PROBLEMS OF ENERGY TRANSFER IN THE KINETICS OF CHEMICAL REACTIONS1

K. N. Hinshelwood, Oxford.

The kinetics of reactions is a borderland area of chemistry and pure physics, in which physicists and chemists do not live in complete agreement. In particular, physicists suppose that chemists ought to perform such decisive experiments as prove too difficult for experiment, while chemists set physicists such theoretical problems as are excessively tangled. Nevertheless, successes have been achieved in this field, since there exist many problems whose solution appears especially important to chemists and which are also, from a purely physical point of view, very interesting. What is at issue here is, in particular, the conditions under which molecules can exert an influence upon one another through collisions or radiation.

First of all I must dwell quite briefly on the question of why exactly we think that the energy of molecules plays the chief role in chemical processes.

There are many chemical reactions that proceed at a measurable rate; consequently, at a given moment in time molecules possess unequal capacity for reaction. Further, almost all reactions depend very strongly on temperature, and this dependence

is expressed by the well-known exponential Arrhenius law. These two facts can be explained by proceeding from the assumption that only the so-called “active molecules” react, i.e., molecules which, by collision, absorption of radiation, or in some other way, have acquired a certain excess of energy relative to its mean value. This assumption leads directly to the Arrhenius equation, from which the magnitude of the activation energy can be calculated. This theory of activation gives us a coherent picture of chemical reactions. Especially simple conditions occur in the case of gaseous reactions, where the number of collisions and analogous quantities can be calculated exactly by the well-known methods of the kinetic theory of gases.

In the case of bimolecular gaseous reactions (for example \(2HI = H_2 + I_2\)), almost without exception it turns out that the number of molecules reacting per unit time is equal—at least as regards order of magnitude—to the number of molecules colliding per unit time, multiplied by the factor \(e^{-E/RT}\), where \(E\) is the heat of activation, which can be calculated from the experimentally found temperature coefficient. Since in different gases the number of collisions does not vary too greatly, it follows that, at temperatures corresponding to the same reaction rate, the factor \(E/RT\) has an almost constant value. This result may serve as evidence that, in simple cases, activation is not only a necessary but also a sufficient condition for chemical transformation.

If we turn to trimolecular reactions, then the number of collisions in this case is significantly smaller; therefore trimolecular reactions which at a given temperature proceed with a still measurable rate are associated with correspondingly smaller heats of activation, which, generally speaking, is confirmed by experiment.

The number of impacts that cause chemical transformation is found by multiplying the number of all impacts by the simple exponential factor \(e^{-E/RT}\). This factor is derived from the Maxwell–Boltzmann law. The application of this

formulas is based on the assumption that the energy of the molecules is distributed over two degrees of freedom or, more precisely, over two quadratic terms. Such an assumption would be exact if one were to imagine, for example, the mechanism of activation as a sufficiently intense collision of two molecules. In that case the two degrees of freedom would be associated with two opposite components of the relative translational motion. Although this is not the only possibility, I would nevertheless like to emphasize that the experimental applicability of the simple equation for bimolecular reactions at least proves that the mechanism of activation is comparatively simple, since only in the case when the energy is confined to a small number of degrees of freedom can our formula be regarded as valid.

From this result one may conclude that, probably, no appreciable interval of time elapses between the impact and the actual chemical transformation. This is, consequently, a supposition that can be made a priori. In accordance with this, it turns out that the reactions which obey the above-mentioned law proceed in strict proportion to the square of the gas pressure, i.e. purely bimolecularly. If, however, a considerable interval of time were to elapse between activation and transformation, the reactions would not exhibit a purely bimolecular course. I shall now try to explain this.

Let us suppose that, by collision, a molecule has been brought into a state with a large store of energy. If the energy can be distributed over many degrees of freedom, it may happen that the molecule reacts only after the lapse of a certain interval of time—and, further, it may happen that this interval proves longer than the interval between two collisions. Thus there is a considerable probability that the activated molecule will again lose its energy as a result of collision before it passes into the stage of reaction. This means that the mutual activation and deactivation of molecules will occur very rapidly in comparison with the actual completed chemical transformation. The statistical equilibrium between

by molecules rich and poor in energy will be almost independent of the disappearance of active molecules as a result of the reaction. The rate of the reaction will in that case be quite different from the rate of activation; namely, it will be equal to:

\[ \begin{array}{ccccc} \text{Total number} & \times & \text{Fraction of molecules possessing} & \times & \text{A small fraction,}\\ \text{of molecules} & & \text{sufficient energy} & & \\ N & \times & f(E) & \times & \chi \end{array} \]

\(\chi\) is a purely probability factor.

It is easy to see that in this case the rate of the reaction will be independent of pressure, for it is natural that the distribution of energies does not depend on pressure, since in general it is not disturbed. We have, therefore, a criterion for a monomolecular reaction. Formerly it was thought that gas reactions independent of pressure—the independence of pressure being our experimental criterion of a monomolecular reaction—must represent the transformation of isolated molecules. This conception has now proved untenable. Reactions that do not depend on pressure and therefore may, seemingly, not depend on collisions (strictly speaking, they do not depend on the number of collisions, but not on collisions) may conveniently be called quasi-monomolecular reactions. It has turned out that simple decompositions proceed bimolecularly, while decompositions of complex molecules proceed quasi-monomolecularly, as is evident from Table I.

TABLE I

In bimolecular reactions we thus have activation and immediate transformation; consequently, a constant disturbance of statistical equilibrium. In monomolecular reactions the situation is quite different: their characteristic feature is a large number of degrees of freedom, the absence of transformation until the molecule enters a favorable phase, and numerous deactivations, i.e. an insignificant disturbance of statistical equilibrium.

The fact that a large number of degrees of freedom plays a role in the mechanism of activation is also revealed in another connection. At low pressures the free path length is large; the lower the pressure, the more probable it becomes that the chemical transformation will occur in the interval between the activating and the deactivating collision. In this case statistical equilibrium already begins to be appreciably disturbed, and the reaction ceases to follow a monomolecular course. The rate coefficient for a monomolecular reaction begins to depend on the pressure, which is excellently confirmed by experiment. Such a case occurs, for example, in the decomposition of dimethyl ether. Thus one may quite justifiably assume that in the region of pressure where the decrease of the rate constant is only beginning, the number of collisions is just sufficient to maintain the reaction rate. This means that the rate of activation can be calculated. In all quasi-monomolecular reactions this rate is considerably greater than the number of collisions, and greater by a factor measured by \(10\) to some power. A sufficient rate of activation for the decomposition of ether is obtained only when the energy is distributed over a number of quadratic terms from 8 to 12. For a larger number of degrees of freedom the formula obtained is

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

where \(Z\) denotes the number of collisions per unit time, and \(n\) the number of quadratic terms. (When the energy is distributed over

several degrees of freedom, the number of activating impacts, owing to the enormous magnitude of the number of permutations, increases by \(10\) to some power.)

TABLE II

I should like to emphasize that, in this way, there exist two independent confirmations of the conclusion we have drawn. On the one hand, we obtained this confirmation from a purely kinetic point of view, from comparing bimolecular reactions (simple molecules) with quasi-monomolecular reactions (complex molecules). On the other hand, the distribution law itself gave us the proof sought.

In monomolecular reactions one can no longer expect a strict parallelism between the heat of activation and the temperature of equal reaction rate, for here other specific factors come into play. Nevertheless, a certain parallelism exists, although it is considerably less distinct than in the case of bimolecular reactions (cf. Table II).

Even this brief survey is quite sufficient to convince us that the conception of the reaction rate as an activation problem is fundamentally correct.

We may now turn to the consideration of unresolved problems with a certain degree of confidence that the difficulties encountered—if one may put it so—are fruitful difficulties.

The first problem of which I shall speak is connected with the famous decomposition reaction of nitrogen anhydride \((N_2O_5)\). In all the other monomolecular reactions that have been investigated at low temperatures, a decrease in the rate coefficient has been established. For nitrogen anhydride, according to new investigations, it has so far not been possible to observe this decrease. If it does occur, then it is at pressures less than one hundredth of a millimeter. At these pressures the maximum calculated rate of activation is considerably less than the observed rate of reaction, however large the adopted number of degrees of freedom may be. Naturally the question arises how the reacting molecules obtain the energy which they must possess, since the reaction does in fact take place.

In order to resolve this difficulty, three assumptions may be made.

First, one may assume that in a collision all the energy of the two molecules remains with one of them. This assumption, which seems to me somewhat extreme, is used by Fowler. It can be shown quite directly that this assumption may be replaced by another, namely, that the diameter of the molecule for deactivating collisions is several times larger than for activating collisions. The diameter of a molecule in an excited state is, naturally, somewhat larger than in the normal state. Whether, however, this difference is as sharp as must be assumed in the present case is still entirely unclear.

We now pass to the second possibility. One might with equal right assume that, also in activation, the diameter of the molecule is much larger than that which is calculated in the known way from internal friction and thermal conductivity. This possibility is of the highest interest; it has already been indicated, and it is connected with a whole series of purely physical observations.

The third possibility is given by the theory of chain reactions.

These possibilities are of considerable interest, extending far beyond the limits of the problem of nitrogen anhydride. I shall therefore dwell briefly on each of them.

As regards the problem of the diameter of the molecule, observations of the quenching of the polarized fluorescence of mercury have shown that mercury atoms exert on one another an influence far exceeding that which atoms of a foreign gas can exert. Moreover, this influence is such that the mercury atoms must be assigned a radius considerably greater than the ordinary radius calculated from the kinetic theory of gases.

Thus the mutual influence of identical molecules should be regarded as one of the fundamental problems of physics.

As for the phenomena of fluorescence in solutions, it is known that in concentrated solutions fluorescence, generally speaking, is quenched. This quenching should be ascribed to the mutual deactivation of the molecules of the fluorescing substance, i.e., to an influence which the molecules of the solvent cannot exert on the molecules of the fluorescing substance.

In the decomposition of acetaldehyde a phenomenon is observed which, apparently, is to a certain extent related to that described. This decomposition proceeds according to the equation:

\[ 2[\mathrm{CH}_3\mathrm{CHO}\to \mathrm{CH}_4+\mathrm{CO}], \]

i.e., bimolecularly. From a purely chemical point of view this bimolecular course of the reaction is by no means inevitable, as, for example, in the case of the decomposition of hydrogen iodide. One might therefore have expected that collisions with foreign molecules would also be able to transfer the necessary activation energy. Experimentally it has been found that collisions between aldehyde molecules and hydrogen or nitrogen, although they can cause decomposition, are nevertheless approximately ten times less effective than collisions between two aldehyde molecules. Here we have analogous actions of identical molecules upon one another.

A detailed discussion of such questions with the aid of the new quantum theory appears highly desirable and would undoubtedly be fruitful.

The general question concerning specific energy transfer is of outstanding interest to chemists. If apparently identical molecules possess this peculiar “sympathy,” then there also exist striking examples of specific physical interactions between dissimilar molecules.

In the well-known reaction of the combination of chlorine with hydrogen, chlorine molecules are excited by the absorption of light and thereby become capable of reaction. Oxygen exerts a quite peculiar inhibiting influence on this reaction. Consequently, it must deactivate the chlorine molecules.

Among the above-mentioned quasi-monomolecular reactions there is one especially interesting example. As I have already said, at low pressures the constant of these reactions falls. The decrease begins from the moment when the number of collisions becomes insufficient to maintain statistical equilibrium. In the presence of hydrogen, however, the constant again assumes its normal value. Hydrogen can hardly exert a truly chemical action, since in no case can it increase the constant to values exceeding its normal magnitude. Its action, therefore, must consist solely in a transfer of energy. Other gases, if they possess this capacity at all, do so to a far smaller degree. Thus hydrogen possesses a specific activating action (it must, of course, also exert a corresponding deactivating action; otherwise we would violate the principle of microscopic reversibility. In the case of the chlorine reaction the matter stands otherwise, since there we have a constant conversion of light energy, i.e., the absence of thermodynamic equilibrium).

I must now turn to the important question of the so-called chain reactions.

The assumption of the existence of chain reactions has proved fruitful in two directions. There are photochemical reactions in which very many reacting molecules correspond to one light quantum. This deviation from

Einstein’s law can hardly be explained otherwise than by making the assumption that the molecules formed in the primary light reaction transfer their activation energy to the molecules of the initial substance and directly activate the latter, thus creating a chain.

The same assumption helps to cope with the difficulties that arise in those cases where the observed rate of ordinary thermal reactions apparently exceeds the possible rate of activation, as I have already mentioned in speaking of the decomposition of nitrogen anhydride. By direct transfer of the initial activation energy, together with the heat evolved in the reaction, new molecules are activated which otherwise could not have been activated at all. Consequently, the maximum calculated rate of the reaction may be multiplied by an arbitrarily large factor.

Some photochemical reactions, and especially those associated with an abnormally high quantum yield, prove to be very sensitive to the specific inhibiting effects of various substances. These inhibiting substances probably act because they break off chain reactions. Recently Bäckström found that some reactions which proceed photochemically with a large quantum yield and are subject to inhibiting effects are sensitive to the same inhibiting substances when they proceed purely thermally. This may serve as evidence that, in chemical reactions as well, a chain mechanism is possible in general.

Christiansen and Kramers, as is known, proposed an analogous hypothesis of chain reactions for solving the problem of nitrogen anhydride. It may be objected to this that the hypothesis of Christiansen and Kramers is shaken by the fact that foreign gases do not inhibit the decomposition, whereas one might have thought that molecules of foreign gases should take energy away from the active molecules participating in the chain. Christiansen and Kramers had to make a special hypothesis that the active molecules transfer their energy only to a special ki-

molecules. Objections could be raised, on which I have no need to dwell here, for I am convinced that these objections, even if they do not disappear, are in any case far less important than one might at first think.

Experiments recently carried out at Oxford, it seems to me, have given proof—or at least an indication—that collisions between active molecules and molecules of foreign gases are in most cases elastic—or, at least, that there is a fairly high probability that, in the development of chain reactions, active molecules can withstand a large number of collisions without losing their activation energy until this energy is transferred to the corresponding molecules. I shall allow myself briefly to report on these experiments, since they have some bearing on the general question. The purpose of the experiments was to study the gas reaction between oxygen and hydrogen. The slow combination of detonating gas is subject to a strong influence from the walls of the vessel. These catalytic influences were studied in detail by Bodenstein. We carried out our experiments by a modified method at temperatures only slightly lower than the explosion temperature, intending in this way to isolate the homogeneous gas reaction. At 500° the reaction proceeds only on the walls of the vessel. Between 540° and 590° a reaction occurs which is evidently of a quite different character from the reaction on the walls; it is distinguished by a very strong dependence on pressure and by a high temperature coefficient. This reaction, without any doubt, proceeds in the gas phase. Most remarkable, however, was the following: the new reaction, in contrast to the reaction on the surface, was not accelerated by increasing the surface of the vessel, but, on the contrary, was very strongly retarded. This fact can be explained only on the assumption that the energy liberated within the gas is transmitted by chains. When the active molecules reach the walls, they are there, owing to an independent reaction, destroyed or deactivated. Thus an increase in surface area causes a shortening of the chains, i.e.

inhibition of the reaction1. Thus it is probable that chains of some kind play a role in these reactions.

Now the question arises: if chains exist at all, what influence do foreign gases have on the reaction rate?

Foreign gases either strongly inhibit the reaction, this occurring when collisions with the active molecules are inelastic; or, if the collisions are elastic, then the foreign gases increase the length of the path which the chain traverses before it reaches the wall. In this case one should expect an acceleration of the reaction. I must emphasize that I by no means wish to predict the result of the experiment. In reality, foreign gases produce a sharply pronounced acceleration, and the action of different gases is different. The sequence of the gases is as follows: He, N₂, A, H₂O, and to them correspond the ratios 1 : 3 : 4 : 5. An excess of hydrogen and oxygen acts in an analogous manner, which greatly complicates the application of the law of mass action. The sequence of the accelerating effects is approximately the same as that of the reciprocal diffusion coefficients, and therefore corresponds to the relative lengthening of the chains.

I cannot find any other interpretation of these results, except that, in the case of the union of hydrogen with oxygen, the active molecules are not deactivated when they undergo collision with foreign gases.

In conclusion I shall also mention some questions whose solution would be important for chemistry. English physicists think that all chemists are fools; perhaps German physicists think the same. Be that as it may, I only wish to illustrate the proverb according to which a fool can ask, in the course of an hour (and I need only two minutes for this), more questions than a wise man can answer in centuries. Thus it would be very important to develop in detail the problem of the diameter of the molecule on the basis

of the new quantum theory, in particular to establish rules that might provide some indication of when and how the “ordinary” radius of molecules is modified.

Secondly, the complete dynamics of the chemical “bond electrons” would help meet an urgent need of chemists. Thus, for example, in chemical decomposition reactions that proceed thermally, we are dealing with vibrations of nuclei, whereas in photochemical reactions—with the activation of an electron. What is the internal relation between these two processes? How is a chemical bond to be activated?

Thirdly, closely connected with the first question is the question of catalysis by “traces of a substance.” If one thinks, together with Bekker, that the first traces of water really shift the equilibrium between simple and polymerized molecules of a liquid, then this shift arises only because the addition of an entirely minimal number of molecules causes a substantially stronger change in entropy than was generally considered possible. The first molecules of a foreign substance must propagate their influence throughout the entire system. I confess that I am somewhat skeptical: it is possible that the whole phenomenon has been interpreted incorrectly. Nevertheless, it would be useful to know—starting, for example, from Schrödinger’s charge cloud—whether such a phenomenon is possible at all.

1. I tentatively consider it probable that the chain of reactions is maintained by the transfer of energy only from the newly formed water molecule to the oxygen molecule. The grounds that lead me to think so I cannot set forth here for lack of time. ↩↩