneous formation of new bonds, for example according to the scheme \(A—B—C=A—C+B\) or \(A—B—C=A—C—B\). In this case, owing to the superposition of processes accompanied by a decrease in potential energy, the final state of the system corresponds to an energy level lying considerably lower, since not only is \(E\) always less than \(D\), but the heat of reaction \(Q\) is always less than \(E\). Very schematically this may be represented by the scheme of Fig. 7, where the state of the molecule before activation corresponds to region \(I\), and the state of the final products to region \(II\). The remaining designations are obvious and need no explanation.

Fig. 7.

For systems of this type quantum mechanics opens up entirely new possibilities.

If in classical physics the probability of transition of a particle from region \(I\) to \(II\) (Fig. 7) is equal to 0 for all states with energy less than the height of the threshold (which may have any form) and is equal to 1 for the level \(E\) and higher ones, then the probability of such transitions according to quantum mechanics always lies between these two limits:

\[ 0 < W < 1. \]

As is known, in this way Gamow was able to explain the emission of \(\alpha\)-particles from the nuclei of radioactive elements; Fowler-

and the Nordheim—Richardson effect, etc. As the most simplified model, this same method was used in one of the previous works by L. Rovenkevich and the author of the present article. The problem is solved quite easily, analogously to Fowler’s and Nordheim’s solution of the electronic problem.^37 We shall not dwell now on this derivation, since it gives only a very rough approximation. Let us note only that in this way we arrive at an exponential dependence of the Arrhenius equation, but the coefficients standing before the exponent cannot be calculated in advance. A somewhat different conception brings us much closer to the same question, according to which monoreactions are, in their mechanism, analogous to predissociation or to the generalized Ožė effect. The essential difficulty of chemical transitions, as compared with physical transitions of electrons and α-particles, consists in the restrictions imposed by the low mobility of atomic nuclei in comparison with electrons. Consequently, in practice, electron transitions constituting the elementary act of bond rupture or migration are probable only in the case when the nuclei are at the distance required for the transition, and the latter is possible without a significant change in their relative positions. If we schematically represent the change of potential energy with the distance between two centers by curve I for the initial molecule and by curve II (Fig. 6) for the final system, then, graphically, what has been said above means that the transition from the first state to the second occurs only at the point of intersection of the curves or at this very point C. It is quite obvious that this probability is indirectly preserved also for molecules with energy considerably greater than level C, since during vibrations such a molecule will necessarily pass through states of potential energy corresponding to C. The level of internal energy (more precisely, the set of vibrational and rotational levels in the n* de—

* Cf. the articles by Fowler, Uspekhi fizicheskikh nauk, 9, 1929; 10, 135, 1930, and by Gamow, ibid., 10, 531, 1930. Ed.

values of freedom corresponding to this level in the very sense of the scheme has nothing in common with \(D\) and, as a rule, lies considerably lower; therein lies the explanation of the relations between \(E\) and \(D\) analyzed above. The heat of activation is that energy corresponding to \(C\) at which the vibrations of the individual oscillators of the molecule have become so intensified that the potential barriers can no longer prevent a new regrouping of the parts of the molecule. This may, generally speaking, occur for different combinations \(E_1\), \(E_2\), etc., making up

\[ E=\sum_{1}^{i=n} E_i. \]

It is easy to see that the scheme developed here very closely resembles Franck’s scheme (Fig. 8) of predissociation, with the only difference that in our case electronic transitions are possible even without optical excitation of the electronic “level.” Let us now see how, for this scheme, the probability of transformation and the rate constant are to be calculated.

Fig. 8.

Since we accept the second part of the Lindemann scheme, which speaks of the summation of the activation energy \(E\) over a large number of bonds, for the magnitude \(K_{\infty}\) in our case there must hold a relation analogous to Eq. (19):

\[ K_{\infty}=\int_{0}^{\infty} F(E)\cdot D(E)dE, \]

where \(F(E)\) is the probability of finding a molecule with internal energy \(E\), and \(D(E)\) is the probability of decomposition of such a molecule. Without changing anything in the first factor, which we shall subsequently take in the simpler Hinshelwood form, in view of the arbitrariness of the assumptions underlying the more complicated expressions of Rice–Ramsperger and Kassel, we may try to determine \(D(E)\) and solve

thus an equation. This was done by us in one of our works, the transformation of the molecule being interpreted as an internally compensated transition, analogous to the Auger effect, in which, as is known, owing to the transition energy of one of the external electrons to a vacant place in an inner group, another electron is ejected, the probability of ejection being incomparably greater than the probability of the analogous process in the absence of the first transition. If the state of one of the parts of the initial molecule is characterized by the wave function \(\psi_\alpha^{\mathrm{I}}\) and of the second by \(\psi_\beta^{\mathrm{II}}\), and their new states after the transition by \(\psi_\gamma^{*\mathrm{I}}\), of the second by \(\psi_\delta^{*\mathrm{II}}\), then the probability of the transition may be written as follows:

\[ \int \int \psi_\gamma^{*\mathrm{I}}\psi_\delta^{*\mathrm{II}}\, H \psi_\alpha^{\mathrm{I}}\psi_\beta^{\mathrm{II}}, \]

where \(H\) is the Hamiltonian operator and \(dv_1\) and \(dv_2\) are the volume elements of the initial and final systems.

In view of the impossibility of directly solving the Schrödinger equation for so complex a case, as a first approximation we took for \(\psi_\alpha^{\mathrm{I}}\), etc., the functions of linear oscillators, taking into account the interaction of the parts according to Heitler and London. The total probability of transition in this case is equal to the sum of the integrals for all possible initial states of the molecule, i.e., for all possible values of the quantum numbers \(m\) and \(n\) of the oscillators; if the molecule is regarded as a system of coupled oscillators, this gives for \(K_0\):

\[ K_0 \sim \sum_m \sum_n e^{-\frac{h\nu m}{kT}} e^{-\frac{E_n}{kT}} \left| \int \int \psi_\gamma^{*\mathrm{I}}\psi_\delta^{*\mathrm{II}}\, H \psi_\alpha^{\mathrm{I}}\psi_\beta^{\mathrm{II}}\, dv_1 dv_2 \right|^2 . \]

Allowing for an increase in the probability of transition with increasing \(m\) and \(n\), we obtain, by virtue of the superposition of two probabilities (one—statistical probability—rapidly decreasing with increasing \(m\) and \(n\); the other, sharply increasing in doing so), a sharp maximum at certain \(m\) and \(n\), which corresponds, roughly speaking, to the crossing scheme in our

of the present scheme. Solving the equation written for \(K_{\infty}\), we obtain the following final expression:

\[ K_{\infty}\sim e^{-\frac{E}{RT}}\cdot e^{\frac{\beta(E+Q)}{\sqrt{E}}}, \]

where \(Q\) is the heat of reaction. If \(Q\) is small in comparison with \(E\), we obtain:

\[ K_{\infty}\sim e^{-\frac{E}{RT}}\cdot e^{\beta\sqrt{E}}. \]

With a somewhat different interpretation one may arrive at the expression:

\[ K_{\infty}\sim e^{-\frac{E}{RT}}\cdot e^{\beta(A+Q)}. \]

The factor \(\beta\) of the Arrhenius equation contains, in the form of a factor in the exponent, \(A+Q\) to the first power, or to a power more or less close to unity. Closer to the intersection scheme is a somewhat different analysis of the question, developed by us in an article appearing in “Z. phys. Chem.” (B).

The transitions may be below and above \(C\). In the first case the probability of transition must be proportional to \(exp(-\int pdq)\). For a certain value of the internal energy \(W\), \(p\sim\sqrt{E-W}\); \(q\) is proportional to \((E-W)\) and, roughly speaking, inversely proportional to \(E\) (for a group of reactions similar in type); consequently the transition probability is written as:

\[ D(W)\sim exp\left(-\alpha\frac{(E-W)^{3/2}}{E}\right), \]

which is very close to the expression \(N\) and gives a maximum at \(W=E\). These transitions, whose number, by virtue of the peculiarities of the Boltzmann distribution, is greater than the number of transitions above \(C\), thus again give an exponential relation of \(\beta\) to \(A\), or, under a more detailed analysis, to \((A-Q)\). Unfortunately, for the time being we cannot indicate with sufficient certainty the dependence of \(D(W)\) on \(W\) above \(C\), i.e., when \(W>E\), but here too the transitions are concentrated near \(C\).

Thus the two principal conclusions of the quantum-mechanical treatment of monoreactions reduce to an exponential relation \(D(W)=B_2\) with \(A\) within groups of related reactions and to the essential significance of compensated transitions. From the data of Table II we can construct a graph of the dependence of \(B_2\) on \(A\) for two groups of reactions (decomposition of azo compounds and decomposition of ethers), for which, with some approximation, the number \(n\) of quadratic terms participating in \(E\) is known. The results are compared in the graph.

[Figure: axes \(V\) and \(R\); curves I, II, III; labels \(A\), \(C\)—transition region, \(D\), \(r_1\), \(r_2\).]

Fig. 9.

We see that three azo compounds give a clearly expressed exponential dependence. The fourth decomposition reaction,
\(\mathrm{CH_3—N_2—NCH_3}\), connected with the breaking of other bonds, has substantially different \(E^{43}\) (\(E\) for azo compounds and ethers within each of the groups is constant, in contrast to \(E_{\text{emp}}\)) and an anomalously small \(n\). This reaction does not fall on the curve. For the ethers \(B_2\) changes, although in the right direction, but so little that we refrain from drawing conclusions. It can only be noted that here the check does not contradict the conclusion, especially since \(B\) as a function of \(A\) does not here give any regular picture. Unfortunately there are no other groups of reactions with known \(n\). In view of the fact that, with increasing \(E\), \(n\), generally speaking, decreases [E] in small molecules, as a rule, is larger than in large ones, which is explained not by a change in \(D\), etc., but by the decrease of \(\left(\frac12 n-1\right)RT\), entering into the full expression for \(\dfrac{d\log k}{dT}\). \(B_1\) varies comparatively little with \(n\) within each group, and therefore one may hope to find, though distorted, an exponential relation directly of \(B\) to \((E-Q)\).

Monomolecular Reactions in Chemical Kinetics

This was carried out in our first work on monomolecular reactions, and the exponential relation of \(B\) to \(A-Q\), or—since \(Q\) is nearly constant within each group—the exponential relation of \(B\) to \(A\), proved to be a very general property of monomolecular reactions. Moreover, this dependence is obeyed much better than could have been expected from the analysis given above for all cases that have been investigated with any care, both for groups of related reactions and for one and the same reaction in different solvents. A number of examples of this type are analyzed in our article. The same dependence, in a somewhat different form, was found purely empirically somewhat later by Gapon\(^{38}\), who showed that for each group of related reactions there exists a so-called temperature at which all \(K\)’s are equal, which clearly follows from the generalized equation:

\[ K \sim e^{-\frac{A}{RT}+\varepsilon A}. \]

The mechanism underlying this very general peculiarity of monomolecular reactions is represented only very imperfectly by the semiclassical considerations given above.

Somewhat less definite are the results of testing the second conclusion of our scheme: the essential significance of the energetic compensation of transitions. This means that \(E-Q\)—the energetic characteristic of a monoreaction—must always be positive and, if possible, large. Statistically this is true. The overwhelming majority of monomolecular reactions known to us are very well compensated: \(Q>0\), or only slightly less than 0, and \(A-Q\) is equal to several tens of large calories; but there are several comparatively simple monomolecular reactions—the decomposition of nitrogen peroxide, the decomposition of nitrosyl chloride, and the decomposition of nitrogen pentoxide—that are sharply endothermic and have a small energetic characteristic. In view of the possibility of large variations in the form and character of the potential curves of equal groups, the presence

of these three reactions does not annul the basic scheme, but, unfortunately, as yet there is not even any possibility of predicting in which case such deviations will appear.

We thus see that the conception of monomolecular reactions developed above has made possible a qualitative interpretation of certain features of monomolecular reactions and has predicted some of their new properties; but as yet the theory is still very imperfect, as, indeed, is the experimental material which it is to explain. A whole series of fundamental elementary questions concerning the transfer of energy in collisions and within the molecule still has no more or less satisfactory theory; at the same time, the number of systematically studied reactions and groups of reactions is insufficient both quantitatively and qualitatively for its inductive characterization. Thus, despite undoubted successes, the theory here is taking only its first, and not very confident, steps.

§ 7. Applications of the Theory to Other Problems of Kinetics

The closest analogue of homogeneous monomolecular reactions is provided by heterogeneous reactions of zero order, in which we have the possibility of disregarding diffusion processes and the influence of concentrations, and observe the spontaneous decomposition of molecules filling the surface of a catalyst. It is natural precisely in this case to seek applications of the results obtained for homogeneous monomolecular reactions. Some indications of this are found in the works of Kremer^39 and Grimm^40, who studied decomposition on catalyzing surfaces: the first—spirans, the second—halogen derivatives. Even before the appearance of the quantum-mechanical theory of monomolecular reactions, for both cases proceeding according to zero order, an exponential relation of \(B\) to \(A\) was found over a very broad interval of variation of both quantities. Their explanation presented serious difficulties and led to the creation of special

artificial schemes, for example the scheme of frozen active points, formed at higher \(T\) in accordance with the distribution of energies at these temperatures and preserved at the temperature of catalysis, which gives a different number of points for different catalysts; moreover, by formally more or less arbitrary assumptions one can obtain the required dependence. From the point of view of resonant intramolecular transitions, in all these ad hoc hypotheses there is no necessity, and it is natural to suppose that on the surface and in monomolecular homogeneous reactions we are dealing with two manifestations of one general regularity. In exactly the same way, the extension of our scheme to reactions of higher orders is quite natural; the interpretation of these in the work of Frank and Rabinovich, Polanyi and Wigner is in many respects very close to our interpretation of monoreactions. This is understandable, since two molecules at the moment of collision may be regarded as one particle existing for a very short time, and the conditions for the spontaneous rearrangement of such a complex are entirely analogous to the conditions for the spontaneous transformation of ordinary molecules. In bimolecular reactions as well, as a rule, we do not observe bond rupture as a special stage of the process, and the reaction proceeds with \(E < D\). It must be noted, however, that in view of the short duration of the collision time and the rather difficult fulfillment of the prerequisites for the intersection of curves, bireactions must be rather rare (at present the number of known bimolecular reactions proceeding without any complications is very small), and the probability of transition must, as a rule, be considerably smaller than in a monoreaction. This may be confirmed by the fact that if the number of collisions of molecules with energy exceeding \(E\), according to the simple activation formula,

\[ k = Z e^{-\frac{E}{RT}}, \]

where \(Z\) is the number of collisions per unit time, gives, for monomolecular reactions, numbers that are far too small

and the introduction of a factor \(F(W)\), increasing \(k\) by \(10^n\) times, where \(n\) is from 4 to 10, is necessary; then for bimolecular reactions of molecules of the same complexity:

\[ k = Ze^{-\frac{w}{RT}}, \]

and only in one case of the bimolecular decomposition of ozone are there indications of the need to introduce a distribution factor. Strictly speaking, it should always be introduced, but since for bimolecular reactions we have no methods whatever for determining \(\eta\), the calculation of \(D(W)\) is likewise impossible. Apparently, the probability of transition is on average \(10^6\)—\(10^8\) times smaller than in monomolecular reactions.

§ 8. Applications of the theory to particular cases

Up to now the theory of monomolecular reactions has been developed in connection with processes having no particular applied significance; however, the possibilities for its application are rather great. The first indications of this are contained in recent works on the investigation of cracking in the gas phase and on the low-temperature decomposition of explosive substances. As has been shown chiefly by the works of American kineticists, the pyrolysis of hydrocarbons proceeds according to the monomolecular law, exhibiting a number of characteristic features of essentially monomolecular reactions. As is known, the character and composition of the products of cracking can change very strongly depending on the nature of the substance being cracked and on the conditions of temperature and pressure.

If, for secondary reactions, the doctrine of monomolecular reactions cannot give anything essential, then for estimating the probability of various spontaneous transformations possible for large molecules, such an answer, provided the theory is developed somewhat further, could in principle be given. Such attempts have already been made, for example, by Rice, Pease, and Henri on the basis of the concept of dissociation into free radicals. At present kinetics faces

the task of developing this technically very urgent problem on a more modern basis. Matters stand somewhat more successfully with the second, equally urgent question. The laws determining the stability of large molecules are still very little known; meanwhile, in a number of cases, in particular for the entire field of explosives, the question of stability is of primary importance, and the provision of a certain theoretical basis for it is a very important task. As was shown in papers printed in the Journal of Physical Chemistry by the author of this review and his collaborators, the primary process of decomposition, at least in the case of nitric-acid esters, is a spontaneous monomolecular decomposition with the liberation of nitrogen dioxide, capable of subsequently initiating secondary autocatalytic processes. Determination of the kinetic characteristics of these processes showed that all explosives whose decomposition kinetics is known are characterized by a great height of the activation barrier, $A$, of the order of 50 Cal., but at the same time $B$ is very large, greater than for the majority of ordinary nonexplosive reactions. This gives a picture characteristic of explosives: sharp temperature sensitivity (a low onset owing to the enormous $B \infty 10^{20}$) and at the same time very sharp temperature limits and sufficient stability at low $T$ owing to the large $A$. Apparently this spontaneous decomposition, very little sensitive to additives, sets the upper limit of stability of the substances investigated by us, whereas the reduced sensitivity of contaminated compounds and the explosion itself are rather closely connected with the autocatalytic reaction.

The two groups cited do not exhaust the range of possible applications of the theory. The varied processes of organic and biological chemistry, connected either with rearrangements, with migrations of groups and bonds, or with low-temperature decomposition, all the innumerable reactions with $E$ constituting only a very small part of the work of bond rupture $D$, probably proceed according to an analogous scheme, just as do the decomposition or transformation of those intermediate complexes of a substrate with an ion whose assumption constitutes a cha-

acteristic feature of modern schemes of ionic catalysis. True, the theory itself is still very imperfect, but the advances made by chemical physics in recent years give grounds to expect very rapid progress in this field.

Conclusion

In the 5 years that have elapsed from the time of the dominance of radiation hypotheses to the beginning of the emerging quantum-mechanical stage of the theory of monomolecular reactions, kinetics has advanced very far; and the difficulties of that period, which seemed insurmountable, are now in the main resolved. But the new stage has put forward new, deeper and more fundamental problems, whose solution is necessary for further progressive movement ahead. The study of the laws of energy exchange and the corresponding reworking of the statistical part of the theory, elementary experiments on the excitation of monomolecular reactions by simple physical impulses (ionic impact, infrared radiation, etc.), the further accumulation and reworking of purely kinetic material, and, finally, the central point—the improvement and development of the doctrine of intramolecular spontaneous transitions—these are the principal tasks of the present day.

Note added in proof. The article had already been sent to press by the time the Polish issue of J. Am. Ch. Soc. arrived with the excellent work of Eyring on the activation energy of bimolecular reactions. This work, and the works of Eyring and Polanyi on the simplest gas reactions, have not been used in the review.

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