On the Nature of Activation Heat and the Behavior of Activated Molecules

L. Rozenkevich, Kharkov

Interaction of Atoms

The question of the interaction of atoms has, in principle, been completely solved by quantum mechanics. We can describe this interaction, conveying, in the case of atoms with a small atomic number, all the features of the formation or decomposition of molecules, determining the dissociation energy of molecules and the activation energy, indicating the dimensions of molecules corresponding to the state with the minimum possible vibrational energy (and, at absolute zero temperature, not equal to zero), and drawing various conclusions about the probability of one transformation or another. However, in the case of molecules built from atoms with even a somewhat larger atomic number, even in the case of diatomic molecules, such calculations become exceptionally complicated. Theoretical physics, concerned with general regularities and striving to understand and explain what is general in experiment, leaves these calculations aside. There are other ways, besides direct calculation, to determine various molecular constants. These ways are indicated, for example, by the theory of molecular spectra, which contributes very much to the solution of specific questions. It is simplest to determine constants by making maximum use of experiment, and not by calculations based only on the known number of electrons in the system and the number and charge of the nuclei.

The interaction of atoms has several special features. Among them, the first to be noted is the discontinu—

applicability of the principle of superposition to the interaction of atoms capable of reacting chemically. The forces acting between two atoms depend on the presence and relative positions of extraneous atoms. In this there is an enormous difference between the interaction of atoms and the interaction of point electric charges or the gravitational interaction of material bodies. The second feature, which must also be emphasized, consists in the fact that homopolar molecules are bound by forces that decrease very rapidly with the distance between the nuclei. If the nuclei are moved away from one another, this force decreases exponentially with distance, giving way, at distances greater than the gas-kinetic dimensions of molecules, to van der Waals forces, which are much weaker and have a polarization origin.

Heat of Activation

For clarity, when considering the mutual motion of atoms, it is very useful to describe this motion by means of curves or surfaces of the total potential energy of the system of atoms. The displacement of atomic nuclei relative to one another can almost always legitimately be represented as motion occurring according to the rules of classical mechanics for a system of material points. (The behavior of the electrons need not interest us at all here. The electrons merely create the forces that attract or repel the atoms. It is clear that the behavior of the electrons in no way obeys classical notions.)

After the remarks set forth in § 1, it is easy to understand what constitutes the essence of activation energy. For simplicity, let us consider an exchange reaction of the type

\[ \mathrm{A}+\mathrm{B}_2=\mathrm{AB}+\mathrm{B}. \tag{1} \]

The magnitude of the activation energy depends on the direction in which atom \(\mathrm{A}\) approaches the molecule \(\mathrm{B}_2\), as will be discussed below. We shall choose the direction for which the activation energy is minimal. Such a direction

is, for the approach of atom A, the direction of the straight line connecting the nuclei of molecule B$_2$.

The bond existing within B$_2$ will not remain constant as atom A approaches the molecule. This bond will weaken as the approach proceeds, while the molecule as a whole resists the approach of atom A, repelling it. However, at a certain mutual distance the repulsion between A and B$_2$ changes into attraction between A and B; atom A combines with atom B, and the second of the atoms of molecule B$_2$ begins to be repelled by the system AB. As a rule, in the case of an exchange reaction we must not associate such a transition with any abrupt change in the state of the electron shell of the three-atom system. The entire process proceeds adiabatically, i.e., by changing the distance between A and B, for each value of $R_{AB}$ we always have one definite state of the system, varying continuously with the change of $R_{AB}$*.

Fig. 1.

The interplay of forces occurring in the indicated process is easy to visualize if one represents graphically the surface of the potential energy for the mutual distances. We proceed as follows. In three-dimensional space, along the $x$ and $y$ axes of a rectangular coordinate system, we lay off $R_{AB}$ and $R_{BB}$. Along the $z$ axis, directed upward perpendicular to the drawing, we lay off the value of the mutual potential energy of the system of three atoms. We shall characterize at each

* For simplicity, we have restricted ourselves here to the case where, for the reaction, there is no need for preliminary excitation of B$_2$.

point of the plane \(xy\) (each point of the plane \(xy\) determines the relative position of all three nuclei) the potential energy of the system of three atoms by a series of equipotential surfaces. The drawing obtained in this way is given here. From it one sees that the most favorable path in the sense of the distances \(R_{\mathrm{AB}}\) and \(R_{\mathrm{BB}}\) lies along the dotted line with arrows: this is how these distances should change as A approaches \(\mathrm{B}_2\) along the straight line joining the \(\mathrm{B}_2\) nuclei. (For a sufficiently slow approach they can in general change only as indicated by the dotted line.)

The activation energy corresponds on the surface to the pass between two directions, between two valleys, marked by the intersection of the line \(MN\) with the dotted line. The magnitude of this energy in simple cases can be determined directly by calculation.

Analogously to what has just been said, one may describe a chemical reaction also in the case when more than three atoms take part in the reaction. Here, however, the visual clarity inherent in the exchange reaction (1) is lost. Indeed, the potential-energy surfaces must then be constructed in a space of as many dimensions as are necessary to describe the relative positions of all the nuclei of the system (the number of dimensions of the surface) plus one, since it is still necessary to construct the axis of mutual energy of all the atoms. It is clear that there can be no question here of any particular visual clarity; however, operating with these surfaces is often very useful.

Let us dwell further on the question of why the activation energy turns out to be substantially different depending on whether atom A approaches molecule \(\mathrm{B}_2\) along the straight line joining the nuclei, or at an angle to this straight line (the activation energy has its maximum value when A approaches \(\mathrm{B}_2\) along the perpendicular to the midpoint of the distance between the B nuclei). In § 1 it was said how rapidly the force acting between atoms changes with distance. Atom A, approaching the \(\mathrm{B}_2\) system, interacts with each of the B atoms. The interaction with the distant B atom is very much weakened when

approach along the straight line connecting the nuclei; atom B practically does not interact with A. The interaction will be considerably greater when the approach is in the perpendicular direction. The consideration given is very simple, and it may be used, although it is not entirely exact. Complete clarity here is provided only by a theoretical analysis of the problem.

Depending on the conditions of the interaction, the activation barrier of reaction (1) may thus differ severalfold in different directions. This should explain, for example, the steric factor introduced by chemists.

In conclusion to this paragraph I shall recall that the curves or surfaces of the potential energy of a system, of which we have spoken up to now, understanding by them the electronic states of atoms separated to infinity, states of the lowest energy, are not the only possible ones. It is often necessary to speak of surfaces constructed for the interaction of ions or for the interaction of excited atoms. In this case there is no fundamental difference in the reasoning. Only in the case of ionic interaction, even at comparatively large distances between the centers (atomic nuclei), can the mutual energy of the system change strongly with a change in distance, since it is inversely proportional to the distance (i.e., it decreases with increasing distance much more slowly than the exponential homopolar interaction).

Behavior of Activated Molecules

It is natural now to pose the question of how a molecule or a group of molecules possessing an energy above the activation barrier should behave. In doing so I shall leave entirely aside those conclusions which follow from classical conceptions and which reduce, for example, to the necessity of triple collisions in the reaction of combination of two atoms in order to remove energy, etc. These considerations are quite obvious. We shall be interested here only in the probability of the elementary process and in what new things quantum mechanics can say in this direction.

If the reaction always proceeded adiabatically, as was set forth in the preceding paragraph, then the reaction rates would, understandably, be determined exclusively by classical considerations such as those just given concerning triple collisions. In reality, however, the adiabatic course of a reaction is by no means the only possible one. Often another course is considerably more probable, one in which, in some configuration of the nuclei, the electronic state of the system must change instantaneously, i.e., we are dealing with a purely quantum phenomenon.

It is easy to say in which cases this quantum transition can and must occur. It can occur only at those points on the potential-energy surface of the system where, during the electronic transition, the kinetic energy of the heavy nuclei does not change by any appreciable amount. This kinetic energy, of course, cannot change discontinuously. Such a condition is satisfied, as is not difficult to see, by all places where two potential-energy surfaces, constructed for two of the electronic states under consideration,* approach each other very closely or intersect one another.

As we shall see somewhat below, deviations from adiabaticity may have a very substantial effect on the probability of a reaction, sometimes changing this probability by tens of thousands of times. In monomolecular decomposition or rearrangement, a deviation from adiabaticity—the “diabatic” course of the elementary process—must be associated with a decrease in its probability. Thus, for example, the decomposition reaction of the simple molecule \(N_2O\) is certainly diabatic, which, incidentally, could easily have been foreseen.** For reactions of high order, on the contrary, the adiabatic course must sometimes be especially taken into account, since it is only this that leads to the reaction; in the case of a course

* On each of these surfaces the energy of the system is uniquely determined by the configuration.

** The activated molecule must live long enough that the rate constant of the reaction does not depend on the number of collisions (the condition of monomolecularity). In the decomposition of \(N_2O\) the spin of the system changes; the system acquires paramagnetic properties (see below).

of an adiabatic atom, during a collision, would approach adiabatically and separate adiabatically, without changing the state of the system. This includes, for example, the probability of transfer of electronic excitation energy by atoms upon their collision. It is quite easy to understand in what cases an adiabatic course increases the probability of a reaction and in what cases it does not lead to a reaction at all. In Fig. 2 there are presented, quite schematically, examples of these cases.

Case I: the adiabatic course corresponds to a high probability of reaction and a larger rate constant

a

b

Case II: adiabatic, with preservation of the function of the curve of potential energy, the reaction does not proceed at all (the system returns to the initial state).

a) At point A an electronic jump occurs; the reaction proceeds with a probability significantly less than unity per one vibration of the activated molecule

b) The reaction proceeds adiabatically with probability equal to unity per one vibration.

Fig. 2.

In more complicated conditions, when the potential-energy curves are replaced by surfaces in the space of configurations and mutual energy, the essence of the matter remains the same.

The curves given in Fig. 2 deserve careful consideration. First of all, it is quite obvious that in many cases, when an adiabatic course is possible, a nonadiabatic one at the same activation energy would have to give a considerably smaller rate constant (always a smaller transition probability). Indeed, in the case of an “diabatic” course the system may pass many times along the potential-energy curve I—I of Fig. 2a, before

than by undergoing an electronic jump to curve II, and the reaction will occur. If one disregards the upper part of the curves or raises it sufficiently, arranging an energy gap as shown in Fig. 2b, the reaction would proceed adiabatically. We know in which cases the gap indicated in Fig. 2b is in fact formed of itself. This always happens when the potential-energy curves I and II do not belong to states of the same symmetry (the same multiplicity, the same parity, etc.). Therefore, knowing the character of the functions corresponding to the two states, one can also speak about the reaction-rate constant, having first determined how the reaction should proceed. It should be noted, however, that the presence or practical absence of a gap is determined not only by symmetry, but in some cases also directly by the magnitude of the interaction between the two states.

The considerations presented here provide exceptionally rich material for investigations in the kinetics of chemical reactions and for the systematization of reactions. Such work has not yet been begun, but it will be continued. Some details also still require theoretical development, although the main question is now quite clear.

Of the popular literature in Russian on chemical kinetics, I shall mention only the excellent and very detailed article by N. Semenov in Advances in Chemistry, first issue for 1932. References to the literature may also be found there.