QUENCHING OF FLUORESCENCE OF SOLUTIONS BY FOREIGN SUBSTANCES AS A METHOD FOR STUDYING THE KINETICS OF BIMOLECULAR REACTIONS IN SOLUTIONS

B. Ya. Sveshnikov

The question of the influence of diffusion on the number of effective collisions between molecules of two dissolved substances was posed and studied by S. I. Vavilov in 1929[^1] in connection with consideration of the kinetics of fluorescence quenching by foreign substances. The fundamental propositions of his theory in subsequent years received full justification and in 1936[^2] were extended by the author to the kinetics of thermal bimolecular reactions.

Analogous considerations were expressed by Fowler and Soper[^3] at a 1938 conference devoted to the kinetics of reactions in solutions. These authors noted in their report that the theory of activated complexes is not able to explain the dependence, observed in a number of cases, of the rate of bimolecular reactions on viscosity, and proposed calculating the number of effective collisions in solutions by means of the methods of diffusion theory.

Finally, in 1942 a paper by Debye[^4] was published, in which he also repeated the conclusions of Soviet authors*.

Since then, articles devoted to the theoretical and experimental study of the influence of diffusion on the rate of bimolecular reactions in solutions have appeared systematically in various journals. These works fully confirm the great role of the viscosity of the solvent in the kinetics of fluorescence quenching and in certain other photochemical reactions. However, with respect to thermal bimolecular reactions, the experimental material confirming the dependence of the reaction rate on viscosity is very limited. The reason for this is the fact that a dependence of the reaction rate on viscosity can exist only when the active states are sufficiently long-lived, and is often masked by other factors.

* This circumstance was noted by Umberger and La-Mer[^5].

In view of this, it seems highly advisable to give a brief survey of the basic propositions of the theory of quenching of fluorescence by foreign substances and to consider once again the possibility of transferring this theory to the kinetics of thermal bimolecular reactions.

Already in the earliest works on the study of the fluorescence of dye solutions it was noted that the addition of certain colorless impurities, which absorb neither the exciting light nor the fluorescence light, can strongly weaken the intensity of luminescence.

Analysis of the causes producing quenching shows that all cases of quenching can be divided into two large classes. The first includes those cases in which quenching is due to processes occurring only with excited molecules (“quenching of the second kind” in S. I. Vavilov’s terminology*); the second includes those cases in which quenching is caused by processes occurring with molecules independently of their excitation (“quenching of the first kind” in S. I. Vavilov’s terminology).

The first class, in turn, can be divided into two groups according to reversibility. The first group includes all those cases in which the molecules return to the normal state within a very short interval of time after the act of quenching, and therefore the concentration of excited molecules does not change in the course of quenching; whereas the second group comprises irreversible photochemical reactions.

A number of experimental criteria make it possible for us easily to distinguish reversible quenching of the second kind from the other types of quenching. These criteria are as follows: 1) in the case of reversible quenching of the second kind, when high concentrations of a colorless quencher (thousands of times greater than the concentration of the luminescent substance) are introduced into a solution of a luminescent dye, no appreciable deformation of the absorption and fluorescence spectra of the given dye occurs; 2) the quenching action does not depend on the concentration of the luminescent substance over a wide range of concentrations; 3) the fluorescence yield*) increases upon dilution of the quenched solution with pure solvent; 4) no dependence whatever of the quenching process on the time or on the intensity of the exciting light is observed. However, the main argument in favor of quenching of the second kind is the decrease in the mean lifetime of the excited state of the molecule as a result of quenching. This phenomenon, as will be shown below, can be detected both by direct measurements of the lifetime of the excited state and indirectly.

*) That is, the ratio of the emitted energy to the absorbed energy.

For the theory of fluorescence quenching in solutions, as well as for the kinetics of chemical reactions, one of the fundamental questions is: how should one count the number of effective collisions in liquid media? This question arose in all its acuity before S. I. Vavilov when he began to create the theory of the quenching of fluorescence of solutions by foreign substances[^9]. Since by that time (1928–1929) chemists had accumulated considerable material on the kinetics of bimolecular solutions, it seemed natural to turn to the chemical literature. However, even in the mid-thirties there was no clear answer there to the question indicated.

The majority of authors[^10] believed that the viscosity of the medium has no substantial influence on the number of effective collisions between reacting molecules and, in considering bimolecular reactions in solutions, used the same expression for the number of collisions as is customary for gas reactions:

\[ Z = (\sigma_1 + \sigma_2)^2 n_1 n_2 \sqrt{8\pi RT\left(\frac{1}{M_1}+\frac{1}{M_2}\right)}, \tag{1} \]

where \(Z\) is the number of collisions, \(R\) is the gas constant, \(T\) is the temperature, \(\sigma_1\) and \(\sigma_2\) are the radii of the colliding molecules, \(M_1\) and \(M_2\) are their molecular weights, and \(n_1\) and \(n_2\) are the numbers of molecules of each kind.

Other authors assumed that the number of collisions between molecules of both components of the reaction is proportional to the viscosity of the medium. Ya. K. Syrkin[^11], for example, proposed the following formula:

\[ Z=\frac{c^2 n_1 n_2}{3ND}, \tag{2} \]

where \(c\) is the mean velocity, \(N\) is the sum of all molecules of the solvent and dissolved substance, \(D\) is the diffusion coefficient; the remaining notation is the same as in formula (1).

We showed as early as 1936[^2] that the latter formula is incorrect, since in its derivation an impermissible extrapolation of the formula for Brownian motion to the time of the free path of a molecule was allowed*). As for formula (1), it completely fails to take into account the peculiarity in the distribution of collisions in solutions between reacting molecules. In a gas, two molecules that have undergone a collision either enter into a reaction or immediately move apart. In a liquid, especially a viscous one, two molecules that have met may undergo a whole series of repeated collisions before they separate. Therefore, if the probability of reaction is very high, then, evidently, only

*) An analogous error is also present in the monograph of Melvin-Hughes[^10]. With the aid of a similar extrapolation of the formula for Brownian motion, he tried to show that the number of collisions between a molecule of a dissolved substance and molecules of the solvent is proportional to the viscosity of the solution.

first collisions. If the probability of reaction is very small, repeated ones must also be taken into account. The determination of the probability of the first encounter is reduced, obviously, to the solution of a diffusion problem.

This peculiarity in the distribution of collisions between molecules of the dissolved substance was taken into account by S. I. Vavilov in his theory of fluorescence quenching by foreign substances. In the first version of the theory^1 S. I. Vavilov assumed that the probability of quenching upon collision is equal to unity. In accordance with this, he confined himself only to counting the first collisions according to the following formula:

\[ Z_1 = 4\pi D r n_1 n_2 dt, \tag{3} \]

where \(Z_1\) is the number of first collisions (encounters) during the time from \(t\) to \(t+dt\), \(r\) is the radius of the molecule’s sphere of action, and the remaining designations are the same as before.

Assuming that deactivation of molecules by radiation and by quenching are two competing processes, S. I. Vavilov obtained the following formula for the change in the fluorescence yield of solutions when quenched by foreign substances:

\[ \frac{L_0}{L} = 1+ \frac{2kTc(\tau_1+\tau_2)\tau_0}{3\eta r_1 r_2}, \tag{4} \]

where \(L_0\) and \(L\) are respectively the luminescence yield in the absence of the quencher and in its presence*), \(\tau_0\) is the mean lifetime of the excited state in the absence of quencher, \(c\) is the concentration of the quencher, \(\eta\) is the viscosity, and \(k\) is Boltzmann’s constant.

Formula (4) provided for a linear dependence of the quenching, i.e. of \(\frac{L_0}{L}-1\), on the concentration, fluidity, and lifetime of the excited state of the molecule. The proof of the last dependence constituted the aim of the indicated work by S. I. Vavilov. Using data on quenching, he succeeded in determining the order of the lifetime of the excited state of the molecule and thereby in justifying the fundamental idea of his theory.

The dependence of quenching on the lifetime of the excited state was confirmed in subsequent years by a large body of experimental material. We shall confine ourselves to three examples:

*) Since in the principal experiments on quenching by foreign substances neither the fluorescence spectrum nor the absorption spectrum of the exciting light changes upon introduction of the quencher, the ratio of yields may be replaced by the ratio of fluorescence intensities. However, this is correct only in the first approximation. The point is that, upon quenching, the degree of polarization of luminescence changes^13,14 and, as a result of this, the change in the sum of the light given by the two components of the electric vector perpendicular to the line of observation is not proportional to the change in yield. The author indicated a method of observation in which the observed change in intensity is proportional to the change in yield.

1. The dependence of quenching on the duration of the excited state of the molecule appears quite distinctly even for luminescent substances of very different chemical composition. An example is Table 1, taken from the author’s dissertation[^15].

Table 1

Dependence of the quenching action \(\left(\dfrac{L_0}{L} - 1\right)\) of potassium iodide on the duration of the excited state of the molecule of the fluorescent substance

1. For one and the same substance, phosphorescence is quenched by foreign substances many times more strongly than fluorescence. For example, for orange rhoduline, whose phosphorescence lifetime in glycerin exceeds the fluorescence lifetime of the same dye in the same solvent by approximately \(5\cdot 10^5\) times, the phosphorescence lifetime is reduced fourfold upon adding \(4\cdot 10^{-5}\) g/cm\(^3\) of hydroquinone to the solution[^16], whereas the intensity (and lifetime) of the fluorescence of rhoduline is reduced, upon addition to a solution of the same viscosity of \(12\cdot 10^{-3}\) g/cm\(^3\) hydroquinone, by only 15%.

Fig. 1. Quenching of an aqueous fluorescein solution by potassium iodide at different dye concentrations.

1. When the duration of the excited state of a fluorescent substance is shortened (for example, by concentration quenching), quenching by foreign quenchers is noticeably diminished[^15]. Fig. 1 gives data obtained by the author.

In subsequent works by S. I. Vavilov and his students, the dependence of quenching by foreign substances on the quencher concentration and the viscosity of the solvent was studied in detail. The experimental data showed that \(\dfrac{L_0}{L} - 1\) increases with increasing

concentration more rapidly and decreases with increasing viscosity more slowly than would be expected from formula (4). To explain these facts, S. I. Vavilov and I. M. Frank\(^{12}\) introduced the concept of a sphere of action exceeding the kinetic one, and in the first approximation the quenching within the sphere of action was regarded as independent of time. Formula (4) then took the following form:

\[ \frac{L_0}{L}=e^{\omega c}\left(1+\frac{2kTc(\sigma_1+\sigma_3)r\tau_0}{3\eta\sigma_1\sigma_2}\right), \tag{5} \]

where \(\omega\) and \(r\) are, respectively, the volume and radius of the sphere of action. The remaining notation is the same as in formula (4).

Fig. 2. Change in the lifetime of the excited state \(\tau\) (○) and the fluorescence yield (×) upon quenching of an aqueous fluorescein solution by potassium iodide.

Formula (5) made it possible to explain the nonlinear dependence of quenching on concentration and viscosity; however, subsequent experiments\(^{14}\) showed that it contradicts the strict parallelism between the change in the lifetime of the excited state and the change in yield. This relation, theoretically substantiated by S. I. Vavilov\(^{8}\), is observed for many cases of quenching with considerable accuracy*). As an example one may cite the results of comparing fluorometric measurements\(^{17}\) of the lifetime of the excited state \(\tau\) under quenching of the fluorescence of fluorescein by potassium iodide with data on the relative change in yield under the same quenching, obtained by us\(^{14}\) (Fig. 2)**).

Unfortunately, fluorometric measurements of \(\tau\) in solutions quenched by foreign substances are very limited; therefore, to verify the parallelism between the change in the lifetime of the excited state and the change in yield, an indirect method is often used, determining \(\tau\) from polarization measurements.

The theory\(^{19}\) of fluorescence polarization of solutions leads to the following dependence between the polarization and the lifetime of the excited state:

\[ \frac{1}{p}-\frac{1}{p_0} = \left(\frac{1}{p_0}-\frac{1}{3}\right) \frac{kT}{v\eta}, \tag{6} \]

*) This is explained by the fact that, for quenching caused by processes developing in time and moreover with excited molecules, longer-lived molecules have a greater chance of being quenched.

**) This result was later confirmed by the experiments of M. D. Galanin\(^{18}\).

where \(p\) is the degree of polarization observed for the solution under study, \(p_0\) is the so-called limiting polarization, i.e., the polarization observed at \(\eta \to \infty\), \(v\) is the molecular volume; the remaining notation is the same as in (4).

If, as we indicated above, there is a parallelism between the change in \(\tau\) and the change in the yield \((L)\), then it is obvious that between the change in \(\frac{1}{p}\) and \(\frac{L}{L_0}\) there must also exist a linear dependence. The validity of this proposition was verified by the author for several cases \({}^{14}\) (Fig. 3) and proved very thoroughly by A. N. Sevchenko \({}^{13}\) on extensive experimental material.

Fig. 3. Change in the degree of polarization of fluorescein upon quenching by potassium iodide: \(a\)—in aqueous-glycerol solution and \(b\)—in water.

A detailed analysis of the kinetics of diffusion processes occurring around the excited molecule allowed S. I. Vavilov \({}^{20}\) and the author \({}^{21}\) to establish another cause of the nonlinear dependence of quenching on the concentration of the quencher. The point is that diffusion during the lifetime of the excited state of a molecule, of the order of \(2 \cdot 10^{-9}\)—\(5 \cdot 10^{-9}\) sec, cannot be regarded as a fully established process, as a result of which an additional term arises in the formula and, instead of (3), one should write

\[ Z_1 = 4 \pi D r n_1 n_2 \left(1 + \frac{r}{\sqrt{\pi D t}}\right) dt, \tag{7} \]

where all the notation is the same as in formula (3*).

* In Smoluchowski’s work, formula (7) was derived with the aid of the assumption of the existence of a concentration gradient around the excited molecule. This assumption is not necessary. A. N. Kolmogorov and M. A. Leontovich \({}^{22}\) showed that the same formula can be obtained by proceeding from the Brownian motion of the colliding molecules.

The physical meaning of the factor in parentheses is quite obvious. We count spheres intersected by the moving molecule only once and do not take account of repeated intersections. At the beginning of the process all the spheres surrounding the molecule have not yet been intersected by it even once, and therefore the number of first encounters per unit time is greater at the first moment than subsequently, when a stationary state is established*).

Formula (7) has an additional term in comparison with formula (3**). Taking this circumstance into account, the quenching formula (4) takes the following form:

\[ \frac{L_0}{L} = \left( 1+ \frac{2p\tau_0 c k T r(\sigma_1+\sigma_2)} {3\eta\sigma_1\sigma_2} \right)\delta, \tag{8} \]

where

\[ \delta= \frac{1} {1-2e^{-\gamma^2} \left( \frac{\sqrt{\pi}}{2} - \int_0^\gamma e^{-x^2}\,dx \right)} \]

and

\[ \gamma = 2p r^{3/2} c^{1/2} \left( \frac{\tau_0}{p\tau_0+T} \right)^{1/2}. \]

In addition, in (8) the probability of quenching upon encounter, \(p\), has been introduced, which in the general case is less than unity. For example, for the generally known case of quenching of an aqueous fluorescein solution by potassium iodide, \(p=0.25\).

We have shown that formula (8) in a number of cases agrees well with the experimental data giving the dependence of quenching on concentration (Table II), but is not able to explain the observed dependence of quenching on viscosity.

The dependence of quenching on the viscosity of the solvent, obtained experimentally, as was first shown by S. I. Vavilov and I. M. Frank\(^{12}\), is represented by a rather complicated curve (Fig. 4). In their experiments the change in viscosity was achieved by replacing one solvent with another—

* It is curious to note that the nonstationarity of diffusion processes entails the consequence that the law of decay of the fluorescence of a quenched solution in the initial stage of decay must differ considerably from an exponential law. This circumstance can be verified only by direct fluorometric experiments.

** Recently Umberger and La Mer\(^{5}\), as well as Montroll\(^{5}\), have attempted to derive a quenching formula for charged particles with allowance for the nonstationarity of Brownian motion. The approximate solution obtained by them leads for uncharged particles to formula (8).

Table II

Quenching of fluorescein by potassium iodide

) and by the composition of binary mixtures. The author23 showed that the same dependence also holds for the case when a change in viscosity is achieved by changing the temperature (Fig. 5), and that the form of the curve does not depend on whether the reacting particles are ions or are uncharged. In Fig. 6 the curve of the dependence on viscosity of the quenching of rhoduline orange by aniline in various solvents is reproduced *).

Fig. 4. Quenching of rhodamine B in various potassium iodide solutions (according to data of C. I. Vavilov and I. M. Frank).

*) It should, however, be noted that, as the author’s experiments23 showed, the range of solvents that can be regarded as a medium possessing only viscosity is very small. These are mainly limiting alcohols, alcohol–glycerol, water–glycerol, and sugar solutions. A very important condition for obtaining a dependence on the viscosity of the solvent in pure form is the constancy of the lifetime of the excited state when passing from one solvent to another or when increasing the temperature, and this is not always the case.

**) The indicated dependence of the quenching of fluorescence by foreign substances on the viscosity of the solvent was subsequently confirmed by a number of authors4,25.

All these facts indicate with sufficient reliability that the observed deviations from a linear dependence between quenching and viscosity are not due to any random causes, but that the number of effective collisions between quencher molecules and excited molecules is in fact a nonlinear function of viscosity.

Fig. 5. Quenching of fluorescein fluorescence by aniline in various solvents and at different temperatures.
$\bullet$ — isobutyl alcohol, $\circ$ — ethyl alcohol, $\odot$ — mixture of glycerin with ethyl alcohol, $\oplus$ — mixture of glycerin with isobutyl alcohol.

S. I. Vavilov indicated two possibilities for explaining this phenomenon: the noncoincidence of molar and molecular viscosity for particles of molecular dimensions^24 and the possibility that the sphere of action (quenching probability)*) increases with increasing viscosity.

“Extrapolation of the concept of macroscopic viscosity to the motion of a molecule,” S. I. Vavilov points out, “has a conditional and indeterminate meaning. Its justification lies in practical success: in the possibility of correctly determining the order of magnitude of one quantity or another, while the physical basis lies in statistical averaging of an enormous number of individual deviations.”

Fig. 6. Quenching of rhodamine orange by aniline.
$\triangle$ — methyl alcohol, $\circ$ — ethyl alcohol, $\square$ — isopropyl alcohol, $\blacktriangle$ — isoamyl alcohol, $\blacksquare$ — isobutyl alcohol, $\bullet$ — glycerin, $\odot$ — mixture of glycerin with ethyl alcohol.

However, the assumption of a noncoincidence of molar and molecular viscosity (at least for viscosities,

*) The notion of a sphere of action expanding with increasing viscosity, and of an increase in the probability within the sphere of action to a certain degree, is plausible. The probability of interaction of two molecules is a continuous function of their distance and interaction time. Therefore, upon transition to more viscous solutions, the action of quencher molecules situated comparatively far from the excited molecule becomes increasingly effective.

not exceeding 1–2 poise) is refuted by the experiments of many authors who studied the change in the polarization of fluorescence upon changing the viscosity of the solvent. The point is that the polarization formula (6) was derived under the assumption of complete applicability to the molecule of Einstein’s formula for rotational Brownian motion, and therefore the good agreement of the constants determined from formula (6), for example the lifetime of the excited state, with the values obtained for these quantities by direct methods, for example fluorometric ones, is a justification of all the assumptions made in deriving it.

For a more careful investigation of the question, we reproduce in Table III the data of A. N. Sevchenko \(^{13}\), who studied the dependence

Table III

Change in the polarization of fluorescence upon changing the viscosity of the solvent (according to A. N. Sevchenko)

| \multicolumn{3}{c}{a) Water–glycerin solution} | \multicolumn{3}{c}{b) Alcohol–glycerin solution} |
|---:|---:|---:|---:|---:|---:|
| \(\eta\) in poise | \(p\) in % | \(\left(\dfrac{1}{p}-\dfrac{1}{p_0}\right)\eta\) | \(\eta\) in poise | \(p\) in % | \(\left(\dfrac{1}{p}-\dfrac{1}{p_0}\right)\eta\) |
| 0.038 | 5.7 | 0.56 | 0.0116 | 7.5 | 0.43 |
| 0.047 | 7.0 | 0.545 | 0.046 | 8.3 | 0.43 |
| 0.060 | 8.0 | 0.588 | 0.054 | 9.2 | 0.45 |
| 0.068 | 9.1 | 0.56 | 0.065 | 10.5 | 0.44 |
| 0.085 | 10.9 | 0.53 | 0.08 | 12.2 | 0.43 |
| 0.098 | 11.5 | 0.588 | 0.10 | 13.9 | 0.46 |
| 0.117 | 13.5 | 0.55 | 0.12 | 16.9 | 0.47 |
| 0.159 | 15.7 | 0.576 | 0.167 | 19.6 | 0.39 |
| 0.22 | 18.9 | 0.57 | 0.265 | 22.5 | 0.46 |
| 0.47 | 23.7 | 0.70 | 0.42 | 28.0 | 0.41 |
| 0.71 | 25.8 | 0.83 | 1.03 | 32 | 0.32 |
| 1.30 | 31.6 | 0.52 | 2.07 | 35.9 | 0.38 |
| 2.2 | 35.4 | 0.28 (?) | \(\sim\) | 33 | |
| \(\infty\) | 37 | | | | |

of the degree of polarization on viscosity for water- and alcohol–glycerin solutions. In the last column of this table are given the products we calculated, \(\left(\dfrac{1}{p}-\dfrac{1}{p_0}\right)\eta\). From (6) it is clear that these products must be equal to a constant value if, in the viscosity interval studied, there are no deviations from Einstein’s formula for rotational motion. Table III shows that this is indeed the case.

The strict fulfillment of the formula for rotational motion gives us grounds to suppose that, in the viscosity interval under consideration, the formula for Brownian displacement is also fully applicable. Consequently, the explanation of the observed deviations must be

one must seek in the second hypothesis: in an increase of the probability of quenching (the sphere of action) with increasing viscosity.

The first attempt to take into account the influence of viscosity on the change in the probability of quenching on the basis of purely mechanical notions was undertaken by the author in 1936.^2 The basic idea of the author’s hypothesis was that the increase in the probability of quenching in viscous solvents is connected with a change in the ratio between the primary and repeated collisions of the reacting molecules in going from slightly viscous to viscous solutions. Indeed, with increasing viscosity the number of quencher molecules which an excited molecule meets during its lifetime decreases, but on the other hand the number of repeated collisions with each quencher molecule encountered increases.* If the probability of reaction in a single collision were equal to unity, repeated collisions would play no role and the rate of reaction (quenching) would be a linear function of the viscosity. If, however, the probability of reaction \(w\) in a single collision is considerably less than unity, repeated impacts will be of great importance and the effectiveness of the encounter of two reacting molecules would be equal to

\[ p = 1 - (1 - w)^{\nu}, \tag{9} \]

where \(\nu\) is the total number of collisions, independent of the viscosity, and \(\nu_1\) is the number of first collisions, inversely proportional to the viscosity.

If the expression for \(p\) is substituted into formula (8), then, depending on the values of \(w\), we obtain: 1) a linear dependence of \(\dfrac{L_0}{L} - 1\) on \(\eta\) (for \(w = 1\)); 2) a nonlinear dependence, usually observed in quenching for \(w < 1\); and 3) independence of the viscosity for \(w \ll 1\).

The same results can be obtained if the purely mechanical notions of repeated collisions are replaced by the notion of a sphere of action exceeding the kinetic one, and a dependence of the probability of reaction on the interaction time of two molecules is introduced.

Indeed, with increasing viscosity the number of molecules entering the sphere of action from outside rapidly decreases, but the effectiveness of these molecules in the sense of quenching will increase.

Concluding this brief review of experimental and theoretical works on quenching by foreign substances, we consider it necessary to note that the theory of quenching set forth above is capable

* The question of the change in the ratio between primary and repeated collisions when the viscosity of the solvent is changed, and of the influence of this factor on the kinetics of reactions, was discussed by Fowler and Sletter^3 at the Faraday Society conference in 1938. From the reasoning presented by them it follows much that is analogous to ours. Our work of 1936,^2 evidently, was unknown to them, although they refer to Rabinovich’s work,^26 in which, in turn, there is a reference to our work.

explain not only those cases in which quenching decreases linearly or nonlinearly with increasing viscosity, but also those cases in which quenching does not depend on the viscosity of the solution or, conversely, increases with increasing viscosity.

Indeed, in formula (8), besides the viscosity, two more parameters enter: \(\tau_0\) and \(p\). If, with respect to \(p\), we can obtain information only from the quenching process itself, then the lifetime of the excited state \(\tau_0\) can also be determined by other methods. The data available in the literature on the influence of the solvent on the lifetime of the excited state clearly show that, for individual substances, the duration of fluorescence changes very little on passing from one solvent to another. However, there are also cases in which \(\tau_0\) has different values for different solvents. Sometimes, for example, \(\tau_0\) systematically increases with increasing viscosity of the solvent. Therefore cases in which the entire influence of the solvent on quenching processes reduces only to a redistribution of collisions between the reacting molecules are sufficiently rare. Nevertheless, these rare cases are very important. They justify the whole theory, which gives them a simple explanation and makes it possible, as S. I. Vavilov\(^1\) showed, to obtain values of the lifetime of the excited state that agree in order of magnitude with values determined by other methods, for example fluorometric ones.

Let us now turn to the kinetics of thermal bimolecular reactions. The following reaction may serve as the clearest example of the dependence of the rate of thermal reactions on viscosity:

\[ \mathrm{CH_3J + (C_2H_5)_2S \to (C_2H_5)_2CH_3SJ,} \]

studied by Ya. K. Syrkin and I. T. Gladyshev\({}^{27}\). This reaction is especially noteworthy because for it a very good linear dependence is observed between the reaction rate and the fluidity (Fig. 7)—a case that is very rare and, according to formula (9), corresponds to the probability of reaction in one collision \(w\) being equal to unity.

It should be noted that taking into account the fact that, when the temperature is raised, the number of effective collisions also changes owing to the change in the viscosity of the solvent makes it possible to show that, in this reaction, the activation energy is the same for all solvents (11–13 kcal). Meanwhile, neglecting this circumstance, the cited authors obtained 13.3 kcal in acetone, where the change of viscosity with temperature is small, and 21.7 kcal in isopropyl alcohol.

For another reaction of the same type\({}^{28}\):

\[ (\mathrm{C_2H_5})_3\mathrm{N} + \mathrm{C_2H_5J} \to (\mathrm{C_2H_5})_4\mathrm{NJ} \]

for the same solvents, excluding isopropyl alcohol, instead of

for which, in this case, isobutyl alcohol is used, a dependence on viscosity is obtained that is quite analogous to that observed in fluorescence quenching (Fig. 8).

Fig. 7. Change in the rate of the reaction
\(\mathrm{CH_3J + (C_2H_5)_2S \rightarrow (C_2H_5)_2CH_3SJ}\)
as a function of the viscosity of the solvent. \(\triangle\) — acetone, \(\bigcirc\) — methyl alcohol, \(\square\) — ethyl alcohol, \(\times\) — propyl alcohol.

Fig. 8. Change in the rate of the reaction
\(\mathrm{(C_2H_5)_3N + C_2H_5J \rightarrow (C_2H_5)_4NJ}\)
as a function of the viscosity of the solvent. \(\triangle\) — acetone, \(\bigcirc\) — methyl alcohol, \(\square\) — ethyl alcohol, \(\bullet\) — isobutyl alcohol.

Since in most cases the solvent by no means plays the role of a “void with viscosity,” the dependence of the reaction rate on viscosity when one solvent is replaced by another may mask—

Table IV

Change in the rate of the reaction
\(\mathrm{C_6H_5NH_2 + CH_2BrCO\cdot C_6H_5 = (C_6H_5\cdot NH_2CH_2CO\cdot C_6H_5)Br}\) upon an increase in temperature

QUENCHING OF FLUORESCENCE BY FOREIGN SUBSTANCES

...be manifested by changes in other parameters that determine the kinetics of the reaction. Nevertheless, in these cases as well it is sometimes possible to draw a conclusion about the presence of a dependence on viscosity by studying the change in the reaction rate with temperature in a series of solvents.

As an example, let us cite literature data on the temperature dependence of the rate of the reaction of aniline with bromacetophenone.^29 In the first column of Table IV are given the values of the reaction rate in various solvents at 27.8°, in the second—the viscosity values at this temperature, in the third—the ratio of the reaction rate at 47.8° to the rate at 27.8°, i.e. \(\frac{k_2}{k_1}\); in the fourth—this ratio with the change in viscosity taken into account, in the fifth—the activation-energy values calculated from the data of the third column, and in the sixth column—the activation-energy values calculated from the data of the fourth column.

If we confined ourselves to considering only the first two columns, we would of course arrive at the conclusion that viscosity does not affect the kinetics of the reaction. However, from the data of the fourth and sixth columns one may arrive at the opposite conclusion. Allowance for the change in viscosity upon heating of the solution makes it possible, for six very different solvents, to obtain practically the same values of the activation energy, and this result can hardly be explained by pure chance*).

An attempt to extrapolate the basic propositions of the theory of quenching by foreign substances to thermal reactions requires special justification. The point is that quenching, like most photochemical reactions, does not require a noticeable activation energy, whereas for thermal reactions of the nonionic type a significant activation energy is necessary. Therefore, if one attempts to apply the kinetics of quenching to thermal reactions, one must require that the active states of the molecules have a lifetime of at least the order of \(10^{-10}\) sec.**). But the idea of a long lifetime of active states of molecules in solutions is considered rather improbable if these states are thought of as high vibrational states of molecules. A certain way out of the difficulty may be found in the supposition that in thermal reactions in which a dependence on viscosity is manifested, the activation of molecules consists in their transition into a special electronic state (biradical). The hypothesis that in a number of reactions the triplet (biradical) state is the principal

*) In benzene the value of the activation energy is noticeably lower than in the other solvents. This circumstance can readily be explained by assuming that in benzene there is a decrease in the lifetime of the active state of the reacting molecules upon heating.

**) This time is necessary so that diffusional processes can develop.

... as a reaction link, has been vigorously discussed recently. Thus A. N. Terenin^30 proposed it several years ago for oxidation reactions. Other authors have extended this hypothesis also to polymerization^31 and isomerization^33 reactions.

Unfortunately, the biradical states of organic molecules have not yet been studied sufficiently. Almost all the available information about them has been drawn from the study of phosphorescence and, consequently, pertains to cases in which the transition to the biradical (triplet) state occurs through an excited singlet state and the energy value for the triplet level is equal to 1.5–3 eV. Recently, however, another route for the optical study*) of biradical states has been emerging—namely, the study of those cases of thermochromism in which, upon heating a solution, a new absorption band appears, shifted considerably toward the long-wavelength side relative to the principal absorption band. In this way it has already been possible to detect biradical states possessing energies of the order of several kilocalories.^39

From what has been said it is clear that at present there are no fundamental objections to attempts to transfer the scheme adopted in fluorescence quenching to the kinetics of certain thermal reactions. It should therefore be recognized as quite expedient to undertake a detailed study of those bimolecular reactions in which there is a sharply expressed dependence of the reaction rate on viscosity, even if only for a limited assortment of solvents, or of those cases in which allowance for the temperature variation of viscosity makes it possible, as was shown above, to obtain activation-energy values that are fairly close for a series of solvents.

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6. R. Williamson and V. La-Mer, J. Amer. Chem. Soc. 70, 717 (1948); K. Hodges and V. La-Mer, J. Amer. Chem. Soc. 70, 722 (1948).
7. F. Collins and G. Kimball, J. Colloid. Sci. 4, 425 (1949); J. Christiansen, J. Colloid. Sci. 6, 213 (1951).
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9. S. I. Vavilov, Zeits. f. Phys. 50, 52 (1928).
10. F. Moelwyn-Huges, The Kinetics of Reactions in Solutions, Oxford, 1933. R. Norrish and F. Smith, Trans. Chem. Soc. (London) 129 (1928).

*) Optical methods of investigation are very convenient, but not obligatory. Biradical states can also be detected by other methods, for example by magnetic measurements. The point is that a triplet (biradical) state with two unpaired electrons must possess a magnetic moment.

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