GAS EXPLOSIONS AND THE THEORY OF CHAIN REACTIONS*

N. N. Semenov, Leningrad

Explosions in general, and gas explosions in particular, belong among the most interesting phenomena of nature. In these phenomena there appear with particular vividness the passive forces of chemical resistance, by means of which nature defends itself against the destructive action of the second law of thermodynamics. In these phenomena one can observe with the greatest clarity the boundary of the chemical stability of matter against the influences of temperature and pressure, i.e., the boundary at which the forces of passive resistance lose the capacity to oppose the external action, and matter almost instantaneously passes into the forms required by the second law.

Analogously to the way in which the limit of mechanical strength is determined by the magnitude of the forces acting between molecules up to rupture (it is thus connected with the phenomena of elastic and plastic deformation), and the limit of electrical strength (dielectric breakdown) is derived directly from an analysis of the electrical conductivity of insulators at voltages not reaching the breakdown voltage, so too must the limit of chemical strength (explosion) be sought by studying the chemical transformations—often infinitely slow—which proceed before the onset of the conditions necessary for explosion. However small the rates of these transformations may be, they reflect in themselves that

* Report at the VII Physico-Chemical Conference in Leningrad, September 25, 1930.

decrease of the forces of chemical resistance, which leads to an explosion when the external action is increased. Therefore the question of the nature of explosions and their causes is closely connected with the study of the kinetics of reactions, often beginning long before the onset of the conditions necessary for an explosion.

In my report I shall confine myself only to an examination of the latest development of the theory of chain reactions, which may perhaps shed some light on questions connected with the mechanism of ignition of gaseous mixtures.

In the development of chemical kinetics one should distinguish two main periods. The first of them was begun by the work of van ’t Hoff and Arrhenius at the end of the last century. A splendid culmination of this period is provided by the work of Hinshelwood in the field of bimolecular and monomolecular reactions. The results of this period may be formulated as follows: only those molecules enter into reaction which, according to the Maxwell–Boltzmann distribution law, possess an energy (potential or kinetic) exceeding a certain value \(E\), characteristic for each given reaction and called the energy of activation. In the case of reactions of higher order, the elementary act of reaction takes place during the collision of such an active molecule with some other molecule. In first-order reactions the elementary act is the result of fluctuations of energy in the molecule itself.

The fundamental numerical law of this period in the development of chemical kinetics is the dependence between the rate of reaction and the temperature, expressed in the following formulas:

\[ w = A e^{-\frac{E}{RT}} \]

or

\[ \log w = -\frac{E}{RT} + B. \]

The value

\[ \frac{d \log w}{d \frac{1}{T}} = -\frac{E}{R} \]

must, according to this theory, be constant for every temperature. This classical *

the theory of homogeneous reactions cannot explain a whole series of cases in which the reaction rate depends far more strongly on negligible traces of impurities than on the concentration of the principal substances (positive and negative catalysis). The simple and monotonic dependence of the reaction rate on pressure and temperature given by this theory is also found to contradict a whole series of cases in which the reaction rate is a very complex and often discontinuous function of pressure and temperature (the limits of oxidation of phosphorus, H₂, CO, etc.). Likewise, the induction period, when time explicitly enters into the expression for the rate, cannot be fitted into the framework of this theory.

On closer examination, it turns out that the number of reactions following the simple laws of the classical theory is incomparably smaller than the number of reactions that stand in contradiction to the classical theory.

The extraordinary variety, complexity, and instability of these anomalies, and the absence of any unifying theoretical idea, compelled many investigators of that period to avoid these anomalies and not to include them among current scientific problems. In the last few years, however, the situation has changed, and interest in these varied and complex processes has begun to grow in geometric progression. This happened because a theory appeared which, although in a very general and vague form, was capable of uniting all these phenomena, whose most characteristic feature was that they did not fit within the framework of the old theory.

The second period in the development of chemical kinetics is closely connected with this new theory—the theory of chain reactions. The roots of this theory lie in photochemistry: for the first time, the idea of a chain was introduced by Bodenstein (1913) in considering the photochemical formation of HCl. According to the classical Einstein theory, the photochemical yield of a reaction $\nu$ should be equal to 1, or, if secondary reactions are taken into account, should have been equal to 2, 3, 4, etc.; that is, in all cases it should be expressed by a small integer. At the same time, for the reaction of photochemical formation of HCl the number $\nu$ proved to be equal to 100,000. By other

In other words, each absorbed quantum causes 100,000 acts of reaction. Such cases of large quantum yield are also observed for many other reactions, if they are exothermic. Bodenstein proposed that the primary reactions caused by the absorption of a quantum of light are only the first link in the reaction chain. In what follows we shall denote by $\nu$ the length of the chain, and shall understand by this the number of all secondary reactions caused by the appearance of a single reaction center.

Bodenstein’s theory has been confirmed from the most diverse sides and may now be regarded as absolutely correct.

Let us note the following remarkable consequence of this theory: if nothing forced the chain to break off, it could be continued to infinity, i.e., a single initial center would be sufficient to make the entire mixture react. Therefore the factor determining the reaction is not so much the pressure of the mixture itself as the presence of minute impurities that enter into reaction with the intermediate products of the chains and thereby break the chains. Thus an explanation is given of negative catalysis. Using the example of oxygen, which plays the role of an impurity, Bodenstein and his school brilliantly confirmed this consequence of the chain theory.

In 1923 Christiansen and Kramers attempted to apply the idea of chains to dark reactions, namely to the decomposition of $\mathrm{N_2O_5}$. In 1924 Christiansen attempted to explain, by means of chains, the phenomenon of negative catalysis in thermal reactions. In 1930 Backström showed, using the oxidation of $\mathrm{Na_2SO_3}$ and benzaldehyde as examples, that reactions which form chains under the action of light give the same chains when they proceed in the dark. Thus, dark reactions and photochemical chain reactions differ only in the mechanism of formation of the initial links of the reaction. In the first case they are obtained as a result of absorption of a quantum of light, in the second—as a result of collisions between molecules possessing a sufficiently large store of energy.

However, the first works concerning chain reactions in the dark did not make a great impression, although the main idea of the chain theory had already been expressed in them in its most general form. In 1928, independently of one another, in two laboratories, in Oxford and in Leningrad, for the first time a whole series of new phenomena was accurately established and studied, phenomena standing in sharp contradiction to the classical theory; and the principal theoretical milestones were also set on the path of the development of the new theory.

Before turning to these new phenomena, we must give the basic mathematical expressions of the chain theory, which, as was already indicated above, was in the main formulated in the work of Christiansen and Kramers in 1923. Let \(n_0\) be the number of initial links of the chain formed under the influence of thermal motion per unit time, and let \(\alpha\) be the probability that the chain will not break at the given link (in other words, \(\alpha\) is the probability of continuation of the chain). Then very simple calculations lead to the following expression for the rate of the reaction:

\[ w = \frac{n_0}{1-\alpha} = \frac{n_0}{\beta}, \]

where \(\beta\) is the probability of rupture of the chain at the given link. The classical theory, which does not take into account the development of chains, gives:

\[ w = n_0 = A e^{-\frac{E}{RT}}. \]

Thus, the result of the chain theory differs from the classical result only by the factor \(\frac{1}{1-\alpha}\). The value of this factor is evidently equal to the number of elementary reactions in the chain, or, according to our notation, to the length of the chain. Note that

\[ \nu = \frac{1}{1-\alpha} = \frac{1}{\beta}. \]

Let us now turn to the new data obtained at the beginning of 1928 and contributing to the rapid development of the whole field of chain reactions.

1. The Role of Walls in Chain Reactions

When the active intermediate products of a chain collide with the wall of the apparatus, the further development of the chain ceases, owing to the adsorption of these intermediate products on the wall. In other words, the chain is broken upon collision with the wall. Since the length of the chain determines the rate of the reaction, we here encounter for the first time the fact that the magnitude of the volume in which the reaction proceeds has a substantial influence on the rate of the reaction. We now know many cases in which the deactivating action of the walls is observed, and this effect is the most characteristic sign of a chain reaction. This phenomenon is closely connected with another fact characteristic of chain reactions—the strong catalytic action of admixtures of inert gases (for example argon, nitrogen, etc.).

These gases themselves do not take part in the reaction, but they prevent the chains from diffusing to the walls and thereby lengthen the chains, and consequently increase the rate of the reaction. There is already a large literature connected with this question.*

If the question of chain breaking is relatively clear, the matter stands much worse with the question of the initiation of chains, i.e. of the formation of initial centers. Above we assumed (following Christiansen and Kramers) that the initial links of the chains appear as a result of thermal motion. It is very possible that in some cases this assumption is correct, but in other cases it encounters serious difficulties. Thus, for example, in the case of the combustion of \(H_2 + O\), the act of chain initiation is evidently connected either with the splitting of the \(H_2\) molecule into \(H + H\) (Marshall), or with the split-

* 1927. Semenov (oxidation of phosphorus).
1928. Hinshelwood (\(H_2 + O_2\)).
1929. Trifonov (\(H_2 + Cl_2\)).
1929. Bodenstein and Wagner (\(CO + Cl_2\)), Yost and Young (\(H_2 + I_2\)), Semenov (\(H_2 + O_2\)).
1930. Bäckström. Oxidation of benzaldehyde. Hinshelwood—oxidation of \(PH_3\). Schumacher and Sprenger. Decomposition of \(ClO_2\).

by the splitting of \(O_2\) into \(O+O\), or by the splitting of \(H_2O\) into \(OH+O\) (Tabor and Bonhoeffer). In all these cases it is not difficult to show (Semenov, 1929) that the number \(n_0\) is not too small to explain the observed magnitude of the reaction rate. The same, apparently, also applies to the reaction \(H_2+Cl_2\), where the primary act must consist in the decomposition of \(Cl_2\) into \(Cl+Cl\). There is also another difficulty, concerning the dissociation of the molecules \(H_2\), \(N_2\), and perhaps others as well; according to as yet unpublished experiments of Schechter, for such dissociation collisions are required in which the relative energy of the particles is much greater than the dissociation energy. This leads to smaller values of \(n_0\) and, correspondingly, makes less probable the mechanism of formation of the initial chain links in the volume.

A much more probable assumption is that the chains arise on the walls (Semenov, 1929, with regard to the reaction \(H_2+O_2\)). This assumption can be represented concretely in two ways:

1) When a molecule of the type \(H_2\) or \(Cl_2\) collides with the wall of the vessel, dissociation may occur according to the scheme \(H_2+\text{wall}\to(H\ \text{wall})+H\) (Semenov and Frankel, 1928). In other words, at the expense of the adsorption energy of one of the atoms of the molecule, the other atom is ejected into the volume. It is not difficult to see that the energy necessary for splitting the molecule is expressed by the difference \(D-F\), where \(D\) denotes the dissociation energy of the molecule in the volume, and \(F\) is the adsorption energy of an atom adsorbed by the wall. Since \(F\) is usually equal to several tens of large calories, it is clear how much, in this case, the possibility of dissociation is facilitated, and consequently also the beginning of the chain.

2) We know that on a surface a reaction almost always proceeds more readily than in the volume. In this case one might suppose that, in individual acts of such a surface reaction, the heat of reaction is not always given up to the wall, but under certain favorable conditions may be transferred to some neighboring molecule, thereby causing dissociation of this molecule and the ejection of the dissociation products into the volume (1930).

GASEOUS EXPLOSIONS AND THE THEORY OF CHAIN REACTIONS

Unfortunately, we have still not been able to obtain direct and indisputable proof of such chain initiation at the walls. The most immediate observation is provided by the experiments of Polyakov (1928), who passed a stream of very pure hydrogen over heated palladium and obtained, at a distance of 20–30 mm from the palladium, in a cold quartz tube, a glow resembling the glow of atomic hydrogen. At the same time the quartz tube became noticeably heated. This phenomenon continues until the palladium becomes saturated with hydrogen. Unfortunately, this phenomenon is very difficult to reproduce, and, owing to Polyakov’s departure from Leningrad, the experiments were discontinued at an initial stage; therefore one cannot be completely certain of their correctness.

Next, we have the very ingenious experiment of Bone and Neumann, who investigated the torque appearing on vanes coated on one side with catalyst. The absence of an effect was interpreted by the authors as evidence that on the surface of the catalyst only the primary act of the reaction takes place, and that the energy thereby liberated creates a chain of reactions proceeding in the volume. If this is so, then the whole theory of catalysis must be rebuilt on new foundations. In 1929 I attempted to find this effect by studying the influence of the diameter of the vessel on the rate of the reaction \(\mathrm{H_2 + O_2}\), in the presence, as catalyst, of a palladium wire stretched along the axis of the vessel, but here a negative result was obtained. Alongside these direct experiments, which from our point of view are not decisive, we have a whole series of indirect arguments in favor of the chains beginning at the wall. The most convincing in this respect are the experiments of Haber (1930), who studied the explosion of \(\mathrm{H_2 + O_2}\) in “wall-free” space by the intersection of jets of \(\mathrm{H_2}\) and \(\mathrm{O_2}\), and showed that under these conditions it is very difficult to obtain an explosion. But it is enough to introduce a quartz filament at the place where the jets intersect in order to obtain ignition under the same conditions as those occurring in ordinary experiments with explosions in quartz vessels. Further, one should note the observations of Garner, the results of which make one think that the walls ...

the vessel have a substantial influence on the conditions of ignition. Among other indirect proofs we mentioned the experiments of Christiansen (1929) with the thermal reaction \(H_2 + Cl_2\); he considers that the regularities obtained by him can be explained only on the assumption that the Cl atoms necessary for the initiation of the chain appear as a result of the dissociation of \(Cl_2\) at the wall. Very convincing in this respect are Schumacher’s experiments (1930) with the thermal decomposition and explosion of \(ClO_2\); the author not only showed that the material and the state of the wall substantially determine the rate of the reaction, but was also able to explain the dual role of the walls: on the one hand, stimulating the reaction (when chains are initiated on them), and on the other hand, retarding it (when chains are terminated on them). Unfortunately, the work mentioned lacks precise quantitative data that could describe the role of the walls by means of a mathematical formula. In this way reliable data on the role of the walls could have been obtained.

It is not difficult to show that the rate of reaction, calculated per unit volume of a cylindrical vessel, can be represented as a function of the radius \(r\) of the vessel in four different ways, depending on four different assumptions about the role of the walls:

1. The chains begin and terminate in the volume. In this case \(w\) does not depend on \(r\).

2. The chains begin in the volume and terminate at the walls—

\(w = kr^2\) (Semenov, Trifonov, 1929).

1. The chains begin at the walls and terminate in the volume—

\[ w = \frac{k}{r}. \]

1. The chains begin at the wall and terminate at the wall—

\[ w = kr. \]

Thus, by varying the diameter of the vessel and studying the rate of reaction, one can quite precisely resolve the question posed. Unfortunately, this has not yet been done.

The chain length itself could depend on the temperature; the number of initial centers \(n_0\) is expressed by the function \(Ae^{-\frac{E}{RT}}\). The quantity \(1 - \psi\) is a function \(\psi(T)\). Hence,

GAS EXPLOSIONS AND THE THEORY OF CHAIN REACTIONS

\[ w=\frac{Ae^{-\frac{E}{RT}}}{\psi(T)} \]

whereas the classical theory gives the value

\[ w=Ae^{-\frac{E}{RT}}. \]

Since \(\psi(T)\), generally speaking, may be any function of temperature, the very idea of a temperature coefficient of a reaction, characteristic of the classical theory, becomes illusory. The temperature coefficient, expressed for chain reactions as

\[ \frac{d\log w}{d\frac{1}{T}}, \]

is a variable quantity depending on temperature (Hinshelwood, \(H_2+O_2\); Schumacher and Sprenger—the decomposition of \(Cl_2O\)). In photochemical reactions \(n_0\) is determined by the absorption of light. The chain length \(\frac{1}{1-\alpha}\) is determined directly as the quotient obtained by dividing the number of molecules that have reacted by the number of absorbed light quanta. As was shown by Kistiakowsky (1929) and by Haber, Harteck, and Farkas (1930), for the photochemical reactions \(H_2+O_2\) and \(CO+O_2\) the chain length increases rapidly with temperature.

It can be shown that the new type of dependence of the reaction rate on temperature does not contradict the fundamental laws of thermodynamics (Semenov, 1929). In 1929 Semenov gave a preliminary theory of the growth of chain length with temperature. In essence it reduces to the following: individual elementary reactions associated with the development of the chain are accompanied by the release of energy. At the first moment this energy is concentrated in the reaction products. Thus particles with increased energy are formed in the gas. Let us take, for example, the chain of reactions in the case of the photochemical formation of HCl:

\[ Cl_2+h\nu=Cl+Cl. \]

\[ \begin{aligned} 1)&\quad Cl+H_2=HCl+H+0\ \text{Cal.}\\ 2)&\quad H+Cl_2=HCl+Cl+45\,000\ \text{Cal per mole.}\\ 3)&\quad Cl+H_2=HCl+H\ \text{etc.} \end{aligned} \]

Thus the main chain develops. However, in each reaction (2), 45,000 cal. per mole are released. This energy is at the first moment entirely concentrated in the reaction products HCl and Cl, in the form of either kinetic or potential energy of these particles. Let us assume that each of the molecules receives half the energy, i.e. 23,000 cal.

Upon collision of these energy-rich particles with some molecule of Cl₂, their energy is insufficient for the dissociation of Cl₂ into atoms. For this, 56,000 cal. are required. However, if the Cl₂ molecule itself, owing to the Maxwellian distribution, possesses an energy greater than 33,000 cal., then upon collision the dissociation of Cl₂ into two atoms will occur, and these atoms themselves will be the initial links of two new chains. Thus branching of the main chain takes place. The probability of such branching is determined by the probability that the Cl or HCl atoms obtained in reaction (2) undergo their first collision with a Cl₂ molecule possessing an energy \(56\,000 - 23\,000 = 33\,000\) cal. This probability is evidently equal to \(e^{-\frac{33\,000}{RT}}\). If the length of the main chain is \(\nu_1\), then the number of branchings in it will be \(\nu_1 e^{-\frac{33\,000}{RT}}\). The length of these secondary chains will again be equal to \(\nu_1\). Thus the total number of reactions in the main chain and in the chains created by its first branchings will be

\[ \nu_1 + 2 \nu_1^2 e^{-\frac{33\,000}{RT}} . \]

Each of the secondary chains in turn creates

\[ 2 \nu_1 e^{-\frac{33\,000}{RT}} \]

branchings, with a total number of reactions

\[ 2 \nu_1^2 e^{-\frac{33\,000}{RT}} . \]

Since the number of secondary chains is \(2 \nu_1 e^{-\frac{33\,000}{RT}}\), the total number of reactions in the main chain, the secondary chains, and the tertiary chains is equal to

\[ \nu_1 + 2 \nu_1^2 e^{-\frac{33\,000}{RT}} + 2 \nu_1^3 e^{-\frac{2 \cdot 33\,000}{RT}} . \]

Continuing these arguments further, we find that the total

the number of elementary reactions in the chain, including all its branchings, will be equal to

\[ \nu=\nu_1(1+2\nu_1\gamma+2\nu_1^2\gamma^2+2\nu_1^3\gamma^3+\cdots) =\nu_1+2\nu_1(\nu_1\gamma+\nu_1^2\gamma^2+\nu_1^3\gamma^3+\cdots) =\nu_1+2\cdot\frac{\nu_1^2\gamma}{1-\nu_1\gamma} =\nu_1\left(1+\frac{2\nu_1\gamma}{1-\nu_1\gamma}\right) =\nu_1\frac{1+\nu_1\gamma}{1-\nu_1\gamma}, \]

where

\[ \gamma=e^{-\frac{33000}{RT}}. \]

As long as \(\nu_1\gamma\) is considerably less than 1, one may use the formula

\[ \nu+\nu_1(1+2\gamma\nu_1)=\nu_1\left(1+2\nu_1 e^{-\frac{33000}{RT}}\right). \]

Thus we see that the total length of the chain with all its branchings increases with temperature only at a sufficiently low temperature, when there is practically no branching: \(\nu=\nu_1\).

The rate of the dark reaction for the case of the combination \(H_2+Cl_2\) is determined by the quantity \(n_0\nu\), where \(n_0\) is the number of Cl atoms initially formed per unit time. If it is assumed that Cl atoms are obtained as a result of the dissociation of \(Cl_2\) into atoms in the volume upon collision of \(Cl_2\) with any other particle having an energy greater than 56,000, then

\[ n_0=Ae^{-\frac{56000}{RT}} \]

and

\[ w=n_0\tau=A\nu e^{-\frac{56000}{RT}}+A\nu_1 e^{-\frac{56000}{RT}}\left(1+2\nu_1 e^{-\frac{33000}{RT}}\right). \]

Or, putting \(56000=Q\), and \(23000=U\), we obtain the general expression for the reaction rate as a function of temperature in the form

\[ w=A\nu_1 e^{-\frac{Q}{RT}}\left(1+2\nu_1 e^{-\frac{Q-U}{RT}}\right), \]

where \(\nu_1\) does not depend on temperature. We thus obtain, instead of the classical law,

\[ w=Ae^{-\frac{E}{RT}} \]

more complex,

\[ w=Ae^{-\frac{E}{RT}}+Be^{-\frac{E_{1}}{RT}}. \]

This law is valid only for small values of \(Be^{-\frac{E_{1}}{RT}}\) in comparison with unity. In a more general form the law is expressed by the formula

\[ w=Ae^{-\frac{E}{RT}}\cdot \frac{1+Be^{-\frac{E_{1}}{RT}}} {1-Be^{-\frac{E_{1}}{RT}}}. \]

We have shown this with the example of the ignition of phosphorus, sulfur, \(\mathrm{PH}_{3}\), \(\mathrm{H}_{2}\), CO, where branching of the chain occurs readily.

However, as we saw in the example of \(\mathrm{H}_{2}+\mathrm{Cl}_{2}\), any chain reaction, if it is exothermic, can occasionally give rise to branching. The number of these branchings, and hence the magnitude \(\alpha\), increases with increasing temperature and pressure, reaching, under certain conditions, a value equal to 1. Consequently, the chain mechanism of explosion may be extended to all chain reactions without exception.

As we saw, the reaction rate is expressed by the formula

\[ w=\frac{n_{0}}{1-\alpha}, \]

where \(\alpha\) is the probability of chain continuation; in other words, each act of reaction gives rise on average to \(\alpha\) new chains, these latter to \(\alpha_{2}\) subsequent ones, and so on. In the example of the reaction \(\mathrm{H}_{2}+\mathrm{Cl}_{2}\) we saw that the magnitude \(\alpha\) would be equal to 1 if the chains were not broken at the walls of the apparatus or as a result of reactions of active centers with oxygen molecules. We can also imagine chain termination as a consequence of emission, by the active molecule, of the energy necessary for chain continuation, as well as as a consequence of a number of other causes. All these causes, generally speaking, make \(\alpha\) less than 1. However, in the example of \(\mathrm{H}_{2}+\mathrm{Cl}_{2}\) we also saw that sometimes processes of chain branching can occur. In this case one elementary reaction gives rise to 3 new ones. Such cases increase the average value of \(\alpha\) and may, under certain conditions, lead to \(\alpha\) becoming greater than 1.

In a whole series of reactions such branching of the chain may occur considerably more often than in the case of \(H_2 + Cl_2\). Thus, in the oxidation of CO the reaction mechanism may be represented in the following form: \(CO + O = CO_2^*\); \(CO_2^* + O_2 = CO_2 + O + O\); \(O + CO = CO_2^*\), etc. Each act of reaction here gives rise to 2 new active centers (O atoms), and \(\alpha\) may reach the value 2. An analogous development of the chain is found in the oxidation of phosphorus, sulfur, arsenic vapors, \(PH_3\), etc. The first act of the reaction apparently consists in the splitting of \(O_2\) into atoms \(O + O\). Although Haber and Bonhoeffer adduce a whole series of considerations in favor of another mechanism for the initiation of the reaction, namely the splitting of \(H_2O\) into \(H + OH\), it seems to me that Garner’s experiments on the ignition of dry and moist mixtures of \(CO + O_2\) contradict this. Whichever of these two assumptions is accepted as correct, it may be asserted that the creation of the initial centers requires a very large energy, and therefore they arise very rarely, i.e. in our formula for the rate of reaction \(n_0\) will be a very small quantity. Therefore, so long as \(\alpha\) is even at all appreciably less than unity, the rate of reaction remains very small. And only when \(\alpha\) is quite close to unity does the rate of reaction become noticeable, and within a very small interval of change of \(\alpha\) becomes practically infinite.

Thus, for

\[ \alpha < 1 \qquad w = 0, \]

and for

\[ \alpha \geqq 1 \qquad w = \infty \]

this is evidently the condition for ignition. And since \(\alpha\) is a function of the pressure \(p\), then at pressures less than some \(p_1\) the reaction practically does not proceed at all, while at pressure \(p \geq p_1\) the rate of reaction becomes very great—ignition occurs.

This remarkable phenomenon was discovered by us in the oxidation of phosphorus vapors, sulfur, CO and \(H_2\), and by Hinshelwood and Dixon in the oxidation of \(PH_3\). The numerical value of the quantity \(p_1\) and its dependence on var—

measures of the vessel, the admixture of an inert gas, and other conditions is in excellent agreement with the quantitative results of the chain theory (Semenov, 1927, 1928, 1929, 1930; Hinshelwood, 1929).

The assumption, still current since the time of van ’t Hoff, that explosion is connected with self-heating of the mixture as a result of the quiet reaction that proceeds before the explosion is not applicable to the cases under consideration. The point is that in all cases ignition occurs at very low pressures, from 0.01 to 1 mm, and therefore the negligible amount of substance converted before the explosion makes it possible directly to establish the absence of heating of the mixture. At first sight it would seem that this fact is also inexplicable from the point of view of the chain theory, since, owing to the great probability of chain termination at the walls at low pressures and owing to the emission of energy by the $\mathrm{CO}_2^{*}$ molecule in the form of light, $\alpha$ must be very small. But one must not forget that in these cases the chains are highly branched, and therefore $\alpha$ may attain a sufficient magnitude. Also not fitting within the framework of the thermal theory is the experimentally established dependence of the ignition pressure on the dimensions of the vessel and on impurities. An indirect proof of the same thing is the circumstance that the critical explosion pressure $p_1$ depends very little on temperature.

A splendid proof of the non-thermal character of these ignitions is also the existence of an upper limit. For more than a hundred years it has been known that oxygen at pressures greater than a certain critical value ceases to react with phosphorus. It is enough, however, to pump out the oxygen to the critical pressure for vigorous ignition to occur. The usual conceptions of chemical kinetics and of thermal explosion, of course, cannot in any way explain the fact that a lowering of the partial pressure of one of the components increases the reaction rate.

Meanwhile the chain theory, as was shown by Semenov and Jorissen, can explain this fact.

Still more astonishing is the presence of an upper ignition limit in the case of $\mathrm{H}_2 + \mathrm{O}_2$ and $\mathrm{CO} + \mathrm{O}_2$, discovered-

to by Dixon, but, so to speak, it was realized as a result of the work of Hinshelwood (1930) and of our laboratory (1930). This fact also plainly contradicts the thermal theory. In general form it may be explained by the chain theory on the assumption that $\alpha$ first increases with pressure, and then begins to fall, reaching the value 1 at the upper pressure limit. However, we do not yet have an answer to the question why $\alpha$ can have such a peculiar course.

Haber carried out experiments in two crossing jets of oxygen and hydrogen, i.e. in a space “without walls.” It turned out that in that temperature region where Garner, Hinshelwood, and we observed the phenomenon of the lower and upper limits, an explosion did not occur at all. It was enough, however, to introduce a quartz rod at the place where the jets met in order to obtain ordinary ignition. This experiment is a brilliant proof that chains originate on the walls, and that without walls there are neither chains nor chain explosion; however, it is hardly possible to conclude from this that the phenomenon of the upper limit is determined by the wall, as Haber tries to do. His point of view amounts to saying that high pressure impedes the diffusion of active centers into the volume and thereby makes explosion difficult. Hence—the upper pressure limit of explosion. However, for the development of the chain there is no need for diffusion of centers over any appreciable distance from the walls; it is not difficult to understand that the number of chains being initiated will in no case depend on the pressure. As regards the destruction of chains on the walls, then, as we saw above, with increasing pressure the length of the chain at first rapidly increases with pressure, and then ceases to increase, but it never decreases with increasing pressure.

Thus, Haber’s explanation should be considered improbable.

It is not excluded, however, that the existence of an upper limit is connected not with the fundamental laws of chains, but with side phenomena.

We have repeatedly had occasion to observe that a mixture of $\mathrm{CO} + \mathrm{O}_2$, admitted into a vessel at a pressure lower than the upper-

of this limit, very quickly after ignition, goes out, so that no more than 5–10% of the gas burns out. When the gas is pumped out of the vessel the mixture ignites again. Another fact relates to the lower limit. If the mixture is admitted into an empty apparatus by a rapid turn of a wide tap, then ignition begins to occur at pressures \(p\) equal to the value of the lower limit \(p_1\), i.e., at the same values as are obtained under the cleanest conditions of a flowing stream. If, however, the mixture is admitted as a very slow stream through a capillary, then in the case of \(\mathrm{H_2 + O_2}\) and \(\mathrm{CO + O_2}\) (with phosphorus this phenomenon is absent) ignition occurs only at pressures considerably exceeding the lower limit, and sometimes (in the case of \(\mathrm{CO + O_2}\)) does not occur at all.

The first experiment shows that during the reaction certain products are formed which poison the reaction. Most probably, these products poison the wall and render it powerless as regards initiating the principal links of the chain. When the pressure of the mixture is lowered, the diffusion of these poisoning products from the wall into the volume increases, the wall is “cleaned,” and the reaction develops again. The second experiment shows that, when the mixture is rapidly admitted into the vessel, the rate of chain development is so great that the walls do not have time to become poisoned. When the gas is admitted slowly, the explosion is preceded by a quiet reaction (at \(\alpha\) close to 1), which develops predominantly near the walls. Near the walls an increased concentration of harmful products is created, which poison the wall before the conditions for explosion are created in the gas (i.e., \(\alpha \geq 1\)).

Such a point of view readily explains the existence of the upper limit. Indeed, the higher the pressure, the more slowly the chains diffuse into the middle of the vessel, and consequently the greater the concentration of harmful products at the walls, and hence the more rapidly the poisoning of the walls proceeds. When the pressure reaches a certain limiting value, the rate of poisoning becomes so great that explosion becomes impossible. The poisoning products diffuse only very slowly into the volume, and instead of an explosion we obtain a slowly proceeding reaction,

which was in fact discovered by Hinshelwood and by us and was studied in detail by Topley. It is possible that precisely this secondary process underlies the phenomenon of the upper limit.

It must be noted that all this reasoning is not included in the usual mathematical formulation of the theory of chains. Indeed, the formula \(w=\frac{n_0}{1-\alpha}\) gives a velocity \(=\infty\) for any arbitrarily small value of \(n_0\), provided only that \(\alpha\) be \(>1\).

But if this is so, then under the condition \(\alpha>1\) an explosion will occur independently of the degree of reflection from the wall, for there will always be found some number of initial centers \(n_0\).

I think, however, that this formula, which does not take into account the influence of the number of initial centers, is insufficient.

We have a number of facts indicating the dependence of the ignition pressure on the number of initial centers; for example, there is no doubt that ignition of sulfur vapors does not occur under the condition \(\alpha>1\), unless a sufficient number of initial centers is produced by an insignificant admixture of ozone to oxygen (Semenov and Ryabinin, 1928). This fact clearly indicates the essential role of the number of initial centers \(n_0\) in the conditions of ignition. The insufficiency of the formula \(w=\frac{n_0}{1-\alpha}\) is especially clearly manifested in the fact that, both in the oxidation of phosphorus and sulfur and in the reactions \(\mathrm{H}_2+\mathrm{O}_2\) and \(\mathrm{CO}+\mathrm{O}_2\), the rate of reaction during combustion, i.e., under the condition \(\alpha>1\), although quite large, is far from \(\infty\) and has some definite value. This also follows from the fact that ignition of \(\mathrm{H}_2+\mathrm{O}_2\) can be obtained at low pressures only in the temperature range \(>440^\circ\). On approaching this temperature the pressure suddenly begins to increase rapidly, and finally, at temperatures lying below \(440^\circ\), ignition becomes altogether impossible at any pressures. This fact can easily be explained by the supposition that the number of initial centers decreases with decreasing temperature. At first, while \(n_0\) is large, ignition is determined by the condition \(\alpha=1\), and since \(\alpha\) is small,

depends on the temperature, the critical pressure likewise changes little with temperature. But when \(n_0\) becomes very small, ignition is impeded, and for it to occur it is necessary that \(\alpha\) considerably exceed 1, which causes a sharp increase, with decreasing temperature, in the critical ignition pressure.

With this increase in pressure, the wall-poisoning factor indicated above begins to act (the phenomenon of the upper limit), as a result of which below a certain limiting temperature ignition proves altogether impossible.

Also in favor of this point of view is the fact that an insignificant admixture of \(\mathrm{NO}_2\) to the hot mixture greatly lowers the critical ignition temperature, reducing it in the case of \(\mathrm{H}_2 + \mathrm{O}_2\) from \(440^\circ\) to \(240^\circ\). The action evidently consists in the fact that the ease of detachment of the atom \(\mathrm{O}\) from \(\mathrm{NO}_2\) contributes to an increase in the number of initial reaction centers and thereby facilitates ignition.

Thus, all the facts indicated argue for the fact that the number of initial centers, together with the magnitude \(\alpha\), somehow enters into the conditions determining ignition.

Posing to myself the question of what exactly is the incorrectness of the formula \(w = \dfrac{n_0}{1-\alpha}\), I initially came to the following conclusion. In all my previous theoretical considerations I included in the quantity \(\alpha\) not only the probability of the continuation of the chain in the volume, but also the probability of chain termination at the wall, assuming that for any link of the chain there is a definite probability, one and the same for all links, of colliding with the wall. Such reasoning is hardly correct, since the probability of colliding with the wall for an active molecule located far from it is negligibly small, and, conversely, for one that is in immediate proximity to the wall, this probability is close to 1. The averaging of this probability, admitted by me in earlier works, reduces to the assertion, evident from Fig. 1: let 1 be the main chain, and 2 and 3 chains that are its branches at links “a” and “b.” My former assertion reduces to the fact that the branching chains begin to develop at the same average

distance from the wall, and that the main chain, hence chains 2 and 3, have the same length as chain 1. In fact, this is hardly the case. A much more probable assumption is that the secondary chains begin at the very place where they arise from the main chain, i.e. at the points “\(a\)” and “\(b\),” and that consequently the length of the 3rd chain is less than that of the 2nd, while the 2nd is less than the first. This may be demonstrated by Fig. 2.

Fig. 1.

Fig. 2.

It is not difficult to show that the new assumption leads to the formula

\[ w=\frac{n_0(1-\alpha^\nu)}{1-\alpha}, \]

where \(\alpha\) is the probability of continuation of the chain in the volume, and \(\nu\) is the number of links in the first chain on its path from the place of origin to the wall. For sufficiently large \(\nu\) and \(\alpha<1\), this new formula coincides with the old one, since \(\alpha^\nu\) is small in comparison with 1. However, not only for \(\alpha=1\), but also for \(\alpha\) greater than 1, the new formula does not give \(w=\infty\).

Thus I posed the question at the VII Physico-Chemical Conference in the autumn of 1930. The question, consequently, reduces to whether infinite chains can or cannot exist when they are broken off at the walls. In other words, is there some critical condition of the type \(\alpha=1\), when we obtain a sharp transition from finite chains to infinite ones—a transition from a slow reaction to an explosion? This question was resolved by the work of Burgstan and Sorokin (1931), who, applying in a very ingenious way the diffusion equation to the calculation of chain reactions, showed with complete clarity that my doubts were incorrect, that chains can

become infinite when passing through a definite critical value, and that, consequently, all my earlier arguments remain valid.

And since \(\alpha\) depends on external conditions and does not depend on the number of initial centers, the experimental considerations set forth above, which point to a connection between the ignition conditions and the number of initial centers, remain theoretically incomprehensible and contradict the theory.

A way out of this situation is indicated by me in a still unpublished work on the interaction of chains. In order to formulate the theory more clearly, let us examine it in the particular example of the combination of hydrogen with oxygen.

According to Bonhoeffer and Haber (1929), the reaction of combination \(H_2 + O_2\) proceeds according to the following chain scheme:

\[ (1)\quad H_2 = 2H \quad \text{(at the wall, and } H \text{ goes into the volume).} \]

\[ (2)\quad H + H_2 + O_2 = H_2O + OH + 100\ \text{cal.} \]

\[ (3)\quad OH + H_2 = H_2O + H,\ \text{etc.} \]

Chain termination occurs, for example, at the wall, i.e.

\[ (4)\quad H + \text{wall} = \frac{1}{2}H_2, \]

\[ (5)\quad OH + \text{wall} = \frac{1}{2}H_2 + \frac{1}{2}O_2. \]

As for chain branching, there is no clarity here. I propose the following mechanism of chain branching. The large energy released in reaction (2) is concentrated in its products \(H_2O\) and \(OH\). Thus we obtain an excited molecule \(H_2O^*\), which only after a large number of collisions loses its excitation, converting the excitation energy into the general heating of the gas, according to reaction \((6)\) \(H_2O^* + M = H_2O + M\). If the molecule \(H_2O^*\), before it loses its excess energy, encounters another molecule \(H_2O^*\), then the following reaction is theoretically possible: \((7)\) \(H_2O^* + H_2O^* = H_2O + H + OH\), i.e. a reaction associated with the appearance of two new initial links of the chain, i.e. associated with chain branching. The concentration of \(H_2O^*\) molecules, and hence the probability of chain branching, will naturally depend on the overall rate of the reaction, i.e. on the number of initial chains (the \(H\) atom), in other words, on the rate of reaction (1).

And hence \(\alpha\) will depend on the number of initial centers \(n_0\), and consequently the number will enter into the condition of ignition \(\alpha = 1\). The smaller \(n_0\), the more difficult ignition will be, which is what was to be proved. I think that the development of this theory of the interaction of chains will be very fruitful, especially for the theory of combustion and detonation.

The new formulation, just like the old one, remains completely helpless before the phenomena of the induction period, which elapses between the beginning of the reaction and the explosion itself. In the explosion of ordinary gas mixtures the induction period is often measured in minutes, whereas the period of complete development of each chain cannot be more than \(0.1\) sec. Still more surprising are the induction periods in explosions occurring under the influence of an external action. Thus, for example, in the ignition of sulfur by an admixture of ozone, the induction period reaches one minute. In the ignition of a sensitized mixture \(H_2 + O_2\), subjected to the action of light (Tabor and Gartek), the explosion occurs several minutes after the illumination has ceased.

There are, however, even more surprising examples of induction periods lasting several hours. Thus Schumacher (1930) observed that \(ClO_2\), under certain conditions, explodes half an hour after its temperature has reached the explosion temperature; Roginskii, with his collaborators, observed that nitroglycerine and trotyl in closed vessels explode spontaneously several hours after heating, the magnitude of the induction period being exactly definite for each temperature.

All these facts show that in the mechanism of explosions there is something which we still do not understand. Chemists call this by the old term autocatalysis. I think that in general they are right. In such a reaction of explosive substances, some products evidently are formed (for example, the already mentioned \(NO_2\)) for which, under favorable conditions, it is easier to start a chain than for the principal substance. Formally speaking, the formation of such products is already the beginning of the formation of a chain. But since these products start the chain not immediately after their formation, but more often after a more

or less long interval of time, we arrive at the following conclusions:..

1. In the formation of such autocatalyzing substances, secondary chains may begin at any point of the volume, and therefore the formula

\[ w=\frac{n_0}{1-\alpha} \]

remains valid..

1. The development of the reaction rate is determined not by the development of a single chain, but by the time elapsing between the appearance of the autocatalyzing molecule and the moment when the reaction chain begins of itself. If \(\gamma(>1)\) is the average number of autocatalyzing molecules arising in the development of each molecular chain, then the reaction rate “\(w\),” as a function of time, is given by the formula

\[ \lg w(t)=n_0\gamma^{\frac{t}{\tau}} . \]

For an explosion it is necessary that the reaction rate reach some definite value \(w_1\), which must be sufficiently large. The time \(T_1\), elapsing from the moment of the beginning of the process to the moment when \(w\) becomes equal to \(w_1\), will be the induction period. Obviously, it will be the greater, the greater \(\tau\) is.

All these considerations show how complex the picture of ignition is, and how much work is still needed in order to bring it to complete clarity. Nevertheless, we also see that chain theory is a reliable instrument, possessing which we need not fear difficulties, and we may hope that the time is not far off when we shall be able to study in detail the nature of ignition and explosions. In studying technical explosions occurring at sufficiently high pressures, we likewise cannot neglect the results of the thermal theory of explosions. At high pressures and temperatures \(n_0\) becomes a very considerable quantity. Of course, the reaction rate

\[ w=\frac{n_0}{1-\alpha} \]

also reaches a considerable value, for \(a < 1\). The heat evolved in the reaction heats the mixture and increases \(n_0\), and consequently \(\omega\) as well. If heat removal to the outside does not compensate the heat evolved in the reaction, the mixture begins to heat up progressively. Such rapid self-heating of the mixture and a very rapid growth of the reaction rate could lead to an explosion, if \(a < 1\). It should be noted that a thermal explosion is theoretically possible for every exothermic reaction, even if it is not connected with the formation of chains. But, if one examines this question carefully, it is not difficult to see that only in the case of a very rapid growth of the reaction rate with temperature can one obtain an increase in the reaction rate sufficiently rapid for an explosion. Thus, for a thermal explosion it is necessary that the temperature coefficient of the reaction, or the value of the activation energy \(E\), be very large; but the reaction rate is given by the expression

\[ Ae^{-\frac{E}{RT}}, \]

in which the larger \(E\) is, the smaller the value of \(e^{-\frac{E}{RT}}\). For an explosion there is needed not only a large value, but also a large absolute value of the reaction rate, i.e. a large value of \(A\). A brief calculation shows that the value of \(A\) in this case is abnormally large, much larger than it ought to be according to the theory of bimolecular and monomolecular reactions. Only the introduction of chain theory can preserve for \(A\) the values required by experiment.

In accordance with this, all explosive reactions whose kinetics have been studied have proved to be chain reactions (Hinshelwood, 1928, 1929, 1930). Good proof of the presence of chains in the case of hydrocarbon explosions is furnished by the action of antiknock agents. According to Egerton’s theory (1928), their action can readily be explained by the breaking of reaction chains.

In conclusion I shall allow myself to say a few words concerning the mechanism of ignition of cold gaseous mixtures. It is known that a spark ignites a gas only in that case—

more often when the gas pressure exceeds a certain critical value. Kowalski carried out several experiments to investigate the dependence of the critical pressure on temperature at different spark powers (the power was determined by the capacitance of the capacitor whose discharge produced the spark). The following series of curves were obtained (1, 2, 3, 4), in order of decreasing spark power (Fig. 3). The curve is the usual curve of self-ignition of the mixture, taken in the very same apparatus.

The first remarkable circumstance is the smooth transition of the lower branch of the self-ignition curve into the system

Fig. 3.

of curves of artificial ignition. We mentioned above that at the point \(T_k\) self-ignition ceases, since thermal motion does not produce initial centers in a quantity sufficient for the development of an explosion.

From this theory one may conclude that an explosion could occur if initial centers were created artificially. The spark creates these centers, and therein lies its action. The smaller the power of the spark, the higher, as can be seen, the critical pressure. This result confirms the new formulation of the theory of chain ignition that we have given—

…the variation according to which, the smaller \(n_0\), the greater the pressure required for the explosion to occur.

It is interesting to note that, in contrast to the phenomenon of spontaneous ignition, in ignition by a spark there is no upper pressure limit. This proves the correctness of Haber’s assumption that the upper limit is connected with the spontaneous formation of chains at the walls. In artificial ignition, the centers appear in the volume, and therefore no upper limit exists.