Chain Reactions

N. N. Semenov, Leningrad

Several decades ago, through the joint work of physicists and chemists, and as a result of the application of the theories of classical physics to chemistry, the principal chapter of physical chemistry was created—the theory of equilibria, which achieved such brilliant theoretical and practical results.

The state of physics at that time, however, could not help chemistry in creating a theory of reaction rates. Now, on the basis of the new electron physics and the theory of atomic structure, a broad field of interaction between physics and chemistry is again emerging, and one of the chief tasks of this joint work is the development of the theory of reaction rates—chemical kinetics.

Far from every collision of two reacting molecules leads to a reaction; only a very small fraction of these collisions proves effective. One might think, as Boltzmann proposed, that not the entire surface of a molecule is active and that only a very small region of it possesses this property; thus only those molecules will react which collide in such a way that their active regions come into contact.

Despite all the ingenuity of this proposal, it must be regarded as incorrect, since it contradicts experiment.

As Arrhenius and Bodenstein showed, the rates of almost all reactions increase very rapidly with temperature (according to the law \(e^{-\frac{A}{T}}\)). This effect cannot be exp—

clear from the standpoint of Boltzmann’s theory. Arrhenius, in the 1880s, gave a theory which to this day serves as the guiding thread of all research in the field of chemical kinetics. According to Arrhenius, not all molecules react, but only those among them that possess a very considerable excess energy, whose magnitude differs for different reactions. The smaller the value of this excess energy of the active molecules, the greater the number of such molecules that will be produced by thermal motion, and the greater will be the rate of reaction.

The higher the temperature, the more energetic the thermal motion of the molecules; the more active molecules will be produced, and the greater will be the rate of the given reaction.

Arrhenius’s theory, supported by a vast body of experimental material, had until recently raised no doubts, and all the efforts of physical chemists were directed almost exclusively toward clarifying the question of what the nature is of these mysterious intermediate modifications—the “active molecules” of Arrhenius. It cannot be said that this question has been finally resolved, but much clarity has nevertheless been brought to it. It has turned out that in many cases active molecules are not chemical modifications of the reacting molecules, but physical modifications, whose excess energy consists either in the increased kinetic energy with which they move, or in the strong oscillation relative to one another of the atoms composing the molecule, or in the transition of the atom’s electrons to higher energy levels.

Thus the question of the rates of chemical reactions has become as much a question of physics as of chemistry.

The fascination with Arrhenius’s theory was so great that, until very recently, no special importance was attached to a whole series of facts that, apparently, stood in complete contradiction with Arrhenius’s theory. The phenomena of homogeneous positive and negative catalysis, in which negligible (often immeasurably small) quantities of foreign substances exert a decisive influence on

the magnitude of the reaction rate, of course, cannot be fitted within the framework of Arrhenius’ theory. These include, first of all, the astonishing facts of the complete cessation of the most energetic reactions (oxidation of phosphorus, cessation of the photochemical reaction \(H_2 + Cl_2\), and many others) when the reacting substances are deprived of the last traces of moisture. Conversely, negligible quantities of certain organic compounds very strongly inhibit oxidative processes in various kinds of readily oxidized liquids. In exactly the same way, impurities of certain organometallic compounds can raise, by many hundreds of degrees, the ignition temperature of mixtures of hydrocarbons with air.

The second class of phenomena that contradict Arrhenius’ theory is called, in older textbooks, the passivity of gases. This includes, for example, the sudden loss of activity of oxygen with respect to phosphorus if the oxygen pressure exceeds a certain critical value. This maximum limit of oxygen pressure depends extraordinarily strongly on impurities of various substances which, when admixed with passive oxygen, cause the combustion of phosphorus; in addition to this upper limit of oxygen pressure in various oxidation reactions, there is known a lower limit, upon reaching which combustion suddenly ceases.

Especially remarkable phenomena of this kind occur in the combustion of \(H_2\) and CO in \(O_2\). In Pyrex glass vessels and at temperatures of \(500\text{–}600^\circ\mathrm{C}\), the reaction \(H_2 + O_2\) does not proceed at all until the pressure of the mixture reaches \(3\text{–}4\) mm of mercury. If the pressure of the mixture exceeds this value, a very rapid reaction suddenly sets in, accompanied by the emission of light; however, by further increasing the pressure to a certain upper limit, the flame can be extinguished by the combustible mixture itself. Here, as in the case of oxygen and phosphorus, there is a certain upper pressure limit of the mixture, above which the reaction becomes impossible.

At a temperature below \(500^\circ\) for \(CO + O_2\) and \(400^\circ\) for \(H_2 + O_2\), ignition is impossible at any pressures.

mixture, since at these temperatures the upper limit (decreasing as \(T\) is lowered) and the lower limit (increasing as \(T\) is lowered) overlap one another. It is sufficient, however, to add thousandths of a percent of NO in order to lower this minimum ignition temperature by a hundred degrees or more.^1

All the facts listed, as well as a number of others similar to them, substantially undermine the generality of Arrhenius’s theory.

Fortunately, a small and quite natural extension of this theory is capable, if not of fully explaining all these deviations, then of indicating the correct paths toward their explanation. According to Arrhenius’s theory, active molecules appear as the result of the action on normal molecules of those rare, but always present in the gas, molecules that possess exceptionally high velocities, far exceeding the mean velocity of thermal motion. It must not be forgotten, however, that every act of chemical reaction is always associated with the release of a considerable amount of energy. After some time this energy, as a result of collisions of molecules, is distributed more or less among all the molecules of the gas and goes into its overall heating. However, at the first moment after an elementary act of reaction this energy is concentrated in several product molecules of the given elementary reaction.

How large the energy of such particles may be can easily be seen from a simple calculation, the result of which is as follows: the energy of each molecule of the products immediately after an elementary act of reaction corresponds to the mean kinetic energy of gas particles at a temperature on the order of ten thousand degrees.

It is therefore not surprising that such energy-rich particles, upon colliding with normal molecules, activate them and make them capable of reaction. These latter, having reacted

^1 These facts with mixtures of CO, \(O_2\), and \(H_2 + O_2\) were obtained very recently by Hinshelwood in Oxford and by us—in the Leningrad Physico-Technical Institute, independently of one another.

again, create, at the expense of the energy of the reaction, new active particles, and so on.^1

Thus, every active molecule initially created by thermal motion does not disappear after the reaction, but is revived again and again, creating a long chain of further reactions.

As is known from certain photochemical data, the number of such subsequent reactions (the chain length) reaches hundreds of thousands of elementary reactions.

This kind of chain character of the reaction makes it possible to explain all those phenomena which did not fit within the framework of the old theory. Substances that promote the initiation of chains, facilitating the appearance of the first links, act as positive catalysts; substances that break chains act as negative ones. The enormous length of the chains is also the reason why negligible quantities of catalysts produce an enormous effect in the sense of changing reaction rates. Finally, as can be shown, under certain conditions the chains become infinite, and herein lies the explanation of the critical values of pressure and temperature in combustion processes. In what follows we shall focus our attention precisely on these chain reactions.

The concept of chain reactions was first formulated with full precision as a result of the study of photochemical reactions. As is known, in 1905 Einstein applied to photochemistry the theory of light quanta created by him and Planck.

According to this theory, light can be absorbed by atoms and molecules only in definite proportions, the energy of which is equal to \(h\nu\), where \(h\) is Planck’s constant, equal to \(6.55 \cdot 10^{-27}\) erg sec, and \(\nu\) is the frequency of the absorbed light. This conclusion of the quantum theory was verified experimentally and has now become a generally known fact.

^1 In some cases, for example in the reaction \(H_2 + Cl_2\), the chain develops somewhat differently (see p. 197 ff.).

In photochemical reactions, where the absorption of light energy causes a chemical transformation, it is natural to suppose that the reacting molecules are precisely those which have absorbed a quantum of light.

In such a case the number of molecules that have reacted will be equal to the number of absorbed light quanta. If \(Q\) is the total energy absorbed by the photosensitive medium, then the number of absorbed quanta, and consequently the number of molecules that have reacted, will be equal to \(\frac{Q}{h\nu}\). This is the essence of Einstein’s law.

In the experimental verification of Einstein’s law, it turned out that it is justified only in very few cases. However, it fails not because it is incorrect, but because the products formed in the primary reaction themselves enter into further reactions. The character of these deviations is analogous to deviations from Faraday’s law due to secondary reactions at the electrodes.

In some cases these secondary reactions were easy to take into account, and then complete agreement between theory and experiment was obtained. For example, in the case of the photochemical decomposition of HJ, a number of investigators found experimentally that the ratio of the number of molecules that had reacted to the number of absorbed light quanta is \(n = 2\).

Since the final products of the decomposition of HJ are \(\mathrm{H}_2\) and \(\mathrm{J}_2\), the excited molecule \(\mathrm{HJ}'\) that has absorbed light, in order to decompose into \(\mathrm{H}_2\) and \(\mathrm{J}_2\), must react with another unexcited molecule \(\mathrm{HJ}\). Hence each absorbed quantum must cause the decomposition of two molecules. In what follows, the quantity \(n\), equal to the number of reacting molecules per one absorbed quantum of light, will be called the quantum yield of a photochemical reaction.

In the study of various photochemical reactions it was found that the quantum yield for some of them is unusually large and reaches millions of molecules per quantum. A classic example of this type of reaction is

photochemical combination of chlorine with hydrogen, for which Bodenstein found that \(n = 10^6\). Bodenstein and Nernst gave two different mechanisms explaining the reason for such a large quantum yield of this reaction. Common to both of these mechanisms is the concept of the chain character of the reaction: a molecule \(\mathrm{Cl}_2\), absorbing light, enters into a reaction, as a result of which a new molecule \(\mathrm{Cl}_2'\) appears; in the reaction of this second one a third is created, and so on. The first molecule is activated by light; and in the subsequent links this energy is not wasted, but together with the heat of reaction serves as a source of further activation.

The scheme of the chain reaction proposed by Bodenstein is the following:

\[ \mathrm{Cl}_2 + h\nu = \mathrm{Cl}_2' \tag{1} \]

\[ \mathrm{Cl}_2' + \mathrm{H}_2 = 2\mathrm{HCl}' \tag{2} \]

\[ \mathrm{HCl}' + \mathrm{Cl}_2 = \mathrm{HCl} + \mathrm{Cl}_2' \tag{3} \]

\[ \mathrm{Cl}_2' + \mathrm{H}_2 = 2\mathrm{HCl}' \quad \text{and so on.} \tag{2} \]

Chlorine, absorbing light, passes into an excited state, characterized by a rise in the energy level of the electron (1). Such an active molecule \(\mathrm{Cl}_2'\) can react with \(\mathrm{H}_2\), and two molecules of \(\mathrm{HCl}'\) are formed. The latter possess a store of energy equal to the sum of the excitation energy of \(\mathrm{Cl}_2'\) and the heat of the reaction of combination of \(\mathrm{H}_2 + \mathrm{Cl}_2\); therefore we mark them with a prime (2). On colliding with a molecule \(\mathrm{Cl}_2\), the excited molecule \(\mathrm{HCl}'\) gives it its energy, forming an excited and, consequently, reaction-ready molecule (3). The process is then repeated a very large number of times, until the excited molecule \(\mathrm{Cl}_2'\) and \(\mathrm{HCl}'\), whether as a result of reaction with some accidentally present impurity molecule or for some other reason, uselessly expends its energy and thereby breaks the chain.

Nernst gives a different chain mechanism:

\[ \mathrm{Cl}_2 + h\nu = \mathrm{Cl} + \mathrm{Cl} \tag{1} \]

\[ \mathrm{Cl} + \mathrm{H}_2 = \mathrm{HCl} + \mathrm{H} \tag{2} \]

\[ \mathrm{H}+\mathrm{Cl}_2=\mathrm{HCl}+\mathrm{Cl} \tag{3} \]

\[ \mathrm{Cl}+\mathrm{H}_2=\mathrm{HCl}+\mathrm{H} \tag{2} \]

The chain scheme is clear without explanation. Reactions of types (2) and (3) may occur at every collision, or not at every collision, if they are exothermic. Knowing the energy of the reaction \(\mathrm{H}_2+\mathrm{Cl}_2=2\mathrm{HCl}\) and the energy of dissociation of \(\mathrm{H}_2\) and \(\mathrm{Cl}_2\) into atoms, it is not difficult to calculate the energy balance of reactions (2) and (3). It turns out that in the second reaction energy is neither released nor absorbed, while in the third 45 large calories are released. Thus reactions (2) and (3) can indeed occur at every collision.

The decomposition of the \(\mathrm{Cl}_2\) molecule into atoms upon absorption of light was proved by Frank’s pupil Kuhn. Thus the experiment fully confirms Nernst’s scheme, and apparently it is precisely this scheme, and not Bodenstein’s scheme, that is valid for the photochemical reaction of formation of \(\mathrm{HCl}\).

The data cited are in themselves sufficiently convincing proof of the existence of chains in chemical reactions; nevertheless, we shall allow ourselves to cite one more brilliant direct proof of this, given by Weigert and Kellermann. These investigators illuminated a mixture of \(\mathrm{H}_2+\mathrm{Cl}_2\) with the light of a spark. The flash had a duration on the order of ten- and hundred-thousandths of a second. They found that the reaction is not limited to the time of illumination, but continues to proceed after it for 0.01 second. This was discovered in the following way. It is well known to everyone that when a gas is heated nonuniformly, then, owing to the difference in the refractive indices of the hot and cold gas, hot streams can be seen. In a special optical setup these streams can be detected and photographed even with negligibly small nonuniformities in the heating of the gas. Weigert and Kellermann photographed a mixture of \(\mathrm{H}_2\) and \(\mathrm{Cl}_2\) in weak, photochemically negligibly active light, at various times after illuminating the gas with a spark. It was thereby found that streams are in fact absent in the photographs taken—

CHAIN REACTIONS

...seconds after \(\frac{1}{400}\) sec after illumination. After longer intervals of time streaks appear; they reach a maximum after \(\frac{1}{100}\) sec. This means that the heating of the gas, and hence also the amount of heat liberated, is negligible at the moment of illumination. Since heat is liberated as a result of the reaction, consequently the number of molecules that have reacted in the same interval of time is small. The chief amount of heat is liberated after illumination during \(\frac{1}{100}\) sec, and then ceases. Consequently, the reaction also continues to develop, after the spark has gone out, for another \(\frac{1}{100}\) sec. The only explanation of this fact can be the idea of chains, according to which light only starts the reaction, while for some time thereafter it can proceed by itself.

Thus, both theoretically and experimentally, the chain character of certain reactions has been established. What is now unclear is not the fact of chain development, but the mechanism of their termination, the mechanism that does not allow the chain to develop indefinitely until all the material is exhausted as a result of the reaction.

The question of chain termination is still little clarified. Three mechanisms for the destruction of active centers are conceivable in the case of the reaction \(\mathrm{H_2 + Cl_2}\): 1) recombination of two hydrogen or chlorine atoms into molecules, occurring in the volume of the gas

\[ \mathrm{H + H = H_2} \]

\[ \mathrm{Cl + Cl = Cl_2} \]

\[ \mathrm{H + Cl = HCl,} \]

2) reaction between atoms \(\mathrm{H}\) or \(\mathrm{Cl}\) and molecules of infinitesimal traces of impurities, always present in gases; 3) adsorption of atoms \(\mathrm{H}\) and \(\mathrm{Cl}\) by the wall of the vessel, on which they then recombine.

The recombination of two hydrogen atoms in the volume, as is known from the earlier experiments of Bonhoeffer, without simultaneous collision with a third particle, is very little prob-

For chlorine atoms, this direct combination, according to the experiments of Kondrat’ev and Leipunsky, is also very unlikely. However, even if one assumes that chain termination is determined by the recombination of H and Cl atoms, it is easy to show that the reaction rate would be proportional to \(\sqrt{J}\) (where \(J\) is the intensity of the light incident on the mixture \(\mathrm{H_2+Cl_2}\)), and not to the first power of \(J\), as is observed experimentally. Thus, we must reject the first assumption concerning the recombination of atoms in the volume. Let us proceed to the analysis of the second assumption.

Let the partial pressure of some impurity \(X\) be \([X]\), and let each collision of H atoms with \(X\) lead to the disappearance of the chain in connection with the reaction \(\mathrm{H+X=HX}\), while each collision of H with \(\mathrm{Cl_2}\) leads to continuation of the chain. In that case the probability of termination will be determined by the sum of the probabilities that the H atom collides with \(X\), and not with \(\mathrm{Cl_2}\), and, correspondingly, that the Cl atom collides with \(X\), and not with \(\mathrm{H_2}\), i.e.

\[ \beta=\frac{[X]}{[\mathrm{H_2}]}+\frac{[X]}{[\mathrm{Cl_2}]}. \]

If the mean chain length (the mean number of elementary reactions in the chain from its initiation to termination) is \(=\nu\), then we obtain the obvious equality \(\beta\nu=1\), or

\[ \nu=\frac{1}{\beta}=\frac{[\mathrm{H_2}][\mathrm{Cl_2}]}{[X]\{[\mathrm{H_2}]+[\mathrm{Cl_2}]\}}, \]

and the reaction rate

\[ w=n_0\nu=\frac{n_0}{\beta}=\frac{kJ[\mathrm{Cl_2}]}{\beta}, \]

where \(n_0\) is the number of Cl atoms (initial links of the chain) created by light per unit time. The quantity \(n_0\) is obviously proportional to the number of light quanta absorbed by chlorine, i.e. it will be proportional to \(J[\mathrm{Cl_2}]\), where \(J\) is the intensity of the incident light.

In this derivation we assumed that each collision \(\mathrm{Cl+H_2}\), \(\mathrm{Cl+X}\), \(\mathrm{H+Cl_2}\), and \(\mathrm{H+X}\) leads to a reaction and that the diameters of all molecules are the same. If such assumptions are not introduced, then we obtain a more general formula of the type

\[ w=\frac{kJ[\mathrm{Cl_2}]^2[\mathrm{H_2}]}{[X]\{k_2[\mathrm{H_2}]+k_2'[\mathrm{Cl_2}]\}}. \]

A formula precisely of this type was obtained experimentally by Tono. Most remarkable is that in this formula the rate

the reaction \(w\) is inversely proportional to the amount of impurity \([X]\) and is, consequently, extraordinarily sensitive to the slightest traces of it. For the reaction \(\mathrm{H_2+Cl_2}\), such an impurity, which breaks chains, is oxygen. At an impurity amount of \(0.01\%\), \(\nu=10^5\); at \(0.1\), \(\nu=10^4\); at \(1\%\), \(\nu=10^3\). A change in the amount of oxygen from \(0.1\%\) to \(1\%\) changes the reaction rate by a factor of 100. This result, paradoxical at first sight, is, as we readily see, explained from the standpoint of chain theory.

If \([X]=0\), then \(\nu\) ought to be equal to \(\infty\). However, this is not so; alongside the quenching mechanism that we have analyzed, there also exists a third one, connected with adsorption of the atoms \(\mathrm{H}\) and \(\mathrm{Cl}\) by the wall. Let us pass to this third type of chain termination.

We shall try to determine what number of elementary reactions \(\nu\) occurs over the entire time of development of the chain until it is broken at the wall (recall that \(\nu\) is called the chain length).

Fig. 1

The path of chain development in the gas is completely analogous to the path of diffusion of some molecule, with the sole difference that during chain development the diffusing particle is alternately a hydrogen atom and a chlorine atom, as shown in Figure 1, where the paths of \(\mathrm{H}\) are shown by a dashed line and the paths of \(\mathrm{Cl}\) by a solid line. To determine the chain length \(\nu\), we must evidently find the total number \(n\) of collisions undergone by our diffusing particle and subtract from it the number of collisions that are useless for the reaction. The total number of collisions is obviously made up of collisions \(\mathrm{H+H_2}\), \(\mathrm{H+Cl_2}\), \(\mathrm{Cl+H_2}\), \(\mathrm{Cl+Cl_2}\) (collisions \(\mathrm{H+H}\) and \(\mathrm{Cl+Cl}\) are not included in the consideration because of their insignificant number). Collisions \(\mathrm{H+H_2}\) and \(\mathrm{Cl+Cl_2}\) are useless in the sense of the reaction; the number of elementary reactions in the chain (equal to \(\nu\)) is equal to the number of collisions \(\mathrm{H+Cl_2}\) and \(\mathrm{Cl+H_2}\).

The numbers \(n\) and \(\nu\) for chains originating at a distance \(x\) from the wall of the vessel will, generally speaking, be very different, depending on the random distribution of the lengths of the free paths. However, on the average we shall have a definite value, equal to

\[ n=\frac{3}{4}\pi x^{2}\left(\frac{1}{4}\sum_{i=1}^{4}\frac{1}{\lambda_i}\right)^3, \]

where \(\sum_{i=1}^{4}\frac{1}{\lambda_i}\) is the sum of the reciprocals of the free-path lengths of H atoms in an \(\mathrm{H_2}\) medium and in a \(\mathrm{Cl_2}\) medium, and of Cl atoms in an \(\mathrm{H_2}\) and \(\mathrm{Cl_2}\) medium at the given pressure. Assuming, for simplicity, all four of these values to be equal to one another, we obtain:

\[ n=\frac{3}{4}\frac{\pi x^{2}}{\lambda^{2}}. \]

A simple calculation makes it possible to estimate what part of these collisions falls on the collisions effective for the reactions \(\mathrm{Cl}+\mathrm{H_2}\) and \(\mathrm{H}+\mathrm{Cl_2}\). Finally we obtain:

\[ \nu=\frac{6\pi x^{2}}{4\lambda_0^{2}}[\mathrm{H_2}]\,[\mathrm{Cl_2}]. \]

Here \(\lambda_0\) denotes the free-path length at a pressure of \(1\ \mathrm{mm}\) of mercury, while \([\mathrm{H_2}]\) and \([\mathrm{Cl_2}]\) are the partial pressures, expressed in \(\mathrm{mm}\) of mercury.

When the entire vessel containing the gas is illuminated, chains originate at the most varied distances \(x\) from the walls. However, the average value of \(x\) will be of the same order as the dimensions of the vessel. For a cylindrical vessel one may take \(x=\frac{r}{2}\), where \(r\) is the radius of the vessel. Then

\[ \nu=\frac{3}{8}\frac{\pi r^{2}}{\lambda_0^{2}}[\mathrm{H_2}]\,[\mathrm{Cl_2}], \]

and the reaction rate is

\[ w=\nu J_{abs}=k\nu J[\mathrm{Cl_2}] =\frac{3}{8}\pi X\frac{r^{2}}{\lambda_0^{2}}[\mathrm{H_2}][\mathrm{Cl_2}]^{3}. \]

The correctness of this formula was confirmed in \(\mathrm{H_2}+\mathrm{Cl_2}\) by Trifonov directly and, somewhat later, also indirectly by Bodenstein and Wagner for the reaction of the photochemical formation of phosgene.

First of all, Trifonov showed that under certain conditions \(w\) is indeed proportional to the square of the vessel radius. To this end he passed a narrow beam of sunlight along the axis first of one wide cylindrical tube, and then of another, narrower one, and measured the amount of HCl formed; the light beam was very narrow and nowhere touched the walls of the tube; the tubes themselves were of equal length (about 1 m). Thus one could be sure that the amount of absorbed light \([J_{abs}]\) was the same in both tubes. Nevertheless, the reaction rate

Fig. 2

was different and, with great accuracy, satisfied the theoretical relation

\[ \frac{w_2}{w_1}=\frac{r_2^2}{r_1^2}. \]

Another conclusion of the theory, namely the proportionality of \(w\) to the product \([H_2][Cl_2]^3\), was tested by Trifonov as follows. He took an equimolecular mixture, where \([H_2]=[Cl_2]=\frac{p}{2}\) (\(p\) is the pressure of the mixture). In this case \(w=Cp^3\), where \(C\) is some constant for the given \(J\). His result is shown in Fig. 2. On the ordinate axis \(w\) is plotted, and on the abscissa axis \(p\); we see that at pressures \(p>0.4\) mm the theory is justified within the limits of experimental error. At pressures \(p<0.4\) mm a striking result is obtained: despite illumination, the reaction ceases altogether. This phenomenon is not entirely clear and requires detailed verification. We shall return to its analysis at the end of the article.

Thus the theory of chain breaking at the walls is confirmed.

It is necessary to note, however, that this mechanism of chain breaking plays an essential role only at comparatively low pressures (10–20 mm Hg and below), when the time required for diffusion of the chain to the wall is not great. At higher pressures, this time is so great that, even before the chain reaches the wall, it will have time to break off as a result of reaction with impurities.

The photochemical reaction of the combination of \(H_2 + Cl_2\) is not the only example of a chain reaction. In gases we know several other cases of reactions with a large quantum yield; for example, the formation of phosgene from CO and \(Cl_2\), where \(n = 10^3\). The greatest number of reactions of this kind is observed in liquids. These include the reactions of iodine and bromine with \(C_2O_4K_2\), the oxidation reactions of various aldehydes, of a solution of sodium sulfite, and many others. The quantum yield in these cases is measured in thousands and tens of thousands of molecules per quantum.

Thus one may confidently consider that many photochemical reactions have a chain character.

Let us now see whether some thermal reactions also have a chain character. We believe that at present we have a series of data that can serve as confirmation of the existence of a very large class of thermal chain reactions. Three cycles of work, which appeared almost simultaneously during the last two or three years (1927–1928), serve as such data.

1. The work of Bäckström on the oxidation of aldehydes, \(Na_2S_2O_3\), etc., carried out in Taylor’s laboratory at Princeton and at the Nobel Institute.

2. Hinshelwood’s work on the reaction of combination of \(H_2 + O_2\) near the explosion temperature.

3. Work on the study of the ignition conditions of sulfur and phosphorus vapors, as well as on the determination of the ignition temperatures of various gaseous explosive mixtures, carried out in the Laboratory of Electronic Chemistry in Leningrad.

1. Unpublished works of Roginsky (GPhTL) on the thermal decomposition of nitroglycerin.

Let us begin with the first. Christiansen expressed the idea that phenomena of negative homogeneous catalysis occur in those cases where the reaction is of a chain character.

He explained the negative-catalytic action of admixed molecules by the breaking of chains, as a result of the reaction of intermediate products with the catalyst (analogously to the influence of \(O_2\) on the reaction \(H_2 + Cl_2\)).

Bäckström studied the influence of various negative catalysts (\(\alpha\)- and \(\beta\)-naphthol, alcohols, etc.) on the oxidation reactions of various aldehydes and sodium sulfite. These reactions may be initiated both by heating and by light. By special experiments he established that the quantum yield in the case of initiation of these reactions by light reaches tens of thousands of molecules per quantum. Consequently these photochemical reactions are chain reactions. He further showed that any negative catalyst which retards the photochemical reaction also retards the thermal reaction (in the dark). Moreover, in most cases the relative retardation of the photochemical reaction is equal to the relative retardation of the thermal one. The only explanation of this parallelism may be the supposition that both the photochemical and the thermal reactions are chain reactions. Moreover, the chains of the one and the other reaction are the same; only the nature of the formation of the initial centers is different. In the first case they are formed upon absorption of light, in the second—under the action of thermal collisions of molecules.

How exactly the law of proportional action of a negative catalyst on the thermal and photochemical reaction is fulfilled may be seen from the following table of rates of oxidation of the solution:

Name of catalysts:	\(C\)	\(w_t\)	\(w_\phi\)	\(w_t/w_\phi\)
No catalyst	0	—	2.8	—
Mannitol	0.002	1.35	0.83	1.6
"	0.01	0.56	0.37	1.5
"	0.04	0.20	0.136	1.5
Methyl alcohol	0.2	0.53	0.37	1.4
Ethyl alcohol	0.2	0.28	0.205	1.4

Here \(C\) is the concentration of catalyst in moles per mole of sodium sulfite, and \(w_t\) and \(w_\phi\) are the rates of the thermal and photochemical reactions.

As another example we shall cite the oxidation of benzaldehyde:

Catalyst	\(C\)	\(w_t\)	\(w_\phi\)	\(w_t : w_\phi\)
\(\alpha\)-naphthol	0.0001	0.013	0.017	0.8
”	0.00004	0.03	0.036	0.8
”	0.000025	0.12	0.112	1.1
\(\beta\)-naphthol	0.0001	0.05	0.046	1.1
”	0.000025	0.2	0.16	1.3

The rate of the photochemical and thermal oxidation reaction in the presence of various alcohols, as negative catalysts, is very accurately expressed by the formulas:

\[ w_t=\frac{k_1}{kC+k_2} \]

\[ w_\phi=k_3 w_t=\frac{k_3 k_1}{kC+k_2}, \]

where \(k, k_1, k_2, k_3\) are constants at the given temperature; moreover, \(k_1, k_2\), and \(k_3\) do not depend on the kind of catalyst, while the constant \(k\) is different for different alcohols, and \(C\) is the concentration of alcohol.

As we have seen, the rate of a chain reaction can be expressed by the ratio of the number \(n_0\) of active molecules initially formed by light or by thermal motion to the probability \(\beta\) of chain breaking at the given link. In the present case

\[ \beta=kC+k_2=\beta_1+\beta_2, \]

i.e., the probability of chain breaking \(\beta\) may be regarded as the sum of two probabilities \(\beta_1\) and \(\beta_2\), corresponding to two possible mechanisms of chain breaking. \(\beta_2\) does not depend on the negative catalyst; it evidently determines chain breaking in the pure substance and corresponds to some unknown mechanism of termination. (For the reaction \(H_2+Cl_2\) this quantity is determined by adsorption of H and Cl by the wall.)

The quantity \(\beta_1\) is proportional to the concentration of alcohol and, consequently, determines the probability of chain breaking as a result of some interaction between active molecu-

with impurity molecules, leading to a loss of activation. In the case \( \mathrm{H_2 + Cl_2} \), the impurity \( \mathrm{O_2} \) broke the chain as a result of the reaction of H and Cl atoms with \( \mathrm{O_2} \). Backström drew attention to the fact that negative catalysts for the oxidation reaction of \( \mathrm{Na_2S_2O_3} \) are substances that are themselves readily oxidized. Hence it is natural to suppose that chain termination is connected with the induced oxidation of the alcohol.

Backström did in fact succeed, by direct experiments, in detecting the oxidation products of the negative catalyst and even in measuring quantitatively the rate of this reaction, although it is negligibly small in comparison with the rate of oxidation of the sodium sulfite itself.

From this point of view, every time the oxidation chain terminates on the alcohol, one molecule of alcohol is oxidized. If the total number of chains arising every second, and consequently terminating every second, is \( n_0 \), then the number of those among them that terminate on the alcohol will be

\[ \frac{n_0 \beta_1}{\beta_1+\beta_2} = n_0 \frac{kC}{kC+k_2}. \]

This quantity will evidently be equal to the rate of the induced reaction of alcohol oxidation

\[ w_n = n_0 \frac{kC}{kC+k_2}, \]

since the rate of oxidation of \( \mathrm{Na_2SO_3} \)

\[ w_m=\frac{k_1}{kC+k_2}, \quad \text{then} \quad w_n=\frac{n_0 kC}{k_1} w_m = a C w_m, \]

i.e. the rate of the induced reaction of alcohol oxidation is proportional to the rate of oxidation of \( \mathrm{Na_2SO_3} \) and to the concentration of the alcohol.

Having measured \( w_m \), one can construct the curve of the dependence of \( Cw_m \) (a quantity proportional to \( w_n \)) on \( C \). On the other hand, Backström was able to measure \( w_n \) directly. In Fig. 3 two curves obtained in this way are given; as can be seen, they correspond very well to one another.

Let us consider the case of large impurity concentrations, when \( k_2 \ll kC \), i.e. practically all chains terminate on

alcohol. In this case, if per unit time \(n_0\) chains of oxidation of \(\mathrm{Na_2SO_3}\) arise, then in the same time \(n_0\) molecules of alcohol must be oxidized, i.e. \(w_n=n_0\). If \(\nu\) is the chain length, then the number of \(\mathrm{Na_2SO_3}\) molecules that have reacted per unit time will be \(n_0\nu\), i.e. the reaction rate

\[ w_m=n_0\nu . \]

Hence

\[ \frac{w_m}{w_n}=\nu . \]

Thus, by determining \(w_m\) and \(w_n\) for large concentrations of the negative catalyst, one can find the chain length \(\nu\).

Fig. 3

As Backström showed, the value of \(\nu\) obtained in the indicated way for the thermal reaction, and that obtained from the quantum yield for the photochemical reaction, coincide excellently. This proves: 1) the correctness of the theoretical assumption concerning the mechanism of action of the negative catalyst, and 2) that the chains of the photochemical and thermal reactions are the same. The difference between these reactions consists only in the manner of creating the initial links of the chain.

Hinshelwood, in his work on the combination of \(\mathrm{H_2+O_2}\) at temperatures close to explosion, gave direct proof of the existence of thermal chain reactions. It turned out that this reaction, being typically heterogeneous at low temperatures, becomes chain and homogeneous near the explosion temperatures \((500—600^\circ \mathrm{C})\).

At low temperatures the reaction rate in a vessel filled with quartz fragments is greater than in an empty one, as should be expected for a heterogeneous reaction. At high temperatures, on the contrary, the reaction in an empty vessel proceeds much more

faster, obviously owing to the fact that the chains break off at the wall, and the narrower the space between the walls, the shorter the chain, and consequently the lower the reaction rate. This is demonstrated still more clearly by experiments with admixtures of inert gases (argon, helium, nitrogen). The more the reacting mixture is diluted with an inert gas, the more difficult is the diffusion of the chain to the wall, the greater the number of links in the chain, and the greater the reaction rate. Generalizing the formula for the chain length (p. 202) to the case in which an inert gas has been added to the mixture, I obtained the following dependence of the chain length \(\nu\) on the pressure of the inert gas \([y]\):

\[ \nu=\frac{6\pi x^{2}}{4l^{2}}[\mathrm{H}_{2}]\,[\mathrm{O}_{2}] \left(1+\frac{[y]}{[\mathrm{H}_{2}]+[\mathrm{O}_{2}]}\right) \tag{1} \]

In deriving this formula it is assumed that the mean free paths of all molecules are the same. If the difference in the lengths of the paths is taken into account and, for the chain reaction, Marshall’s scheme is used,
\(\mathrm{H}_{2}=\mathrm{H}+\mathrm{H};\ \mathrm{H}+\mathrm{O}_{2}=\mathrm{HO}_{2};\ \mathrm{HO}_{2}+\mathrm{H}_{2}=\mathrm{H}_{2}\mathrm{O}_{2}+\mathrm{H}\), etc., then a more accurate formula can be obtained.

Taking the reaction rate in the pure mixture to be

\[ w_{0}=n_{0}\nu_{0} \]

and, in the presence of an inert gas, \(w=n_{0}\nu\), we have

\[ \frac{w}{w_{0}}=\frac{\nu}{\nu_{0}}. \]

Having calculated \(\nu\) and \(\nu_{0}\), I was able to obtain theoretically \(\frac{w}{w_{0}}\). Let us compare these numbers with Hinshelwood’s data:

Nitrogen impurity:	Nitrogen impurity:	Nitrogen impurity:	Helium impurity:	Helium impurity:	Helium impurity:
\(\mathrm{N}_{2}\)	\(\frac{w_{\mathrm{N}_{2}}}{w_{0}}\), calc.	\(\frac{w_{\mathrm{N}_{2}}}{w_{0}}\), obs.	He	\(\frac{w_{\mathrm{He}}}{w_{0}}\), calc.	\(\frac{w_{\mathrm{He}}}{w_{0}}\), obs.
100 mm	2	2.23
200 mm	3	3.64
300 mm	4	4.61	300 mm	1.43	1.75
500 mm	6.1	10.4	500 mm	2.35	3.29

It should be noted that the discrepancy between the calculated and experimental values is not accidental, but is explained by the fact that the formula for \(v\) (p. 202) is only approximately correct.

Let us turn to Roginsky’s data on the thermal decomposition of nitroglycerin. This reaction was chosen because, according to our experiments with explosive gas mixtures, the decomposition of explosives is usually associated with the presence of long reaction chains. In this work it was found that the decomposition of nitroglycerin proceeds quite regularly, at a rate proportional to the amount of nitroglycerin not yet decomposed; consequently, this reaction is not autocatalytic. Its activation energy proved to be unusually large, \(E = 54\,000\) cal. Such a large activation value is usually associated with an unusually low reactivity of the substance.

However, nitroglycerin already decomposes very rapidly at a temperature of \(170^\circ\), which can be explained only by the presence of unusually long chains. Indeed, according to Polanyi’s theory of monomolecular reactions, the reaction rate is

\[ w = C n e^{-\frac{E}{kT}}, \]

where \(C\), for all substances, should have an order of magnitude of \(10^{14}\), and \(n\) is the number of molecules that have undergone decomposition. This formula is correct only in the case when there are no chains. When chains are present, this formula gives the correct expression only for the number of primary links of the chain appearing each second, i.e. for the quantity \(n_0\).

If the chain length is \(\nu\), then the true reaction rate will be

\[ w = n\nu = \nu C n e^{-\frac{E}{kT}}. \]

If, on the basis of this formula, one calculates \(\nu\), then for nitroglycerin it will be \(=10^8\).

Let us now proceed to the analysis of the results obtained in the Laboratory of Electron Chemistry on the oxidation of phosphorus vapor. If oxygen is admitted in a slow stream into a vessel containing phosphorus and well evacuated, the luminescence does not appear at once, but after some time, i.e. as though a certain minimum pressure of oxygen were necessary for ignition; moreover, if

if the oxygen pressure exceeds a certain definite value, which we shall call the maximum, then the burning of phosphorus again ceases. Both these phenomena, of the minimum as well as the maximum limiting pressure of oxygen in the oxidation of phosphorus, were noted by a number of authors many years ago. However, until recently there had been no detailed study of these phenomena, and only during the last two years have they been investigated in the Laboratory of Electron Chemistry.

It should be noted that these phenomena do not constitute some exceptional peculiarities of the phosphorus-oxidation reaction, but are considerably more general.

The same fact of a minimum and maximum pressure was established in England for the burning of arsenic vapors and in the Laboratory of Electron Chemistry for the burning of sulfur vapors in oxygen.

The data obtained in the Laboratory of Electron Chemistry amount to the following. If oxygen is introduced into a vessel with phosphorus at a pressure below the minimum \(p_{\min}\), then the reaction practically does not proceed at all. It is enough, however, to reduce the volume of the reaction vessel with the aid of a bulb filled with mercury, and thereby to raise the pressure \(p\), making it greater than \(p_{\min}\), for the phosphorus vapors to ignite and the reaction to begin. The burning continues until the oxygen has reacted to such an extent that its pressure decreases to the value \(p_{\min}\), after which the burning ceases.

If oxygen is admitted into a vessel with phosphorus through a capillary, then as long as \(p < p_{\min}\), the pressure in the vessel increases proportionally to time, at a rate corresponding to the rate at which oxygen enters. When \(p\) reaches the value \(p_{\min}\), a flash occurs, and any further increase of pressure ceases—the whole of the inflowing oxygen burns out, forming \(P_2O_5\). The course of the phenomenon may be followed in Fig. 4 (the arrow marks the moment of the flash). If the part of the vessel containing phosphorus and oxygen, at a pressure below \(p_{\min}\), is cooled to the temperature of liquid air, then, as was to be expected, the pressure in the vessel does not change. If the pressure in the vessel were due not to oxygen, but to some per-

a higher oxidation product of phosphorus, e.g. \(P_2O_5\), then on cooling the entire gas should have condensed, which was not observed.

By experiments of this kind it was finally established that, indeed, at pressures lower than a certain \(p_{\min}\), oxygen does not react with phosphorus.

This minimum pressure, to an accuracy of \(10\text{–}15\%\), does not depend on temperature at a given pressure of phosphorus vapor.

Fig. 4

phorus. With increasing pressure of phosphorus vapor, the minimum pressure of oxygen decreases according to the law

\[ p_{\min}=\frac{\mathrm{const}}{p_{\mathrm{P}_4}}, \]

where \(p_{\mathrm{P}_4}\) is the pressure of phosphorus vapor. At a given pressure \(p_{\mathrm{P}_4}\), the critical pressure depends very strongly on the dimensions of the reaction vessel, decreasing from the value \(1.7\cdot 10^{-1}\) mm of mercury at diameter \(d=0.46\) cm to \(6\cdot 10^{-4}\) at diameter 18 cm. The law of variation of \(p\) with \(d\) is expressed by one of the following formulas:

\[ p_1=\frac{B}{d^2} \tag{2} \]

or

\[ p_1=\frac{B}{d^{3/2}}. \tag{2'} \]

It is difficult to say which of these is in fact correct.

If some inert gas, e.g. argon, is introduced into the vessel with phosphorus, then the minimum pressure of oxygen diminishes—

CHAIN REACTIONS

decreases,^1 and this decrease is the more considerable, the greater the pressure of the inert gas. The relation between \(p_{\min}\) and the pressure of argon \((A)\) is expressed by the experimental formula

\[ p_1\left(1+\frac{[A]}{p_1+[P_4]}\right)=\mathrm{const}. \tag{3} \]

This is what the main results obtained reduce to.

Of the two possible dependences (2) and \((2')\), taking dependence (2) as the correct one, we can combine all the material obtained into a single formula

\[ p_1[P_4]\left(1+\frac{[A]}{p_1+[P_4]}\right)d^2=\mathrm{const}=c. \tag{4} \]

Let us turn to the interpretation of the results obtained. First of all it is necessary to understand why, in general, under certain conditions phosphorus vapor does not react with oxygen. It is natural to assume that this occurs as a consequence of the very large activation energy required by this reaction. Suppose that, for the initial act of the reaction, oxygen must be split into atoms and that the reaction develops according to the scheme:

\[ \mathrm{O}_2=\mathrm{O}+\mathrm{O} \tag{I} \]

\[ \mathrm{O}+\mathrm{P}_4=\mathrm{P}_4\mathrm{O} \tag{II} \]

\[ \mathrm{P}_4\mathrm{O}+\mathrm{O}_2=\mathrm{P}_4\mathrm{O}_2+\mathrm{O} \tag{III} \]

and so on, until the O atom is adsorbed by the wall. Denoting by \(n_0\) the number of oxygen atoms formed spontaneously as a result of thermal motion, we obtain for the reaction rate, according to the formula on p. 202, the expression

\[ w=n_0v=\frac{6\pi a^2}{4r_0}[\mathrm{O}_2][\mathrm{P}_4]\,n_0. \]

Since the heat of dissociation of \(\mathrm{O}_2\) is very large \((160\,000)\), the number \(n_0\) will be so small that the reaction rate is practically equal to zero.

Let us now take secondary activation into account. The molecule \(\mathrm{P}_4\mathrm{O}_2\), being further oxidized to \(\mathrm{P}_4\mathrm{O}_{10}\), releases energy which may promote the splitting of new

^1 The pressure was measured with a MacLeod manometer, which does not show the elasticity of saturated vapors. Therefore condensation of phosphorus vapors is not detected by the manometer.

atoms. Let the largest portion of the energy \(Q\) liberated in the reaction \(P_4O_{2n} + O_2 = (P_4O_{2n+2})\) be concentrated in the form of excess energy of the molecule \((P_4O_{2n+2})\). Upon its collision with an \(O_2\) molecule, dissociation of \(O_2\) into atoms occurs and, consequently, the appearance of new chains. This property of the chain to branch is also the cause of the existence of a minimum ignition pressure for phosphorus vapor. If the probability of such branching of the chain, or, what is the same thing, the probability of the reaction \((P_4O_{2n+2}) + O_2 = P_4O_{2n+2} + O + O\), is equal to \(c\), then this reduces the chances of chain termination. Indeed, if the probability of termination of the main chain is \(\beta = \dfrac{1}{\nu}\) (where \(\nu\) is determined by the formula on p. 202), then the possibility of its continuation as a result of side branchings reduces the probability \(\beta\) of termination of the chain with all its branchings by the amount \(c\); in this case \(\beta = \dfrac{1}{\nu} - c\), and for the reaction rate \(w\) we obtain the expression:

\[ w = \frac{n_0}{\beta} = \frac{n_0}{\frac{1}{\nu} - c} = \frac{n_0\nu}{1 - c\nu}. \tag{5} \]

The length of the chain increases with increasing oxygen pressure \([O_2]\); therefore, by increasing \([O_2]\) at a given temperature, one can reach the value \(c\nu = 1\). In this case the denominator turns to zero, and the reaction rate to infinity. This condition is precisely the condition for ignition of phosphorus, and the pressure \(p_1 = [a]\) at which this condition is fulfilled is the critical minimum ignition pressure. For small \(n_0\), the reaction rate will remain negligible up to pressures very close to \(p_1\), and then, with a very small change in \(p_1\), it rapidly increases to infinity, as shown in Fig. 5. The condition

\[ c\nu = 1 \]

makes it possible to determine theoretically the dependence of the minimum pressure \(p_1\) on the dimensions of the vessel and the pressure of the inert gas \((A)\). According to formula (1) (p. 209), the length of the phosphorus oxidation chain in this case is

\[ \nu = \frac{6\pi d^2}{64l_0^2}\,[O_2]\,[P_4]\left\{1+\frac{A}{[O_2]+[P_4]}\right\}. \]

According to condition (5), burning begins at \(cv=1\) or at

\[ d^2[\mathrm{O}_2],[\mathrm{P}_4]\left\{1+\frac{A}{[\mathrm{O}_2]+[\mathrm{P}_4]}\right\}=\mathrm{Const}, \]

thus we have derived a theoretical formula entirely coinciding with (4), which was obtained from experiment. Here \([\mathrm{O}_2]\) is the minimum ignition pressure of oxygen at the given \([\mathrm{P}_4]\).

All the results obtained by us with phosphorus were confirmed by Hinshelwood and Dalton on the reaction of oxidation of phosphine.

Let us now dwell on the following question: according to our theory, under the condition \(cv=1\), the reaction rate is equal to infinity. Meanwhile, in reality, the rate of combustion of phosphorus, although comparatively large, is far from being equal to \(\infty\). Several minutes are needed for an amount of oxygen measured by several mm of pressure to burn in phosphorus vapor. The matter here is that the formula

\[ w=\frac{n_0}{\frac{1}{\nu}-e^{U/kT}} \]

will give actual values of the rate only after an infinitely long time has elapsed after the gases have been mixed. In reality the reaction rate continually increases with time, and here one must distinguish three cases:

Fig. 5

1) \(\beta=\dfrac{1}{\nu}-c>0\).

In this case the reaction rate increases with time according to the formula \(w(t)=\dfrac{n_0}{\beta}\left(1-e^{-k\beta t}\right)\), where \(k\) is a certain quantity determined by the pressure of the components. The graph of this curve is shown in Fig. 6 (curve \(a\)).

2) \(\beta=0\).

In this case \(w(t)=kn_0t\) (curve \(b\)).

3) \(\beta<0\).

Then \(w(t)=\dfrac{n_0}{(\beta)}\left(e^{k\beta t}-1\right)\), where \((\beta)\) is the absolute value of \(\beta\) (curve \(c\)).

In this case, as \(t\to\infty\), likewise \(w(t)\to\infty\), and the larger \((\beta)\) is, the faster this occurs.

The smaller the number of initial centers \(n_0\), the more slowly the reaction rate increases. Since in the case of phosphorus oxidation \(n_0\) is probably very small, the increase in rate occurs sufficiently slowly, and therefore the reaction proceeds under conditions in which \(t\) is far from infinity.

Fig. 6

Let us now turn to the case where \(n_0\) is exceptionally small. If \(n_0\) were equal to zero, then the reaction could not proceed at all even under the condition \(\alpha=1\). If \(n_0\) is very small, then the increase in rate may proceed so slowly that several hours would be required for it to reach an appreciable magnitude. If this time of “induction” is measured in hours, i.e. if the increase in the number of initial centers proceeds exceptionally slowly, it is sufficient to assume the existence in the volume of a negligible amount of an impurity that breaks the chains, so that its negative action would be sufficient to compensate for the growth of chains. Therefore induction periods of the order of hours hardly have any meaning, and we should rather assume that, with a very small number of initial centers, the combustion reaction will not proceed at all, despite the fulfillment of the condition. However, by decreasing \(\beta\), i.e.

by taking a pressure exceeding the critical one, we accelerate the process of growth of the rate and can obtain ignition. Another conceivable way of producing ignition consists in artificially creating the initial links of the reaction in sufficient quantity. If combustion has begun, it will continue until the pressure falls below the critical value, i.e. until \(\nu = 1\).

All these phenomena, predicted by the theory, were actually obtained in the Laboratory of Electron Chemistry in the reaction of oxidation of sulfur vapor.

Before these experiments it was believed that sulfur ignites in oxygen at \(285^\circ\). At \(200^\circ\) the reaction of sulfur oxidation was studied by Rideal, and it turned out that its rate is negligible and that the reaction proceeds only on the surface of the sulfur. In fact, however, sulfur can ignite at a considerably lower temperature; only for it, as for phosphorus, there exists a maximum oxygen pressure above which the reaction does not proceed at all. This maximum pressure decreases as the vapor pressure of sulfur decreases. At an oxygen pressure equal to atmospheric pressure, in order for combustion to be possible, a sulfur-vapor pressure corresponding to \(280^\circ\) is required.

At temperatures around \(100^\circ\), sulfur can also burn if the oxygen pressure is less than \(60\)—\(90\) mm. The absence of experiments at reduced oxygen pressure, as well as the difficulties connected with ignition (see below), explain why the combustion of sulfur with visible volumetric luminescence had never before been observed even at temperatures of \(50^\circ\) C. At still lower temperatures combustion could not be noticed not because of the low temperature, but owing to the negligible vapor pressure of sulfur under these conditions.

Thus it turned out that the combustion of sulfur, like the combustion of phosphorus, can be induced only at an oxygen pressure lying between a certain minimum and maximum. However, whereas phosphorus within these limits of oxygen pressure always ignites spontaneously, ignition of sulfur under the same conditions is difficult to bring about; but once ignition has occurred, combustion continues until

oxygen pressure is not made equal to the minimum. After this, damping and complete cessation of the reaction occur, as for phosphorus. The phenomenon proceeds as if \(n_0\) for sulfur were extremely small.

The results of the experiment could be summarized as follows:
1) spontaneous ignition of sulfur occurs (and not always, probably owing to fluctuations in the purity of the preparations) only in the case where the oxygen pressure exceeds the minimum many times over, i.e. \(p > p_{\min}\) (the minimum pressure was determined from the conditions of extinction and artificial ignition; its value is measured in tenths of a mm, whereas the pressure at which spontaneous ignition occurs is measured in centimeters); 2) in the case of spontaneous ignition the flash does not occur immediately after mixing the gases, but after a certain interval of time, from 5 to 60 sec. (the induction period); 3) if traces of ozone, obtained by an electrodeless discharge, are admixed with the oxygen admitted into the vessel with sulfur, or if a single flash of an electrodeless discharge is passed through the mixture, then ignition always occurs, provided the oxygen pressure is greater than the critical one.1

As can be seen, all these results are a direct confirmation of the theory. Experiments on igniting sulfur by passing a discharge or by admixing ozone to the oxygen allow one to think that in this case as well the initial centers are oxygen atoms. If this is so, then why are there more of these O atoms in the reaction of oxygen with phosphorus than with sulfur? There is as yet no direct answer to this question, but one can imagine that the appearance of O atoms occurs during adsorption or reaction of \(\mathrm{O}_2\) with solid phosphorus and sulfur. In that case, if the adsorption energy of the \(\mathrm{O}_2\) molecule by solid phosphorus is greater than by sulfur, then—

It is clear why in the first case we have a larger number of O atoms than in the second.

Turning to the phenomena of maximum pressure, we must, unfortunately, state that here the causes are less clear. Nevertheless, chain theory here too indicates the path along which the investigator’s thought must proceed. The quantity \(\beta\), as a function of the partial pressure of oxygen, apparently has the form shown in Fig. 7, i.e., it has a minimum and intersects the abscissa axis at two points. In this case the reaction rate \(w\), as a function of the pressure of oxygen, will be expressed by the curve in Fig. 8.

Fig. 7

Fig. 8

Why \(\beta\) at high partial pressures of oxygen begins to increase again is difficult to say; however, in any case this is possible. Indeed, suppose, for example, that in oxygen there is some impurity reacting with O atoms and thereby breaking the chain. The concentration of this impurity \((X)\) will be proportional to the concentration of oxygen \((X):(O_2)=\mu\), where \(\mu\) is probably very small. In that case the probability of chain breaking \(\beta\) must be increased by the magnitude of the probability of destruction of O through reaction with impurities, and not through reaction with \(P_4\), i.e., by the amount

\[ \frac{[X]}{[P_4]+[X]}=\frac{\mu[O_2]}{[P_4]+\mu[O_2]}. \]

Thus, instead of the expression \(\beta=\frac{1}{\nu}-c\), we obtain

\[ \beta=\frac{1}{\nu}-c+\frac{\mu[O_2]}{[P_4]+\mu[O_2]}=\frac{1}{\nu}-c+f. \]

At small pressures \([O_2]\), the quantity \(f\) is very small and may be neglected. Thus the condition \(\beta=\frac{1}{\nu}-c=0\) still determines the minimum pressure of oxygen. At large \([O_2]\), \(\frac{1}{\nu}\) becomes negligibly small, but the quantity \(f\) increases. At a certain \([O_2]\), when \(f\) becomes greater than \(c\), the reaction again becomes impossible. The condition

\[ \frac{\mu [O_2]}{[P_4]+\mu [O_2]}-c=0 \]

therefore determines the maximum possible pressure of oxygen, above which the combustion of phosphorus vapor does not occur.

Thus,

\[ [O_2]_{\max}\simeq \frac{c}{\mu}[P_4]. \]

We obtain, therefore, in agreement with our experiments, a direct proportionality between \([O_2]\) and \([P_4]\).

From the experimental value we find that \(\mu\) must be of the order of \(10^{-5}\), i.e. about \(0.001\%\). Such a quantity of impurity we can hardly detect by any chemical means.

In favor of this theory also speak the known facts that the maximum pressure is extraordinarily sensitive to the slightest traces of certain foreign impurities. In our practice we have also had occasion to encounter a number of such cases.

The expressions we have obtained for the ignition limits

\[ P_{\min}=\frac{K_1}{[P_4]} \quad \text{and} \quad P_{\max}=K_2[P_4] \]

are correct only approximately and only at comparatively large \([P_4]\). At very small values of \([P_4]\), the upper and lower limits must be determined as the two roots of the quadratic equation \(\frac{1}{\nu}-c+f=0\), since at small \(P_4\) all three terms here have the same order of magnitude. The roots of this equation become imaginary when \([P_4]\) reaches a certain definite minimum value \([P_4]!\) And, indeed, Kovalsky proved experimentally that at \([P_4]\) less than mm, igni...

change occurs at any oxygen pressure. In Fig. 9 the theoretical curve of the dependence of both roots of the equation \(P_{\max}\) and \(P_{\min}\) on \([P_4]\) is given; on the ordinate axis \([O_2]\) is plotted, and on the abscissa axis \([P_4]\). The points correspond to the experimental data; as can be seen, theory and experiment coincide.

The phenomena of minimum pressure occur not only in the case of the combustion of sulfur, arsenic, and phosphorus. Apparently they are much more widespread. In our laboratory the existence of a minimum pressure was established for the reactions \(H_2 + O_2\) and \(CO + O_2\).

The results of the work of Dixon, Zagulin, and Hinshelwood proved that the ignition region of \(H_2 + O_2\) and \(CO + O_2\) has a very peculiar form, as shown in Figure 10, where several ignition curves are given, corresponding to different percentage compositions of the mixture. Temperature is plotted on the abscissa axis, and the pressure of the mixture on the ordinate axis. The ignition regions lie to the right of the curve. If along the upper part of the curve we obtain a typical explosion, then along the lower part, where the pressure is low, ignition has the character of a weak glow, analogous to the glow of phosphorus and sulfur when they burn in oxygen.

Fig. 9.

This lower limit was discovered by Zagulin, and the question arose before us whether we did not have here a phenomenon analogous to the ignition of phosphorus, and whether the minimum pressure is not determined by the ignition of \(H_2 + CO\) in oxygen, below which these reactions do not proceed at all. We undertook a study of the reaction \(H_2 + O_2\) at pressures lying below it and, indeed, found manometrically a complete absence of reaction up to a definite pressure.

If one takes a mixture of \(\mathrm{H_2 + O_2}\) at a pressure lying above, then the reaction proceeds rapidly, but stops at once when the partial pressure of \(\mathrm{H_2 + O_2}\) decreases to a certain quite definite value, i.e. everything proceeds in the same way as in phosphorus.

In conclusion I must point out that recently facts have appeared which, apparently, cannot be explained by the chain theory and which signal the necessity of a radical reconsideration of all the foundations of chemical kinetics.

Fig. 10

It is true that so far there are only two such facts, and they have not been sufficiently verified. One of them is the result of Trifonov indicated above, according to which the photochemical reaction \(\mathrm{H_2 + Cl_2}\) ceases at pressures lying below a certain minimum. The second fact was discovered by Sprenger and consists in the fact that the thermal decomposition of \(\mathrm{N_2O_5}\) stops completely at very low pressures; at first glance both these facts are very similar to the phenomenon of the minimum pressures for the combustion of phosphorus, sulfur, etc. However, there we are dealing with ignition reactions, self-accelerating reactions, and the phenomenon of the critical pressure was easily explained by the transition from negligible

slow stationary reaction to a nonstationary one, and was connected with the denominator becoming zero.

Here, however, the reactions are certainly not self-accelerating: both in the case of \(N_2O_5\) and, all the more so, in the photochemical combination \(H_2 + Cl_2\), they proceed at a constant, definite rate and have no tendency whatsoever toward self-acceleration.

Thus, here the vanishing, under certain conditions, of the denominator in the formula for the reaction rate cannot explain the reason for the stopping of the reaction. If, in these cases, the explanation must be different, then perhaps also in the case of the stopping of the reaction in the combustion of phosphorus, sulfur, etc., it is precisely this explanation, and not the one we gave, that is correct.

Indeed, what grounds do we have for asserting that all processes in nature tend toward equilibrium?

Ya. I. Frenkel drew my attention to the fact that the atom itself is a nonequilibrium system, in a state of false equilibrium, since it consists of \(+\) and \(-\) electricity, which nature, for some reason, does not allow to combine. What do we know about these prohibitions that nature places in the path toward attaining equilibrium? Do we even have grounds to assert that a mixture of \(H_2 + O_2\), which can stand for millennia without visible signs of reaction, is in fact, albeit very slowly, advancing toward equilibrium? Would it not be more honest to say that the reaction of water formation in this case does not proceed at all, absolutely does not proceed—and that the system is in false equilibrium? And perhaps, in fact, the exceptions in nature are certain cases of the equilibrium state, to which we ascribe such universal significance only because, theoretically, they can be analyzed more simply.

1. Here we obtain for the first time a theoretical approach to the phenomena of positive homogeneous catalysis (e.g. the action of negligible traces of moisture). The action of these catalysts is similar to the role of ozone in the reaction under consideration and consists in creating the initial links of chains, which leads to an extraordinary acceleration of the reaction. ↩