Phenomena of Vibrational Energy Exchange in Molecular Collisions

V. Kondrat’ev, Leningrad

According to modern ideas about the nature of chemical activation, vibrational energy must constitute a significant part of the energy reserve of active molecules. Here it is sufficient to mention the works of London¹, Eyring and Polanyi², Frank and Rabinowitch³, Kassel⁴, and a number of others. Therefore, processes of vibrational-energy exchange are of exceptional interest from the standpoint of chemical kinetics. The study of these processes will undoubtedly shed light on a whole series of features of chemical reactions and, in particular, on the specific action of inert admixtures, which in one way or another affect the reaction rate.

Investigations of the elementary processes of vibrational-energy exchange were begun only two or three years ago. Nevertheless, thanks to the rather considerable number of published works, it is already possible to draw certain general conclusions that bring some clarity to the extremely complex mechanism of these processes, which is still far from being solved.

Let us first turn to the results of these works and to a brief description of the research methods employed. As regards purely theoretical works, here one must first of all point to the work of Zener⁵, who, applying the Born perturbation method to the process of collision of a nitrogen molecule possessing one vibrational quantum \((v = 1)\) with a nonvibrating molecule \(N_2\) \((v = 0)\), found that vibrational energy is converted into the kinetic energy of relative molecular motion only in one out of 25,000 collisions. Thus, in the given case, the probability of the process of conversion of the energy of a vibrational quantum into the energy of translational motion turns out to be equal to \(4 \cdot 10^{-5}\). This result is in good agreement with the experimental data (see below).

Among theoretical works it is also necessary to mention the work of Oldenberg⁶, who, considering the process of molecular collision from the classical point of view, comes to the conclusion that the probability of conversion of the energy of translational motion into vibra-

VIBRATIONAL ENERGY EXCHANGE PHENOMENA

the vibrational or rotational energy must be the smaller, the greater the difference in the masses of the colliding molecules. Oldenberg’s theory, however, is of a purely qualitative character.

Among the experimental works on the study of exchange processes between vibrational and translational energy, we must first of all consider the works of Kneser, Eucken, Richards and Reid, and others, based on the method of dispersion and absorption of sound. As follows from Einstein’s theory of sound propagation in polyatomic gases[^7], at sufficiently high sound frequencies, when the relaxation time becomes greater than the period of the acoustic oscillations, the state of the gas at the moment the sound passes deviates from equilibrium; the result of this is sound dispersion, i.e. a dependence of the propagation velocity of sound oscillations on their frequency, and also anomalous absorption of sound by the gas, differing from the classical one both in its magnitude, exceeding the latter by 10–100 times, and in a different dependence of the absorption coefficient on the sound frequency.

Here we are interested only in those gases whose thermal equilibrium is entirely determined by the distribution of energy among the various degrees of freedom of molecules that are unchanged in their composition. Since experiment[^8] and theory[^5] unambiguously show that the exchange of translational energy between molecules, as well as the conversion of rotational energy into translational energy (and conversely), takes place practically at every gas-kinetic collision of molecules,* the nonequilibrium states of the gases under consideration may be connected only with delays in the exchange of molecular vibrational energy, on the one hand, and rotational and translational energy, on the other. A simplified expression for the velocity of sound of frequency \(\omega\) in such a gas has the following form[^12]:

\[ V^2=\frac{p}{\rho}\left(1+R\frac{C+\omega^2\beta^2 C_\alpha}{C^2+\omega^2\beta^2 C_\alpha^2}\right), \tag{1} \]

where \(p\) and \(\rho\) are the pressure and density of the gas, \(R\) is the gas constant, \(C\) is the molecular heat capacity of the gas, \(C_\alpha\) is the part of the heat capacity due to the translational and rotational degrees of freedom, and \(\beta\) is a constant depending on the probability of transformation of vibrational energy into the energy of translational or rotational motion in a collision of molecules, and also on the probability of the reverse process.

Denoting these probabilities respectively by \(\gamma_1^0\) and \(\gamma_0^1\) (in the cal—

per one collision), the relation between them and the quantity \(\beta\) can be expressed by the following formula:

\[ \frac{1}{\beta}=\left(\gamma_0^1+\gamma_1^0\right)zN, \tag{2} \]

where \(N\) is the number of molecules per unit volume in the stationary gas, and

\[ z=\frac{4}{\sqrt{\pi}}\sigma\sqrt{\frac{RT}{M}} \]

(\(\sigma\) is the gas-kinetic cross section, \(T\) is the absolute temperature, and \(M\) is the molecular weight of the gas).

However, since the quantities \(\gamma_0^1\) and \(\gamma_1^0\) are in a known relation to one another, we can express the quantity \(\beta\) in another way as well, namely either through the quantity \(\gamma_0^2\) or through \(\gamma_1^0\). Indeed, denoting the total number of gas molecules per unit volume by \(N\), and the number of molecules possessing one vibrational quantum by \(n\)*, in the state of equilibrium we shall evidently have:

\[ \frac{dn}{dt}=\gamma_0^1 zN(N-n)-\gamma_1^0 zNn=0, \]

whence it follows that:

\[ \frac{\gamma_0^1}{\gamma_1^0}=\frac{n}{N-n}. \]

Since, further, the right-hand side of the last equality is equal to

\[ \frac{n}{N-n}=\frac{s_1}{s_0}e^{-\frac{\varepsilon}{RT}}, \]

where \(s_1\) and \(s_0\) are the statistical weights of the “excited” and normal molecule, respectively, and \(\varepsilon\) is the magnitude of the vibrational quantum, equal to \(\varepsilon=N_0h\nu_\kappa\) (\(\nu_\kappa\) is the vibrational frequency and \(N_0\) is Avogadro’s number), we obtain

\[ \frac{\gamma_0^1}{\gamma_1^0}=\frac{s_1}{s_0}e^{-\frac{\varepsilon}{RT}}. \tag{3} \]

On the basis of (3), expression (2) can be rewritten in the following form:

\[ \frac{1}{\beta}=\gamma_1^0\left(1+\frac{s_1}{s_0}e^{-\frac{\varepsilon}{RT}}\right)zN =\gamma_0^1\left(1+\frac{s_0}{s_1}e^{\frac{\varepsilon}{RT}}\right)zN. \tag{4} \]

Since usually \(s_1=s_0\) and \(\varepsilon\gg RT\), we may approximately put:

\[ \frac{1}{\beta}=\gamma_1^0 zN. \]

Thus the quantity \(\frac{1}{\beta}\) turns out to be equal to the number of collisions experienced by an “excited” molecule in 1 sec. and leading to the conversion of the energy of the vibrational quantum into other forms of energy. Consequently, the quantity \(\beta\) can be determined

* We shall agree henceforth to call these molecules “excited.”

as the mean lifetime of an oscillatory quantum.

\(\beta\) is a quantity found directly from experiment. According to expression (1), the dispersion curve has two asymptotes at \(\omega=0\) and at \(\omega=\infty\), determined by the following values of the velocity of sound:

\[ V_0^2=\frac{p}{\rho}\left(1+\frac{R}{C}\right)\quad \text{and}\quad V_\infty^2=\frac{p}{\rho}\left(1+\frac{R}{C_\alpha}\right). \]

The part of the curve lying between these asymptotes has an S-shaped form. It is not difficult to see further that the quantity \(V^2\) is equal to the arithmetic mean of the limiting values \(V_0^2\) and \(V_\infty^2\), i.e.,

\[ V^2=\frac{1}{2}\left(V_0^2+V_\infty^2\right) \]

at a frequency value equal to

\[ \bar{\omega}=\frac{1}{\beta}\frac{C}{C_\alpha}. \tag{5} \]

This latter relation is what is usually used to determine the quantity \(\beta\).

This quantity may also be determined from experiments on the absorption of sound. Theory, in complete agreement with experiment, shows that the coefficient of sound absorption \(\mu\), entering into the expression

\[ J_x=J_0 e^{-\mu \frac{x}{\lambda}}, \]

has a maximum at the frequency value

\[ \omega_{\max}=\frac{V_0}{V_\infty}\bar{\omega}, \tag{6} \]

where the maximum value of \(\mu\) is equal to

\[ \mu_{\max}=\pi\left(\frac{V_\infty}{V_0}-\frac{V_0}{V_\infty}\right). \]

Thus expression (6), like (5), may serve for calculating the quantity \(\beta\), and consequently also the quantities of interest to us, \(\gamma_0^1\) and \(\gamma_1^0\).

All the data available in the literature, obtained by means of the methods of dispersion and absorption of sound in homogeneous gases, are given in Table 1, where, alongside the values of the probabilities \(\gamma_1^0\) and \(\gamma_0^1\) calculated by us, there are also indicated the quantities \(\beta\) (for 760 mm Hg), the experimental temperatures (see below), the frequencies of the natural oscillations of the corresponding molecules, which chiefly determine the part of the heat capacity \(C-C_\alpha\), and the method of investigation.

We supplement this table with the data of Pool[^17], relating to nitrogen and obtained by another method. Pool’s method, which is of considerable interest as a new method, not yet extensively used, for studying processes of energy transfer, is as follows. According to the investigations of Gaviola[^18], excited atoms

mercury (\(^3P_1\)), practically at every collision with nitrogen molecules pass into the metastable state (\(^3P_0\)), which underlies the process of quenching of mercury fluorescence by nitrogen. In this case

TABLE 1

Values of the probabilities of “excitation” (\(\gamma_0^1\)) and loss (\(\gamma_1^0\)) of a vibrational quantum upon collision of molecules of one and the same gas

as a result of collisions with excited atoms, vibrating \(\mathrm{N_2}\) molecules (\(v=1\)) arise; thus here the energy of electronic excitation of the Hg atom (\(^3P_1 — ^3P_0\)) passes into vibrational energy of the \(\mathrm{N_2}\) molecule \(^{19}\). If, further, the metastable Hg atoms (\(^3P_0\)) were destroyed only as a result of ordinary impacts of second kind with nitrogen molecules, then, as is not difficult to show, their concentration would have to decrease with time according to the law*

\[ N = N_0 \exp(-\alpha t), \tag{7} \]

where \(\alpha\) is a quantity depending on the number of quenching collisions experienced by the metastable atoms and on their diffusion velocity. The concentration of metastable atoms was determined by absorption of the \(4047\ \text{Å}\) line (\(^3S_1 — ^3P_0\)), and it turned out that the decrease of concentration with time deviates from the indicated

* For convenience of notation, in what follows we shall use the conventional notation \(\exp x = e^x\).

of the law, while satisfying well the following, more complicated regularity:

\[ N=N_0 \exp\{-\alpha t+A[\exp(-\beta t)-1]\}, \tag{7'} \]

obtained under the assumption that, along with the destruction of metastable atoms as a result of collisions of the second kind \((\mathrm{Hg}' + \mathrm{N}_2 \to \mathrm{Hg} + \mathrm{N}_2)\), there occurs their reverse transfer into the state \({}^3P_1\), taking place when these atoms collide with vibrating (“excited”) nitrogen molecules:

\[ \mathrm{Hg}'({}^3P_0)+\mathrm{N}_2^{*}(v=1)\to \mathrm{Hg}'({}^3P_2)+\mathrm{N}_2(v=0). \]

In expression (7′), the quantities \(A\) and \(\beta\) depend on the efficiency of collisions of metastable mercury atoms with excited nitrogen molecules, on the diffusion rate of the latter, and also on the probability of the process

\[ \mathrm{N}_2^{*}(v=1)+\mathrm{N}_2\to \mathrm{N}_2(v=0)+\mathrm{N}_2(v=0), \]

leading to the conversion of the molecule’s vibrational energy into the energy of translational or rotational motion, i.e., to the destruction of “excited” nitrogen molecules. The latter quantity, which is of chief interest to us, can be determined from the experimentally observed dependence of the constants \(\alpha\) and \(\beta\) on the nitrogen pressure. According to Pool’s measurements it is found to be

\[ \gamma_1^0=8\cdot 10^{-5*}. \]

The values of the probabilities \(\gamma_1^0\) (more precisely, the common logarithms of these quantities) for the eight gases studied are compared in Fig. 1, where the magnitudes of the corresponding vibrational quanta (in \(\mathrm{cm}^{-1}\)) are also indicated. From these data we see first of all that the probability of converting the vibrational quantum of a molecule into the energy of translational or rotational motion upon collision of molecules is, on average, of the order of \(10^{-3}\)—\(10^{-4}\), which is in excellent agreement with the result of Zener’s theoretical calculation\(^5\), who obtained, for the case of nitrogen, \(\gamma_1^0=0.4\cdot 10^{-4}\). Further, Fig. 1 shows that the probability \(\gamma_1^0\), apparently, is not in any definite relation to the magnitude of the vibrational quantum. To be convinced of this, it is sufficient to compare the values of \(\gamma_1^0\) for similar molecules, for example \(\mathrm{N}_2\), \(\mathrm{O}_2\), and \(\mathrm{Cl}_2\), or \(\mathrm{CO}_2\), \(\mathrm{N}_2\mathrm{O}\), and \(\mathrm{CS}_2\) (the last three molecules are linear and practically have no dipole moment). The molecules \(\mathrm{SO}_2\) and \(\mathrm{C}_2\mathrm{H}_4\) possess the maximum values of the probability \(\gamma_1^0\). In the first case the reason for this must undoubtedly be sought in the considerable dipole moment \((1.61\cdot 10^{-18}\) abs. units) of the molecule (see below), whereas the large

* Calculating from this value, and from the magnitude of the vibrational quantum of the nitrogen molecule, by formula (3), the value \(\gamma_{10}^{1}\) (for room temperature), we find \(\gamma_{10}^{1}=7.58\cdot 10^{-10}\).

the probability of energy exchange in ethane is apparently connected with the polyatomic character of the molecule \(C_2H_4\).

Before turning to the experimental data relating to gas mixtures, let us dwell on the question of the probability of the simultaneous transfer of a large number of vibrational quanta. Unfortunately, this question encounters great difficulties in the very method of dispersion and absorption of sound, since, as a rule, at ordinary temperatures the “excitation” of two or more vibrational levels is associated with such a negligible part of the heat capacity of the gas that its influence on the dispersion and absorption of sound escapes observation. However, in some cases, when the vibrational quantum is not very large, on the basis of the study of dispersion curves one may still hope to draw certain conclusions

Fig. 1. Probability of loss of a vibrational quantum \((\gamma_{12}^{0})\) by various molecules upon collision with molecules of the same kind.

regarding the probability of the transfer of at least two quanta. If the probability of transferring two quanta differs significantly from the probability of transferring one quantum, then in all these cases two dispersion regions should be observed, and the dispersion curves should have a more complicated form than that described above. The investigation of this “fine structure” of the dispersion curve, however, requires very precise measurements and extreme purity of the gas under study, which hardly any of the published works can boast. Therefore, for the solution of the question of the transfer of two or more vibrational quanta in a single collision we shall turn to other methods, which are more convenient for this purpose (see below). Here we shall only also note that, in the opinion of some authors\(^{16,15}\), the probabilities \(\gamma_1^{0}\) and \(\gamma_2^{1}\), i.e. the probabilities of the processes

\[ M^*(v=1)+M \to M(v=0)+M \quad \text{and} \quad M^*(v=2)+M \to M^*(v=1)+M \]

must have the same order of magnitude, since otherwise

case, for a number of gases the region of dispersion should have been broader than is observed in reality.

To clarify the mechanism of the processes of energy exchange in molecular collisions, experiments on the investigation of the dispersion and absorption of sound by gas mixtures are of enormous importance. Among the few works devoted to investigations of this kind, we shall mention the work of Eucken and Becker \(^{14}\), Richards and Reid \(^{10}\), and also the works of Knudsen and Knezer \(^{20}\). The first of these works, the richest in factual material, is devoted to the study of the dispersion of sound in mixtures of chlorine with Ar, He, \(H_2\), HCl, and \(CH_4\), and in mixtures of carbon dioxide with Ar, Ne, He, \(CH_4\), \(H_2\), HCl, and \(H_2O\). The results of this work are so interesting that we shall present them in full (Tables 2 and 3).

TABLE 2

Dispersion of sound in mixtures of chlorine with various gases

In these tables, along with the indication of the composition of the mixture, are given the values of the quantity \(\beta\) observed by Eucken and Becker and, calculated from these values as mean values, the probabilities \(\overline{\gamma}_1^{\,0}\), as well as the probabilities of conversion of the vibrational energy of the \(Cl_2\) and \(CO_2\) molecule into translational and rotational energy as a result of collisions of these molecules with molecules of the foreign gas (last column). The figures in the last column are of special interest to us, since they clearly show what an enormous role the nature and physicochemical properties of the colliding molecules play in the processes of energy exchange.

Comparison of the figures given in Tables 2 and 3 makes it possible to distinguish various factors influencing the probability of energy exchange. First of all, comparing the action of the noble gases He, Ne, and Ar (Table 3), we see that collisions of “excited” \(CO_2\) molecules with Ne and Ar atoms are less effective than collisions of these molecules with \(CO_2\) molecules, whereas collisions

molecules of CO$_2^*$ with He atoms are extremely favorable for energy exchange: the effectiveness of these collisions is about 18 times greater than the effectiveness of CO$_2^* +$ CO$_2$ collisions. We have an analogous case

TABLE 3

Dispersion of sound in mixtures of carbon dioxide with various gases

in mixtures of chlorine with Ar and chlorine with He (Table 2), where the effectiveness of Cl$_2^* +$ He collisions is about 280 times greater than the effectiveness of Cl$_2^* +$ Cl$_2$ collisions! As we see, these data are in obvious contradiction with the mechanical considerations of Oldenberg$^6$, since according to the latter the greatest effectiveness should be possessed by impacts of identical particles, while impacts of He atoms, owing to their small mass, should be the least effective.* Thus we see that the mass of the colliding molecules is far from always the decisive factor, as can also be verified by comparing, for example, the effectiveness of collisions of “excited” CO$_2$ molecules with Ne and CH$_4$, which have identical masses: the effectiveness of CO$_2^* +$ CH$_4$ collisions is more than 10 times greater than the effectiveness of CO$_2^* +$ Ne collisions (cf. also below).

Among other physical factors increasing the probability of energy exchange, the presence of a dipole moment is undoubtedly of great importance. An excellent illustration of this is provided by the

* Eucken and Becker relate the great effectiveness of helium, observed in their experiments, to the small dimensions of He atoms, without, however, substantiating this explanation.

usually the high efficiency of collisions of CO$_2^*$ molecules with polar molecules HCl and H$_2$O (Table 3). However, simple electrostatic interaction of the colliding particles is undoubtedly insufficient for energy exchange. The obviousness of this assertion becomes especially clear if one compares the efficiencies of collisions Cl$^*_2$ + CH$_4$ and Cl$^*_2$ + HCl, which prove to be identical despite the great difference in the polar properties of the molecules HCl and CH$_4$.

This fact alone is already sufficient for assigning to the forces of their chemical interaction the principal, predominant role in the process of energy exchange upon collision of molecules. It is precisely this point of view that is held by Franck and Eucken$^{21}$, according to whom the essentially physical process of energy exchange in a system of colliding particles must be regarded as the beginning of the chemical process of molecular transformation.

Fig. 2. a—motion of atoms in the collision of HCl and CO$_2$ molecules; b—ClCOOH molecule (after Eucken and Becker).

From this point of view, for example, the loss of vibrational energy by a CO$_2$ molecule upon its collision with an HCl molecule is interpreted as the beginning of the chemical process of formation of a ClCOOH molecule, which can be clearly illustrated by Fig. 2, borrowed by us from the work of Eucken and Becker$^{14}$.

The results of the remaining works on gas mixtures in the main agree with the results of the work of Eucken and Becker that we have examined in detail. Thus, Richards and Reid$^{10}$, studying the dispersion of sound in binary mixtures of ethylene with Ar, N$_2$, He, and H$_2$, arrive at the following conclusions: the collision efficiency of “excited” ethylene molecules with Ar, N$_2$, and He is less than one tenth of the collision efficiency C$_2$H$_4^*$ + C$_2$H$_4$, whereas the latter, in turn, proves to be approximately 10 times smaller than the collision efficiency C$_2$H$_4^*$ + H$_2$. Thus the latter is characterized by a value $\gamma_1$ of the order of $10^{-2}$ (Table 1). We see that collisions with H$_2$ molecules, as also in the experiments of Eucken and Becker, possess high efficiency, which, according to Franck and Eucken, must be due mainly to the presence of an undoubted chemical affinity between C$_2$H$_4$ and H$_2$ (C$_2$H$_4$ + H$_2 \to$ C$_2$H$_6$). Finally, the works of Knudsen

and Kneser’s^20 are devoted to the study of the absorption of sound by moist air and oxygen (Knudsen) and to the interpretation of the results obtained (Kneser). It is found that the absorption of sound in air is due exclusively to oxygen, while the anomalous increase of the absorption coefficient observed in the presence of water vapor is connected with the high efficiency of collisions of vibrating \(O_2\) molecules with \(H_2O\) molecules, in the sense of loss of vibrational energy by oxygen molecules. Thus here, too, water retains those properties which are manifested in the high efficiency of collisions \(CO_2^* + H_2O\) (Table 3). Further, as a result of a theoretical treatment of Knudsen’s data, which revealed a maximum of the absorption coefficient in moist oxygen at a certain definite concentration of water vapor (at constant temperature and sound frequency), Kneser arrives at the conclusion that the loss of vibrational energy by an oxygen molecule (in contrast to all the preceding cases) occurs as the result of a triple collision \(O_2^* + H_2O + H_2O\). This result, however, is so unexpected that it involuntarily raises doubts as to the correctness of Kneser’s theory.

Starting from the conception of the energy-transfer process as the initial stage of a chemical process, we must expect to find in energy-transfer processes certain features characteristic of chemical processes in general. Thus, it is quite appropriate to suppose that, like most chemical processes, the process of energy transfer requires a certain activation. In that case the observed small values of the probabilities of energy transfer could be connected with the presence of an activation energy, assuming

\[ \gamma = b e^{-\frac{a}{RT}}, \tag{8} \]

where \(a\) is the activation energy, and \(b\) is the steric factor.

Indeed, the assumption of the existence of an activation energy in the process of energy exchange is confirmed both by theoretical investigations of these processes^22 and by direct experiments. Here we may point to the experiments of Richards and Reid, who studied the dispersion of sound at various temperatures, given in the previously cited works^10,15. Thus, for example, in the case of ethylene the values of \(\beta\) at two different temperatures, according to the measurements of these authors, have the following values (at 760 mm):

\[ \beta' = 1.93 \cdot 10^{-7}\ \text{sec. at } 45^\circ C \]
\[ \beta = 2.73 \cdot 10^{-7}\ \text{sec. at } 15^\circ C. \]

On the basis of the relations

\[ \frac{1}{\beta} = \gamma_1^0 zN, \]

(see above) and (8), we have

\[ \frac{\beta}{\beta'} = \sqrt{\frac{T'}{T}}\, e^{\frac{a}{R}\frac{T' - T}{TT'}} . \]

Substituting here the above-given values of the quantities \(\beta, \beta', T\) and \(T'\), we find

\[ a = 1800 \ \text{cal.} \]

On the basis of the value of the activation energy \(a\) obtained by us, using formula (8), we can calculate the steric factor \(b\), which turns out to be equal to 0.02. This unusually small value of the steric factor may perhaps not seem so strange if one takes into account the considerable complexity of the ethylene molecule.

Analogous calculations give, in the case of \(\mathrm{CO}_2\) and \(\mathrm{CS}_2\), respectively \(a = 6000\) and \(5400\) cal. and \(b = 0.04\) and 1 (quoted according to Richardson and Reid). Thus we see that the activation energy in processes of energy exchange not only is of the order of magnitude of the energy transferred, but sometimes even exceeds the latter several times (\(\sim\) by a factor of 3), as occurs in the case of \(\mathrm{CO}_2\) and \(\mathrm{CS}_2\). Let us also note that, according to Kneser \(^{20}\), the activation energy of processes of energy exchange in collisions of \(\mathrm{O}_2^*\) molecules with \(\mathrm{H}_2\mathrm{O}\) molecules is equal to zero, which, however, is quite natural in view of the great effectiveness of impacts by water molecules.

Fig. 3.

For the description and visual representation of processes of transformation of the vibrational energy of molecules we may use the method of potential-energy curves, which is successfully applied to the description of chemical processes. In simplified form, curves of this kind are shown in Fig. 3, where the lower curve represents the interaction of two non-vibrating molecules, while the upper one represents the interaction of a normal and an “excited” molecule (in constructing these curves the van der Waals forces have not been taken into account). Since the repulsive forces between two non-vibrating molecules are greater than the repulsive forces between a normal and an “excited” molecule, the upper curve must rise upward less steeply, so that intersection of the two curves is not excluded. Further, since the probability of transition from the upper curve to the lower one, corresponding to the process \(M^* (v = 1) + M \to M + M\), must have an appreciable value only in the region of sufficient approach of the curves (the Franck–Condon principle), then, as follows from Fig. 3, the energy of the relative motion of the colliding molecules must exceed a certain minimum value \(a\), which obviously has the meaning of the activation energy. Constructing a system of curves analogous to the curves of Fig. 3 for different \(v\),

It is easy to convince oneself that the activation energy \(a\) must be the smaller, the larger \(v\) is. This conclusion, however, is not only a consequence of the applicability of the method of potential curves to processes of energy exchange. It may be regarded as an absolutely indisputable fact that molecules situated at high vibrational levels and, consequently, possessing bonds loosened to one degree or another, enter into closer interaction in comparison with non-vibrating molecules. The consequence of this closer interaction is the increased probability of energy exchange and the inseparably associated lowered value of the activation energy. Since the interaction of molecules, as a rule, is also strengthened by their electronic excitation, we must likewise expect high values of the probabilities of vibrational-energy exchange in collisions in which excited molecules participate (here electronic excitation is meant) \(^{21}\). This conclusion is in excellent agreement with experimental data.

The experimental data relevant here were obtained by the fluorescence method, which is based on the change in the spectrum of resonance fluorescence of iodine (visible region) in the presence of foreign gases, first discovered by Wood and Franck \(^{23}\). When excited by monochromatic light, for example by the green mercury line \(\lambda 5461\,\text{\AA}\), the fluorescence spectrum of iodine is a resonance series of bands \(v' = 26 \to v''\). In the presence of a foreign gas, along with the resonance series, other series appear in the iodine fluorescence spectrum, namely the series \(v' = 25 \to v''\), \(v' = 24 \to v''\), \(v' = 27 \to v''\), and \(v' = 28 \to v''\) \(^{24}\). This effect is especially distinct in the case of helium, which quenches iodine fluorescence comparatively weakly. The newly appearing series increase in intensity with increasing pressure of the foreign gas at the expense of a decrease in the intensity of the resonance series. The cause of these changes in the fluorescence spectrum of iodine is collisions of excited \(J_2\) molecules with molecules of the foreign gas, as a result of which the \(J_2'\) molecule from the initially excited vibrational level \((v' = 26)\) passes to one of the neighboring levels \((v' = 25, 24\) or \(27, 28)\); moreover, in the transition to lower levels \((v' = 25, 24)\) the vibrational energy of the \(J_2'\) molecule is converted into energy of translational or rotational motion, whereas in the transition to higher levels \((v' = 27, 28)\) the energy of relative motion of the colliding molecules is converted into vibrational energy of \(J_2'\). Comparison of the relative intensities of the various series in the iodine fluorescence spectrum at a known pressure of the foreign gas makes it possible to determine the probabilities of transfer of vibrational energy in the collision of excited \(J_2\) molecules with molecules of the foreign gas by calculating the number of effective collisions, i.e., the number of collisions leading to energy exchange.

The efficiency of collisions of helium atoms with excited iodine molecules, in the sense of the transfer of vibrational quanta, may be roughly estimated by comparing the intensities of various series in the fluorescence spectrum of iodine at different helium pressures, from the spectrograms given in one of Wood’s papers.²⁵ It follows from these spectrograms that, whereas at \(2\ \mathrm{mm}\) He the new series appearing in the fluorescence spectrum of iodine alongside the resonance series are appreciably weaker than the latter, at \(10\ \mathrm{mm}\) He all the series already have comparable intensity. Assuming that the number of iodine molecules emitting per second and situated on one of the new vibrational levels corresponding to \(v' = 24, 25, 27\), or 28—for example the molecules \(\mathrm{J}'_2\),²⁵—is equal to the number of molecules \(\mathrm{J}'_2(26)\) disappearing in the same interval of time as a result of the process

\[ \mathrm{J}'_2(v' = 26) + \mathrm{He}\quad \mathrm{J}'_2(v' = 25) + \mathrm{He}, \]

we may evidently write

\[ \frac{1}{\tau}[\mathrm{J}'_2(25)] = \gamma zN[\mathrm{J}'_2(26)], \]

where \(\tau\) is the mean lifetime of an excited iodine molecule, \(\gamma\) is the probability of conversion of one vibrational quantum of the molecule \(\mathrm{J}'_2\) into the energy of translational motion upon its collision with a He atom (in the present case \(\gamma = \gamma^{25,26}\)), and \(zN\) is the number of collisions with He atoms experienced by the molecule \(\mathrm{J}'_2(26)\) in 1 sec. (\(N\) is the number of He atoms in \(1\ \mathrm{cm}^3\)). At \(10\ \mathrm{mm}\) He, owing to the comparability of the intensities of the series \(v' = 26 \to v''\) (the resonance series) and \(v' = 25 \to v''\), the concentrations of the molecules \(\mathrm{J}'_2(26)\) and \(\mathrm{J}'_2(25)\) must evidently be equal, and we obtain:

\[ \gamma = \frac{1}{\tau zN}, \]

where \(N\) corresponds to a helium pressure of \(10\ \mathrm{mm}\).

At this pressure the quantity \(zN = 2 \cdot 10^8\), and, consequently,

\[ \gamma = \frac{1}{\tau}\, 5 \cdot 10^{-9}. \]

Thus, in order to find the value of \(\gamma\), it is necessary to know \(\tau\). This latter quantity was determined by Gunther²⁶ and proved to be equal to \(1 \cdot 10^{-8}\) sec. Substituting it into the expression for \(\gamma\), we find

\[ \gamma = 0.5. \]

The value we have obtained for the probability \(\gamma\) shows that the efficiency of collisions of excited iodine molecules with helium atoms leading to energy transfer is very high, which is in excellent agreement with the supposition expressed above concerning an increase in the efficiency of collisions upon excitation of the molecule. It turns out to be still higher if helium is replaced by hydrogen or nitrogen, as follows from the experiments of El’yashevich,²⁷ who studied in detail the influence of these gases on the fluorescence spectrum of iodine. An exact determination of the probabilities \(\gamma\) is complicated in the case of iodine

because the intensity of the bands in each series in the fluorescence spectrum of iodine changes from band to band in a most intricate manner. This difficulty, however, can be overcome by measuring the total intensity of each series at definite pressures of the foreign gas (\(\mathrm{H}_2\) or \(\mathrm{N}_2\)). El’yashevich showed that the probabilities \(\gamma^{25}_{26}\) and \(\gamma^{27}_{26}\), i.e. the probabilities of the processes \(J_2'(26)+M\to J_2'(25)+M\) and \(J_2'(26)+M\to J_2'(27)+M\) (\(M=\mathrm{H}_2,\mathrm{N}_2\)), are practically identical in the case of one and the same gas, as are also the probabilities \(\gamma^{24}_{26}\) and \(\gamma^{28}_{26}\), which is quite natural, since the magnitude of the vibrational quantum transformed in this case amounts to only 255 cal, i.e. a value more than two times smaller than the value \(RT=575\) cal at the temperature of the experiment (room temperature). However, the values of \(\gamma\) for hydrogen and nitrogen differ noticeably from one another, as is seen from the following data (El’yashevich):

\[ \gamma^{25}_{26}\sim \gamma^{27}_{26}= \begin{cases} \mathrm{H}_2: 25,\\ \mathrm{N}_2: 130, \end{cases} \qquad \gamma^{24}_{26}\sim \gamma^{28}_{26}= \begin{cases} \mathrm{H}_2: 18\\ \mathrm{N}_2: 35 \end{cases} \]

These data show that the exchange of energy in collisions of an excited iodine molecule with hydrogen and nitrogen molecules is an extremely probable process, and the values of the probabilities, considerably exceeding 1, indicate that this process occurs with effective cross sections significantly (by tens of times) exceeding the ordinary gas-kinetic cross section. The difference in the probabilities \(\gamma\) for the cases \(\mathrm{H}_2\) and \(\mathrm{N}_2\) is explained by El’yashevich in the spirit of Oldenberg’s mechanical theory\({}^{6}\), according to which the transfer of vibrational and rotational energy into the energy of translational motion of molecules, as well as the reverse process, should occur the more readily the smaller the difference in the masses of the colliding particles.

From the point of view of Oldenberg’s theory, the efficiencies of collisions with the molecules \(J_2'\), He, \(\mathrm{H}_2\), and \(\mathrm{N}_2\) should be arranged in the following order:

\[ \mathrm{H}_2<\mathrm{He}<\mathrm{N}_2. \]

In reality, however, a different relation of the efficiencies of these gases is observed, namely

\[ \mathrm{He}<\mathrm{H}_2<\mathrm{N}_2, \]

as follows from the figures given above. Thus here we again encounter facts that contradict the simple mechanical theory. The interchange of \(\mathrm{H}_2\) and He, apparently, must be interpreted in the spirit of the theory of Franck and Eucken\({}^{21}\) as due to the increased interaction, of chemical order, of the molecules \(\mathrm{H}_2\) and \(J_2'\) in comparison with the interaction of He and \(J_2'\).

By means of the fluorescence method, furthermore, Geil\({}^{28}\) studied the transfer of vibrational energy in collisions of excited molecules \(\mathrm{S}_2\), \(\mathrm{Se}_2\), and \(\mathrm{Te}_2\) with molecules He, Ar, and \(\mathrm{N}_2\). However

only data relating to helium have been published, and these, moreover, are of a purely qualitative character. Experiments with sulfur show that, in agreement with earlier data obtained by Rompe \(^{22}\), the resonance series in the fluorescence spectrum of \(S_2\) with an admixture of helium are supplemented by new series, the number of which increases with increasing pressure of the latter. In each effective collision of an excited sulfur molecule with a helium atom, one vibrational quantum is transferred; the appearance of a large number of series in the fluorescence spectrum at considerable helium pressures is explained by successive transitions of the molecule \(S_2'\) to various vibrational levels as a result of its repeated collisions with He atoms, this being favored by the almost complete absence of quenching of sulfur fluorescence by this gas. Geil notes a great similarity in the behavior of the fluorescence spectra of sulfur and iodine in the presence of helium. The only difference between the two substances is that the same effects (in the sense of complication of the fluorescence spectrum) are observed, in the case of sulfur, at helium pressures approximately 10 times greater than in the case of iodine. According to Geil, this difference is due to the different values of the mean lifetime of the excited molecule \(S_2\) and \(J_2(\tau)\), Geil assuming that for sulfur \(\tau\) has a normal value, close to \(10^{-8}\) sec., whereas for iodine it is \(10^{-7}\) sec. And since the total number of collisions experienced by an excited molecule with foreign molecules during its lifetime is proportional to the product \(\tau p\), where \(p\) is the pressure of the foreign gas, then, for equal efficiencies of impacts \((\gamma)\), the pressures \(p_1\) and \(p_2\) producing the same effect in two different gases must be related as the reciprocal values of \(\tau_1\) and \(\tau_2\), i.e.

\[ \frac{p_1}{p_2}=\frac{\tau_2}{\tau_1}. \]

Substituting here \(\tau_1 = 10^{-8}\) (sulfur) and \(\tau_2 = 10^{-7}\) (iodine), we obtain \(p_1\) (= the pressure of helium in the case of \(S_2\)) \(= 10\,p_2\) (= the pressure of helium in the case of \(J_2\)).

However, the value \(\tau_2 = 10^{-7}\) sec. adopted by Geil contradicts the data of a direct measurement of this quantity by Gudfeld \(^{26}\), who found \(\tau_2 = (1 \pm 0.1)\cdot 10^{-8}\) sec. Gudfeld’s measurements compel us to recognize Geil’s arguments concerning the efficiency of collisions of excited sulfur and iodine molecules with helium atoms as incorrect. Assigning equal values to the mean lifetimes of the molecules \(S'_2\) and \(J'_2\) (\(\tau_1 = \tau_2 = 10^{-8}\) sec.), we obtain from Geil’s data, for the probability of transfer of vibrational energy in collisions \(S'_2 + \mathrm{He}\), a value 10 times smaller than the corresponding value for \(J'_2 + \mathrm{He}\). Thus the probability \(\gamma\) for sulfur (+ helium) proves to be equal to:

\[ \gamma = 0.05. \]

As for Se₂ and Te₂, investigated by Geil along with S₂, then, owing to the strong quenching of the fluorescence of both substances (especially Se₂) by helium, the new series arising in their fluorescence spectrum in the presence of helium are very weak. Thus, even in the case of Te₂, where the fluorescence-quenching effect is much weaker than in Se₂, at a helium pressure of 100 mm the new series are still considerably weaker than the resonance series. This, however, does not mean that the transfer of vibrational energy in collisions of excited Se₂ and Te₂ molecules with He atoms is in itself less probable than in the case of S₂′ + He. In all likelihood, the probability values for Se₂ and Te₂ have the same order of magnitude as for S₂; however, the comparatively weak transfer of vibrational energy observed in the case of Se₂ and Te₂ is explained by the competing process of fluorescence quenching, which, according to Geil, is connected with the decomposition of excited Se₂ and Te₂ molecules as a result of their collisions with He atoms. In view of this circumstance, the apparent or effective values of the mean lifetime of the excited Se₂ and Te₂ molecules under conditions of strong quenching turn out to be considerably smaller than their true values, which also accounts for the apparently small probability of energy transfer. An analogous effect was discovered by Oldenberg ³⁰ in the case of the ultraviolet fluorescence of iodine, where, because of very strong fluorescence quenching, the effect of vibrational-energy transfer proves to be practically unobservable.

As a result of strong quenching in the presence of helium, in the fluorescence spectra of Se₂ and Te₂, alongside the resonance series, only two new series are found, corresponding to the transitions \(v' - 1 \to v''\) and \(v' + 1 \to v''\), associated with the processes:

\[ M'(v') + \mathrm{He} \to M'(v' \pm 1) + \mathrm{He}; \quad M = \mathrm{Se}_2, \mathrm{Te}_2. \]

Thus the vibrational energy of these molecules, in their collisions with He atoms, changes practically by only one quantum. Let us note that the transition of an excited molecule to a higher vibrational level \((v' \to v' + 1)\), associated with the conversion of the kinetic energy of the relative motion of the particles \(M'\) and He into vibrational energy of the molecule \(M'\), under Geil’s experimental conditions (gas temperature 600–800°C), should have a probability comparable with that of the reverse transition, since at the temperature of these experiments the quantity \(RT = 1200\)—\(1600\) cal exceeds the magnitude of the transferred quanta (the vibrational quanta of the molecules S₂, Se₂, and Te₂ are respectively: \(< 1200\), \(< 700\), and \(< 460\) cal).

The study of processes of vibrational-energy transfer in collisions of excited molecules with other molecules is also partially the subject of the works of Kistiakowsky and Nelles ³¹, Gradshtein ³², and Lotmar ³³. The first two authors studied the fluorescence of benzene vapor, excited by the mercury line \(\lambda 2537\) Å, in the pressure range 0.01–25 mm. Investigations of the fluorescence spectrum of benzene at various vapor pressures of the latter show that pure

resonance spectrum, observed at low pressures (0.01–0.1 mm), with an increase of pressure to 1 mm gradually changes, becoming more complex. With a further increase of pressure (from 1 to 25 mm), however, the spectrum practically does not change. In the authors’ opinion, the fluorescence spectrum of benzene in this pressure interval (in general at pressures above 1 mm) is due to transitions of the excited benzene molecule from zero vibrational levels \((v'_i = 0 \to v''_k)\), on which it finds itself as a result of collisions with other benzene molecules during its lifetime as an excited molecule, to the normal state. In these collisions the primarily excited molecule, located on certain vibrational levels, loses its entire reserve of vibrational energy, for which even the number of collisions that it experiences during the time \(\tau\) at a pressure of 1 mm proves quite sufficient. These considerations make it possible to estimate roughly the probability \(\gamma\) of transfer of vibrational energy by an excited benzene molecule to the normal molecules colliding with it.*

Assuming that at a pressure of 1 mm the time interval between two effective collisions is equal to the value \(\tau\) (cf. formula 9), we have

\[ \tau = \frac{1}{\gamma z N}, \]

where \(N\) is the number of benzene molecules in \(1\ \mathrm{cm^3}\) at a pressure of 1 mm. Calculating the quantity \(z\) from the gas-kinetic diameter of the benzene molecule and assuming \(\tau = 10^{-8}\) sec., we find from the preceding formula

\[ \gamma = 14.5. \]

An analogous effect was discovered by Gradshtein, who studied the fluorescence spectrum of vaporous formaldehyde at pressures of 50–100 mm. This author found that the fluorescence spectrum of formaldehyde at these pressures does not depend on the wavelength of the exciting line, which again indicates that the observed spectrum is connected with the transition to the normal state of non-oscillating excited \(\mathrm{H_2CO}\) molecules. In the author’s opinion, these molecules arise as a result of the surrender of vibrational energy (the magnitude of the vibrational quantum is 3,380 cal) by the primarily excited molecules to the molecules colliding with them (during the time \(\tau\)). Since, however, the investigations of the fluorescence spectrum in this case were carried out only at comparatively high pressures, we have no data for calculating the probability \(\gamma\) of transfer of vibrational energy. It may, however, be asserted with confidence that this quantity here is at least of the order of 1.

* The question of whether the energy of the vibrational quanta of the \(\mathrm{C_6H_6'}\) molecule is transformed into energy of translational and rotational motion or into vibrational energy of the \(\mathrm{C_6H_6}\) molecules themselves must here be regarded as open (see below).

Finally, the last of this series of papers—the work of Lotmar—is devoted to the study of the fluorescence spectrum of \(\mathrm{SO_2}\). Like the preceding authors who studied the fluorescence of complex molecules, this author found that, in the pressure range \(0.5\)–\(20\) mm, the fluorescence spectrum does not depend on pressure. In this case, however, the relations are more complicated, owing to the strong quenching of the fluorescence of \(\mathrm{SO_2}\) both by \(\mathrm{SO_2}\) itself and by foreign gases \((\mathrm{Ar}, \mathrm{H_2}\), and \(\mathrm{CO_2})\). In this respect \(\mathrm{SO_2}\) resembles tellurium. According to Lotmar, the independence of the fluorescence spectrum both from the pressure of \(\mathrm{SO_2}\) itself and from foreign admixtures (270 mm He, 100 mm Ar, 100 mm \(\mathrm{H_2}\), 90 mm \(\mathrm{CO_2}\), and 110 mm \(\mathrm{O_2}\)) must here be attributed both to the strong quenching action of collisions (cf. the cases of \(\mathrm{Te_2}\) and \(\mathrm{S_2}\)) and, in part, to the high probability of transitions of excited molecules into a non-oscillating state as a result of their collisions with normal molecules of \(\mathrm{SO_2}\) itself. In favor of the latter supposition is the fact that some exciting lines in the fluorescence spectrum are extremely weak, as well as the presence in the fluorescence spectrum of lines separated from the exciting line by distances equal to one of the vibrational quanta \((\nu_b = 1080\ \mathrm{cal})\) of the excited \(\mathrm{SO_2}\) molecule. The author believes that the quenching action of foreign gases is stronger than the energy-transfer effect, and only in the case of \(\mathrm{SO_2}\) itself are these two effects in the opposite relation. If the explanation given by Lotmar for the facts he observed is correct, then here too the probability of transfer of vibrational energy by excited molecules proves to be very large. Thus, taking \(\tau = 10^{-8}\) sec. (Lotmar assumes \(\tau = 10^{-7}\) sec.), from formula (9) for the probability \(\gamma\) in the case of collisions \(\mathrm{SO'_2 + SO_2}\), at a pressure equal to 0.5 mm, we obtain:

\[ \gamma = 43 \quad \text{(according to Lotmar, 4.3).} \]

The probabilities \(\gamma\) for the other gases investigated apparently have smaller values.

The data presented above, relating to the exchange of vibrational energy in collisions of excited molecules with other molecules, may be supplemented by citing still unpublished results of Yakovleva’s work on this question, carried out at the Institute of Chemical Physics. As Yakovleva showed\(^{34}\), as a result of the photodissociation of cyanogen iodide molecules under illumination by an aluminum spark, excited CN molecules arise, situated on the first and zero vibrational levels \((v' = 1,0)\). It turns out that, when a foreign gas \((\mathrm{Ar}, \mathrm{N_2}, \mathrm{CO}, \mathrm{H_2})\) is admixed, the ratio of the intensities of the CN bands \(v' = 1 \to v'' = 1\) and \(v' = 0 \to v'' = 0\), observed in the fluorescence spectrum, shifts toward a relative strengthening of the latter as the pressure of the foreign gas increases. This effect is undoubtedly due to the conversion of a vibrational quantum of CN \((v' = 1)\) molecules into the energy of translational motion as a result-

Phenomena of Vibrational-Energy Exchange

the collisions of this molecule with molecules of a foreign gas. A preliminary calculation gives the following values for the probability of this process:

\[ \mathrm{Ar}:\gamma = 0.002, \]

\[ \mathrm{N}_{2}:\gamma = 0.02. \]

Finally, let us also mention the work of Richardson \(^{35}\), who found that the intensity distribution in the Fulcher bands of \(\mathrm{H}_{2}\) changes with the addition of helium in such a way that the bands associated with transitions from high vibrational levels (large \(v'\)) become weaker, while at the same time in the band \(v' = 0 \to v'' = 0\) new rotational lines appear, corresponding to high rotational levels (large \(J\)) of the hydrogen molecule. According to Oldenberg \(^{36}\), this effect is due to collisions of excited \(\mathrm{H}_{2}\) molecules with He atoms, as a result of which the vibrational energy of the excited molecules situated at high vibrational levels is partly converted into the translational energy of the relative motion of the \(\mathrm{H}_{2}\) and He molecules, and partly into the rotational energy of the \(\mathrm{H}_{2}\) molecules themselves. The latter is the cause of the appearance of the new rotational lines observed by Richardson in the discharge spectrum in the mixture \(\mathrm{H}_{2} + \mathrm{He}\). Insofar as can be judged from Richardson’s data, the transfer of vibrational energy by excited molecules in this case also represents a very probable process. As Oldenberg emphasizes, the study of the intensity distribution in discharge spectra in various gas mixtures is a new method for investigating processes of energy exchange in molecular collisions.

Summarizing all that has been set out above concerning energy exchange by excited molecules, we must state that in the presence of excitation the processes of energy exchange in molecular collisions proceed with appreciable probability, in many cases exceeding 1. The numerical data relating to this are collected in Table 4.

Comparison of the data in this table with the data given in the preceding tables indicates a considerable increase in the probability of energy exchange upon excitation of the molecules, which, in accordance with the considerations expressed above, is due to the closer interaction of the excited and the normal molecule (as compared with the interaction of two normal molecules), leading to a reduction of the activation energy to a minimum value.

The data of Table 4 further indicate the same specific role of the nature of the colliding molecules that is found in collisions of normal molecules, and also the absence of any definite dependence of the probability on the mass of the molecules and on the magnitude of the quantum transferred. Thus the data relating to excited molecules, evidently, also compel us to abandon a simple mechanical interpretation of the exchange process

energy, bringing to the fore that peculiar specificity which is characteristic of the chemical process.

The data obtained by the fluorescence method, together with this, show that upon collisions of molecules with a noticeable

TABLE 4

Probabilities of transfer of vibrational energy by excited molecules

probability only one vibrational quantum is converted predominantly into energy of translational or rotational motion (or conversely), more rarely two quanta (the iodine case). The simultaneous conversion of many quanta is in practice never observed. The reason for this is not yet entirely clear to us; however, on the basis of classical mechanics we can form a certain qualitative, visual picture of the process of transfer of vibrational energy, which to some extent explains the fact that one vibrational quantum is preferentially transferred. The amount of energy transferred must undoubtedly, to one degree or another, depend on the phase of the vibrations in which the vibrating molecule finds itself at the moment of its collision with its partner. The maximum transferred energy should correspond to such a phase in which the vibrational energy has the form of kinetic energy, i.e., when the vibrating atoms are separated from one another by a distance close to the equilibrium one. However, since in this phase the velocity of the relative motion of the vibrating atoms is maximal, the molecule remains in it for only a very short time. Therefore the probability of collision of a molecule that is in this most favorable phase of vibration with another molecule must be exceedingly small. The most

particular, on the contrary, must be such collisions in which the relative velocity of the vibrating atoms is close to zero; however, from mechanical considerations it follows that in these collisions only a negligible portion of vibrational energy can be transferred, the minimum value of which is obviously equal to one quantum (cf. Oldenberg \(^{36}\), and also Kondrat'ev \(^{37}\)).

These simple mechanical considerations, however, are not applicable to the case of a prolonged interaction of colliding molecules, when the collision process must be regarded as a chemical process of formation of a quasi-molecule. The redistribution of energy in such a quasi-molecule, capable of independent existence during an interval of time considerably exceeding the duration of an ordinary collision \((10^{-13}—10^{-12}\) sec.), is subject to special statistical laws \(^{38}\). The probability of one or another distribution of energy within the quasi-molecule depends both on the energy reserve of the vibrating molecule and its distribution among the degrees of freedom of the latter, and, in the first place, on the chemical nature of the initial molecules forming the quasi-molecule and on the magnitude of their interaction. Therefore the calculation of the probability of interest to us in this case is associated with great difficulties \(^{39}\); however, it seems to us that the calculation of the probability of energy exchange by this method is, for most cases, the only correct one. Almost all the experimental facts which were cited above and part of which were generalized by Franck and Eucken \(^{21}\), who regard every process of energy exchange as the first initial stage of a chemical process, speak in favor of this.

Very valuable information on the processes of energy transfer is also provided by studies of monomolecular, or, in Hinshelwood’s terminology \(^{40}\), quasi-monomolecular reactions. At the same time, the field of monomolecular reactions is a fruitful area for applying to chemical processes the results obtained by means of purely physical methods of investigating processes of energy exchange in molecular collisions. In the kinetics of monomolecular reactions the main role is played by processes of activation and deactivation of molecules, occurring in double collisions (a bimolecular process) and consisting, respectively, in the concentration or loss of energy by the so-called active molecules, chiefly vibrational energy. The competition between the rate of activation and deactivation, on the one hand, and the rate of monomolecular decomposition, on the other, leads to the fact that the rate constant of a monomolecular reaction, at sufficiently low pressures, when the rate of activation becomes equal to or less than the rate of decomposition, ceases to be independent of pressure. The experimental establishment of the critical pressure region, i.e., the region of transition from the monomolecular law of reaction to the bimolecular one, also makes it possible to estimate the order of magnitude of the probability of energy transfer in the process of chemical activation (or deactivation). Further, the fact that various foreign chemically inert gases, being

admixed with the principal (reacting) substance in sufficient quantities, restore the monomolecular law, which has been disturbed by reduced pressure; shows that, in activation processes, foreign molecules may also participate as partners with which the molecules of the principal substance collide. This makes it possible to study not only the effectiveness of collisions of identical molecules, but also that of different molecules.

Unfortunately, the investigations of the majority of monomolecular reactions are not sufficiently detailed for one to draw from them definite conclusions concerning the probability of energy transfer in activating collisions of molecules. Therefore the quantitative factual material suitable for our purposes is very meager here. From the purely qualitative side, however, we have here the same picture as in the region of the simpler “physical” processes of energy exchange considered in detail above. Here the same specificity of the activating action is observed, manifesting itself in a sharp dependence of the effectiveness of impacts on the nature of the colliding molecules. As an example we may point to the considerable effectiveness of the activating impacts of hydrogen in the decomposition of a number of organic substances. Such an action of hydrogen was discovered by Hinshelwood^41, in particular in the case of the decomposition of methyl ether, where the effectiveness of impacts of hydrogen molecules proves to be almost the same as that of the molecules of the ether itself. At the same time, in this case the action of the other gases studied by Hinshelwood (He, N₂, CO, and CO₂) proves to be very small. On the other hand, in the decomposition of nitrosyl chloride, hydrogen no longer stands out among the other gases (O₂, Cl₂, CO₂, N₂, and NO₂), as shown by the investigations of Schumacher and Sprenger^42; the greatest activating action here is possessed by the mixture of chlorine and nitrogen dioxide—the products of the decomposition of nitrosyl chloride. Facts of this kind are present in large numbers in the extensive literature on monomolecular reactions. However, most of these facts cannot be used for the study of elementary processes of energy transfer without additional detailed investigations of the reaction kinetics.

The kinetics of the decomposition both of pure nitrous oxide and of its mixtures with various foreign gases has been studied most thoroughly by Volmer and his collaborators^43. The results of these works are given in Table 5, where the relative effectivenesses (with respect to N₂O itself) of the activating impacts of molecules of various gases are indicated (second column), as well as the probabilities of activation per collision (γ) calculated by Volmer and Bogdan^43.

Comparing the values of γ presented in this table with the quantities γ (γ₁⁰) presented in the four preceding tables, we note that, in order of magnitude, the values of γ in Table 5 approach the maximum values of γ₁⁰ in Tables 1, 2, and 3 (normal molecules) and the minimum values in Table 4 (excited molecules). However, taking

taken into account that the data of Table 5 were obtained at a temperature of \(665^\circ\mathrm{C}\) and, consequently, owing to the presence of an undoubted temperature coefficient (see formula 8), when recalculated to room temperature they must be considerably smaller, we must conclude that these data are rather closer to the values of \(\gamma_1^0\) for normal molecules. Let us compare, for example, the value \(\gamma = 0.0053\) for \( \mathrm{N_2O} \) itself (Table 5) with the probability of transfer of one vibrational quantum by the same molecule, measured at room temperature: \(\gamma_1^0 = 2 \cdot 10^{-4}\) (Table 1). We see that the two values differ by approximately a factor of 26, and this difference corresponds to the temperature difference \(938 - 290 = 648^\circ\). Calculating from this temperature coefficient the value of the activation energy \(a\), we find

\[ a = 2\,720\ \text{cal}. \]

TABLE 5

Activating effect of various gases in the reaction of monomolecular decomposition of nitrous oxide

Comparing the values of the quantity \(a\) obtained by us with the values calculated earlier, we see that in order of magnitude these values agree with one another, although, bearing in mind the close analogy of the molecules \( \mathrm{N_2O} \) and \( \mathrm{CO_2} \), for which, according to Richard and Reid[^10], \(a\) is found to be equal to \(6000\ \text{cal}\), we would be justified in expecting a greater closeness of the value calculated by us for \( \mathrm{N_2O} \), \(a = 2720\ \text{cal}\), to the number \(6000\ \text{cal}\). Here, however, it is necessary to take into account the rather essential consideration that with an increase in the vibrational level of the molecule the activation energy for the transfer of a vibrational quantum (or quanta), which is the process that underlies the deactivation of the \( \mathrm{N_2O} \) molecule, must decrease. And since an active \( \mathrm{N_2O} \) molecule, possessing an energy reserve of \(52\,500\ \text{cal}\), must be at very high vibrational levels, it is highly probable that this circumstance fully explains the somewhat low value of the activation energy \(a\) obtained by us.*

* Let us also note that the value \(\gamma = 0.0053\) was obtained by Fowler and Frelikh[^43] on the assumption that, in an active \( \mathrm{N_2O} \) molecule, 3 degrees of freedom are excited. However, the possibility cannot be excluded that transverse vibrations of the \( \mathrm{N_2O} \) molecule do not participate in its activation, and the number of excited degrees of freedom is equal to 2. Calculations show[^44] that in this case the value of \(\gamma\) proves to be \(\sim\)10 times larger, which consequently leads to a \(\sim\)10 times larger temperature coefficient of the probability \(\gamma\), from which \(a\) is obtained equal to \(4600\ \text{cal}\).

Returning to Table 5 and comparing the efficiencies of the various gases with one another, we see that water has the maximum efficiency. This property of water has already been noted by us more than once in connection with experiments on the dispersion and absorption of sound (see above). Here, however, the action of water is not manifested in so sharp a form as in the preceding cases, apparently owing to the considerable loosening of the bonds of the active molecule $N_2O$, which in this respect resembles an excited molecule. The somewhat increased efficiency of $CO_2$ in comparison with $N_2O$ itself is difficult to explain; however, the rather sharp decrease in efficiency on going from triatomic molecules ($H_2O$, $CO_2$, and $N_2O$) to diatomic ones ($N_2$ and $O_2$) must apparently be ascribed to the decrease in the number of degrees of freedom, which undoubtedly hampers the exchange of energy. In Table 5 one is also struck by the generally insignificant difference between the efficiencies of diatomic and triatomic molecules, on the one hand, and monatomic ones (the noble gases), on the other. Since, in the collision of an oscillating molecule with diatomic and triatomic molecules, its vibrational energy may pass both into the translational and rotational energy and into the vibrational energy of the foreign molecules (and conversely), whereas in its collision with atoms the vibrational quanta can be transformed only into the energy of translational motion (and conversely), the coincidence in the orders of magnitude of the efficiencies of molecules and atoms can be explained only by the fact that the probabilities of the conversion of vibrational energy into vibrational energy itself or into the energy of translational motion must be comparable. This conclusion is also supported by the results of Yakovleva’s work, according to which the efficiency of collisions of Ar atoms in depriving the molecule $CN'$ of its vibrational quantum is only 10 times smaller than the efficiency of collisions of $N_2$ molecules. Since the vibrational quantum of the latter, equal to $6\,650$ cal, differs only insignificantly from the vibrational quantum of the excited $CN$ molecule, equal to $6\,090$ cal (the difference $6\,650 - 6\,090 = 540$ cal $\simeq 537 = RT$ at the temperature of Yakovleva’s experiments), then in collisions of $CN' + N_2$ molecules the possibility is not excluded of “exciting” the vibrational level of $N_2$ molecules, i.e. of converting the vibrational energy of the $CN'$ molecule into the vibrational energy of the $N_2$ molecule itself.

Our conclusion concerning the approximate equiprobability of the conversion of vibrational energy into vibrational and translational energy*—a conclusion which still requires final verification—has an extremely great significance for the theory of energy exchange in molecular collisions, especially since the experimental study of the processes of trans—

* The probability of conversion of vibrational energy into vibrational energy, apparently, can significantly exceed the probability of conversion of vibrational energy into translational energy only in the case of collision of identical molecules situated at neighboring vibrational levels (for example, the first and the zero), owing to the resonance effect.

rotation of vibrational energy into vibrational energy itself encounters great methodological difficulties.

Finally, in Table 5 it is interesting to compare with one another the values of \(\gamma\) for various noble gases. We see that here too, with respect to the activating and, correspondingly, deactivating action, helium stands in first place, while the effectiveness of the remaining gases decreases monotonically with increasing atomic weight. Whether this change in effectiveness is due to the masses or sizes of the atoms of the various noble gases, or to some other factors, existing theories of energy exchange are unable to decide. We note only that attempts to establish an analytical relation between the effectiveness of a noble gas and its mass lead to the following expression:

\[ \gamma = \beta e^{-\alpha \sqrt[3]{A}} \]

(\(\alpha\) and \(\beta\) are constants, \(A\) is the atomic weight), which, apparently, is satisfied most closely by the data of Table 5. At present, however, it is difficult to say whether this empirical regularity has any physical meaning.

In conclusion to the present review, let us summarize the results of our study of the empirical material relating to the question of the exchange of vibrational energy in collisions of gas molecules. These results may be summarized in the following conclusions:

1. The processes of energy exchange between normal molecules proceed with a comparatively small probability (in most cases of the order of \(10^{-3}\)—\(10^{-4}\) per collision).

2. This small probability is mainly due to the presence of activation energy in the processes of energy exchange.

3. With excitation of the molecule, the probability of energy exchange increases, often reaching values of 1 and higher.

4. The “physical” process of energy exchange is the initial stage of a “chemical” process, in which the principal role is played by the forces of a purely chemical interaction. This circumstance explains the observed specific dependence of the probability of energy exchange on the nature of the colliding molecules, the increase in the probability of exchange with temperature (activation energy) and with excitation of the molecules, and also, as a rule, the absence of a direct dependence of this probability on the masses of the molecules (Franck and Eucken).

5. The processes of conversion of vibrational energy into vibrational, rotational, and translational energy have comparable probabilities.

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1. Some delay in the exchange of rotational and translational energy should be expected only in the case of an exceptionally unfavorable ratio of the masses of the colliding molecules[^9,6]. We also note that the existence of delays in the transfer of rotational energy of \(H_2\) molecules, assumed by Richards and Reid[^10] on the basis of their experiments with mixtures of ethylene and hydrogen, is disputed by Kneser and Vollmann[^11], who found no appreciable sound dispersion in pure hydrogen. ↩