FREQUENCY FACTORS OF UNIMOLECULAR DIBBOCIATION PROCESS108
939
DISCUSSION M. BURTON(University of Notre Dame): It beems to me that in interpretation of experiments of the type cited by Dr. Noyes it would be very useful to have systematic information on the effect of the ratio (light beam volume): (cell volume) on the relative rates of the pertinent reactions aa a function of temperature. Such data should reveal to what extent chain-ending steps are bimolecular in the gas or occur on the wall. Is any infomiation available on this point? A. J. C. NICHOLSON (University of Rochester), in aniwer to Dr. Burton’s question: Above 120OC. Trotman-Dickenson and Steacie found an activation energy of 9.7 kcal., using a light beam that almost filled the reaction veasel. I have used a light beam with a diameter about 0.56 of that of the vksel, which waa similar to that used by Dorfman and Noyes, and obtained excellent agreement with this activation energy at these temperatures. This and other experiments in which the light volume was varied indicate that above 120°C. the correct volume to use is the light volume. On the other hand, at room temperature it seems as if the correct volume to use in the work of Dorfman and Noyes is the vessel volume. This would bring the activation energy of Dorfman and Noyes up from 7 kcal. to about 8.5 kcal. This very nearly explains the discrepancy, and the remaining difference could be due either to abstraction of hydrogen from biacetyl or to the fact that the ratio Rca,/R’d,2a6is not quite linear with acetone pressure, as required by the postulated mechanism.
T H E FREQUENCY FACTORS OF UNIMOLECULAR DISSOCIATION PROCESSES1 M. SZWARC
Chemistry Department, University of Maneheskr, Manchedter, England Received August 10, 1060
This discussion of the frequency factors of unimolecular reactions does not claim any considerable originality. It intends only to sum up the ideas well known, to stress a few problems which require further clarification, and to point out experimental achievements as well as experimental uncertainties. I n the theoretical treatment of unimolecular reactions we have to distinguish between two aspects of this phenomenon: activation and decomposition. By activation we understand the process of energy transfer from the molecules of the system to some particular molecule which thus becomes “activated.” 1 Presented before the Symposium on Anomalies in Reaction Kinetics which was held under the auspices of the Division of Physical and Inorganic Chemistry and the Minneapolis Section of the American Chemical Society a t the University of Minnesota, June 19-21, 1950.
940
M. BZWARC
This process is the result of a successful series of collisions in which the molecule eventually activated participates, and it is, therefore, essentially a bimolecular process obeying second-order kinetics. Hence the rate of activation, irrespective of the mechanism, is proportional to the partial pressure of the compound which yields the activated species and to the total pressure in the system.* The proportionality coefficient depends on the nature of the molecules composing the system, different molecules having different specific power of accepting or transferring energy. By decomposition we understand the process of spontaneous dissociation of the “activated” molecule into products. The probability of this dissociation is a characteristic of the “activated” molecule, being independent of the total pressure. It follows, therefore, that the rate of decomposition is proportional to the concentration of “activated” molecules. A discussion of the mechanism and rate of the energy-transfer process is beyond the scope of this paper. We assume that in all cases discussed further the rate of activation is much higher than the rate of decomposition and that most of the “activated” molecules are deactivated by subsequent collisions with surrounding molecules. This assumption, which is the basis of Lindemann’s theory, leads to the conclusion that the concentration of “activated” molecules is given approximately by the equilibrium distribution of the energy of the system amongst all the molecules of which it is composed. Provided the pressure in the system is high e n ~ u g hthe , ~ energy-transfer process will be sufficiently rapid to maintain the appropriate stationary concentration of “activated” molecules. In order to calculate the rate of a unjmolecular decomposition we apply the transition-state method. There is, however, some difficulty in defining the transition-state complex (sometimes called “activated complex”) and in this respect two different types of unimolecular decompositions should be distinguished: 1. Decompositions leading to the formation of two products, recombination of which involves an activation energy, e.g.: C2HbBr -+ C2Ha
+ HBr
2. Decompositions leading to the formation of two fragments, recombination of which requires no activation energy. This case is particularly important for us, as it covers the dissociation of the molecule into two radicals. It need be realized that this type of unimolecular dissociation represents the simplest case of unimolecular reactions, since the process requires a rupture of one bond only. In the dissociation processes of the first type we do not encounter any difficulties in the definition of the transition-state complex. On plotting the energy of the system as a function of the reaction path coordinate one obtains the curve
* If the system contains only one type of molecule, the rate of activation is proportional t o the square of the pressure. * I t seems that, for sufficiently complex molecules, the energy-transfer process is rapid even a t pressures as low as a few millimeters of mercury. However, for small molecules containing three or four atoms the pressure required seems to be much higher.
FREQUENCY FACTORS OP UNfMOLECUWR DISSOCUTION PROCESSES
941
shown in figure 1. The hump of this curve represents the transition state which is, therefore, completely defined by the coijrdinates of this point. Figure 2, on the other hand, illustrates a decomposition of the second type, and because it does not show a hump a similar interpretation of the transition-state complex is impossible. I n order to avoid the difficulties arising from the absence of a de-
INITIAL STATE
TRAN: ST,
FIG.1. Plot of energy of the system as a function of the reaction path coijrdinate
FIG. 2
scription of the transition-state complex, we adopt for the latter case a slightly modified treatment of the transition-state method, as described below. Let us consider all the energy levels corresponding to various modes of motion of some particular bond in some particular molecule. These energy levels can be classified into two groups: (A) energy levels which correspond to the proper vibration of the bond, Le.. for which the energy is smaller than the bond
942
M. SZWARC
dissociation energy D4 (see figure 3); (B) energy levels which correspond to the translational mode of motion, Le., for which the energy is greater than the bond dissociation energy D.‘ These two classes of energy levels are denoted in figure 3 by A and B, respectively. It is obvious, of course, that the bond in question can be ruptured only if it is in a state which corresponds to any energy level belonging to class B.
FIQ.3
Let us now assume that there is no interaction between energy levels which correspond to various degrees of freedom of the molecule. For such cases, the total partition function of the molecule can be represented as fats1
=
II i-1
fi
where fi denotes the partition function corresponding to the itbdegree of freedom of the molecule. We can represent ftotnl in a slightly different way ftotsl
=
f“fk
where f‘ = n’fi represents the product of the fi’s for all i values with the exception of i = k, and fk represents the partition function corresponding to the vibrational degree of freedom of the bond in question. The partition function fk may be represented by 2 ,-‘d&T + e-D/kr‘ftr*”e* = 2 e - r f / k * 4 fk The value of D used here is the dissociation energy per molecule. The energy levels Corresponding to the translational mode of motion could be disciete only if the motion is limited in space, i.e., if the bond is enclosed in a “box.” The length of this “box” is chosen arbitrarily at ds. This approximation is permissible for ds sufficiently small. 4
1
FREQUENCY FACTORS OF UNIMOLECULAR DISSOCIATION PROCESSES
943.
the summation being taken over all vibrational energy levels, the jthof them corresponding to the energy e j , taking co = 0; D is the dissociation energy of the bond in question, Le., the difference between the vibrational zero-energy level and the convergence limit of the vibrational levels; and ftran.l denotes the partition function of the translational levels belonging to class B and measured from the energy level of the convergence limit taken aa a zero. If we confine our attention to the molecules for which the center of gravity of the bond in question is restricted to some segment ds along the direction of the bond,? then the partition function ftransl may be represented, with a fair degree of accuracy, as a translational partition function of "a particle in a box," Le., ftr.n.1
(2~pkT)'/'h-' ds
The fraction of the molecules which can decompose by the rupture of the bond in question (Le., which contain the requisite amount of energy in the bond to be broken) and for which the center of gravity of this bond is confined to the segment ds, is given by: Number of molecules which can decompose Total number of molecules
This expression can be further simplified if we assume that the energy levels e, correspond to a harmonic oscillator so that ze-ajILT - 1 + e-hro/kT + e - U r o / k T + . = (1 e-hp~IkT)-~
..
-
where u0 denotes the fundamental vibration of t h e bond in question. We arrive, therefore, at the expression: Number of molecules which can decompose Total number of molecules We can now assume that half of the molecules which can decompose (and for which the center of gravity is confined to the segment ds) are moving in the direction of decomposition with an average thermal velocity ( 2 k T / ~ p ) ~The /~. rate of decomposition is given, therefore, by the number of these molecules for which the center of gravity will pass the segment ds in a unit of time, Le., a
k = rate constant of decomposition = ( k T / h ) ( l - e-h"o'kT)e-D'kr
Thus we derive the expression, well known from the transition-state theory of unimolecular decomposition, which links the frequency factor of unimolecular
' We assume the origin of the coordinates to be fixed at one end of the bond in question.
944
M. SZWARC
dissociation with the fundamental vibration frequency of the bond in question. Inspection of this expression shows that the expected frequency factor should be of the order of 10'8 set.-' Let us now survey briefly the available experimental meterial. The experimental frequency factor is computed from two sets of data which are provided by experiment: the absolute rate constant and the experimental activation energy. The latter entity is obtained as
One has to recall that k =
ye-z/RT
Y denoting the theoretical frequency factor and E denoting the potential energy barrier. On the other hand
k
= yap
e-Zexp/RT
where vexp stands for experimental frequency factor, and E,,, is aa defined above. It has been shown elsewhere (2) that in unimolecular dissociation processes in which one bond only is broken the relationship
av
ZF-- 0 is a fair approximation. In such a case
Eexp = E and Yap
= Y
All the data derived below, and listed in table 1, are obtained by assuming aY
T-a
=0
and hence Y*xp
= v
The following conclusion should be drawn from table 1. The frequency factors , of the dissociation processes listed are of the order lo1*to loL4 set.-', as required by theory. They are not, however, constant values and the interesting problem is to find the cause of the variation observed. One would expect a decrease in frequency factors with increasing mass and decreasing dissociation energy. The data listed in table 1 do not confirm such a simple relationship. It is necessary to stress, however, that the experimental errors in estimation of frequency factors are usually factors of 2-3, since the computation requires the estimation of a In k/aT and this entity is very susceptible to experimental error.
FREQUENCY FACTORS OF UNIMOLECULAR DI860CIATION PROCEMEB
945
Let us examine this point in a more detailed way. Although no absolute values of the rate constants are required for the computation of the temperature coefficient, it is nevertheless essential to obtain a very high degree of accuracy in estimating relative rate constants. The following example illustrates this point. The rate constants of a unimolecular dissociation were estimated a t two temperatures, TI and T2,for which l/T1 - l/Tz = lo-'. This corresponds to a reasonable temperature range of about 50" if the experiments are conducted in the vicinity of 500°K. and to a range of about 100" for experiments carried out in the region of 1000°K. Let us assume that both rate constants, estimated a t T Iand T?,respectively, are uncertain by about 20 per cent each; then the maxiTABLE 1 Frequency factors for various unimolecular dissociation processes i n which one bond is ruptured PPOCESS
I
Y
E kcol./molr
tCC.-I
0.7 X 0.7 X 0.8 X 0.8 X 1 x
loLs
63
1
x
1013
59
6 8
X loiz X loL3
4 6 2
x
77.5 77.5 76 75 50.5
63 60 60 34 47.5
11 27-33
31 31
3 6 6 6
X X X X
10l3 loi3 lOI3
1013
10'2
X 10"
x
1013
5 x 1011 5 x 10'' 10'~-10'4 8 X 10" 5 x 1014
mum experimental error of the computed activation energy is E = 2 x 2.3 x log (1.2/0.8)/(10-* X 1OOO) kcal./mole = 8.1 kcal./mole. To improve the accuracy of the computed activation energy it is necessary either to extend the temperature range or to increase the accuracy of the estimated rate constants. The extension of the temperature range is limited by technical difficulties. The reaction a t high or low temperatures may be unsuitable for experimentation, being either too rapid or too slow. Alternatively, the mean value of the rate constant can be made more reliable by frequent repetition of individual runs, but this leads to an improvement of the results only when the experimental errors are of the haphazard type. On the other hand, if the determination of the rate constant involves a systematic error, which is itself temperature dependent, then the deviation of the temperature coefficient, and consequently of the "activation energy," is of a permanent nature and cannot be
946
M. BZWARC
eliminated by mere repetition of runs. Such a situation is created if, for example, the main reaction, which is the subject of investigation, is accompanied by some side reaction the relative extent of whioh continuously increases or decreases with the temperature. It is essential, therefore, to iind experimental conditions under which all side reactions are suppressed as far aa possible. Only under these circumstapces can one expect to be able to determine accurately the correct activation energy of the process from the temperature coe5cient of the rate constant. It seems from the above discueaion that the insufficient accuracy of the experimental material, as available a t present, prevents us from drawing any further conclusions in regard to the factors affecting the magnitude of frequency factors. One point, however, may be observed. It seems that in a series of similar processes the frequency factor remains constant for the whole series. For example, identical values (within experimental error) were obtained for frequency factors in a series of dissociations of the type: RH+R+H where R = CsH5CH2,m-CHsCsH4CH2,pCH8CsH4CH2,and o-CHsCsH4CH2. The constancy of frequency factors in such a series may be demonstrated by a still more exact method. It is extremely improbable that in a series of similar decompositions the variation in activation energy would be just balanced by the variation in frequency factor, thus leaving unchanged the rate constant of the reaction. On the other hand, it was observed that the rate constants of decomposition of toluene, m-xylene* (7), p-, m-,and o-fluorotoluenes (8), and of ypicoline (5) were all equal within 25 per cent. It was demonstrated that all these reactions are of the same type: RH+R+H There is no reason to expect changes in the C H bond dissociation energies in any of these compounds (see reference S), and the equality of all these rate constants is, in the writer’s opinion, the strongest argument in favor of the assumption of the constancy of the frequency factors in a series of similar decompositions. It seems that the above conclusion may be generalized for any series of similar reactions. Thus, C. K. Ingold and W. S. Nathan (4) estimated the activation energies for the hydrolysis of various substituted benzoic esters. The plot of the estimated activation energies versus log k (k being the rate constant of hydrolysis) gives a straight line, proving that the frequency factors remain constant throughout the whole series. These authors also drew attention to the results obtained by E. G. Williams and C. N. Hinshelwood (9) for the kinetics of benzoylation of various substituted anilines. A similar plot of E versus log k obtained by the latter authors gave a straight line which was parallel to that obtained by Ingold and Nathan. The idea of the constant frequency factors in a series of kindred reactions was developed further by L. P. Hammett (3), who 8 The rate of decomposition of m-xylene and that of 1,6-dimethylnaphthslene are halved, and thus they represent the “rate of decomposition per one methyl group.”
947
DISCUSSION
devised a system of p and u factors; p represents an entropy change constant for the same type of reaction, and u represents the change in activation energy characteristic for each member of the series. It may be concluded, therefore, that the frequency factor is not affected by the molecule as a whole, but rather by the character of the reacting center. Hence, it seems that the rate of energy flow inside the molecule is not the ratedetermining step. Finally, it must be observed that since in a reaction k
RR’ c 2 -- R
+ R’
ki
the equilibrium constant is given by The “normal” frequency factor for kl requires a “ n ~ r m a l ”collision ~ factor for kl, provided the equilibrium constant has a “normal” value (Le., the vibrational degrees of freedom in the radicals are not unusually “soft”). This requires, of course, “normal” probability factors in association processes involving radicals or atoms (1, 6). REFERENCES (1) EVAXS, M.G.,A N D SZWARC, M.: Trans. Faraday SOC.46, 940 (1949). (2) EVANS, M.G., A N D SZWARC, M.: Trans. Faraday SOC.,in press. (3) HAMMETT,L. P.: Physical Organic Chemistry. McGrsw-Hill Book Company, Inc., New York (1940). (4) INGOLD, C. K.,A X D NATHAN, W. S.: J. Chem. SOC.1936, m. (5) ROBERTS, J. S., A N D SZWARC, M.: J. Chem. Phys. 16, 981 (1948). (6) STEACIE,E. W. R., el al.: Discussions of the Faraday Society 8, 80 (1947). (7) SZWMX,M.:J. Chem. Phys. 16, 128 (1948). (8) SZWARC, M., A N D ROBERTS,J. S.: J. Chem. Phys. 16,609 (1948). E. G., A N D HINSHELWOOD, C . K.:J. Chem. SOC.1934. 1079. (9) WILLIAMS, 9
-
S o h added i+a proof: The adjective “normal” is used in the sense that
~
kz is in fair agree-
ment with the expectations of transition-state theory, i. e., k2 IO“lO” exp(-E/RT) for recombinations involving atoms, kz- 10”-10” exp(-E/RT, for recombinations involving small rrtrlicals (e. g., CHI), and kz is still lower for reactions involving larger radicals.
DISCUSSIOK 0. K. RICE (University of North Carolina): The occurrence of the frequency factor 10’3 in so many unimolecular reactions implies, according to the Eyring activated complex theory, that the activated complex is like the undissociated molecule except in the one degree of freedom or bond which actually breaks. However, actually one would suppose that if two of the atoms in a molecule are widely separated, as they are in the activated complex, other frequencies also would be greatly loosened. The fact that they are apparently not loosened may well have to do with matters concerning the rate of energy exchange between
948
DISCUSSIOX
the various degrees of freedom, especially as this energy exchange is affected by changes in the conditions of quantization as the reaction occurs. This has been briefly discussed in the symposium paper by R. A. Marcus and myself. Such effects could be formally taken into Eyring’s transmission coefficient, but may indicate that the formulation of Rice and Gershinowitz (J. Chem. Phys. 2, 853 (1934); 3, 479 (1935)), in which only general considerations concerning the change-over between vibration and rotational degrees of freedom enter, adheres more closely to what is actually known about these reactions.
K. J. LAIDLER(Catholic University of America): I t does not seem to be quite correct to say, as Szwarc has done, that because the frequency factors of unimolecular dissociations are “normal” (Le., lo1*to lo“), those of the reverse associations must also be “normal.” The ratio of frequency factors for forward and reverse reactions is, of course, e A e i R ,where A S is the total entropy increase in going from the associated to the dissociated state. If the dissociated products are complex this entropy increase may be large, so that the frequency factor for the association may be abnormally low. This situation is in fact found with many of the reactions studied by Wassermann, and may be true of some of the freeradical reactions under discussion here.
S.W. BENSON(University of Southern California): The disagreement between the paper of Szwarc on the one hand and those of Noyes and TrotmanDickenson and Steacie on the other leads one to consider the following points of connection with both sets of data which might bring them into closer conf ormity. It is difficult to believe that there is no dependence of the activation energies of these reactions on temperature. Simple statistical considerations such as those discussed quite generally by Tolman show that there must be a dependence of the observed activation energy on temperature, since the observed rate is an average rate of decomposition for a distribution of molecules over a spread of energy states. Since the activation energies for the reactions discussed (Le., hydrogen abstraction b9 methyl radicals) are quite low (-10 kcal.) and the temperature ranges used by these groups differ considerably (Le., -1000’K. for most of Szwarc’s work and -400°K. for the others), there is some reason for believing that there could well be a change of a few kilocalories per mole in the activation energy going from the low to the high temperatures. Such a change would bring the steric factors into better agreement. If, in addition, we consider the possibility that there is some chance that the temperature of the methyl radicals produced in the photolysis is greater than thermal equilibrium, owing to the excess energy of the light, then this too could result in raising the apparent activation energy observed in the photochemical work. The primary process which presumably produces CH, CH&O produces a “hot” methyl radical a t 2537 b.,and the acetyl radical which decomposes probably by collisions a t the higher temperatures may again produce a hot methyl radical. It is difficult to believe that this extra-thermal energy does not manifest itself in the rate of reaction of these radicals and so ultimately in
+
PHOTOLYSIS AND PYROLYSIS O F ACETALDEHYDE
949
observed activation energy. These observations could easily be tested by using high concentrations of rare gas in the photolysis and seeing if the activation energies remain unchanged. I t is also quite conceivable that a part of the small amount of methane which is produced at 60°C. and lower is produced by the reaction CHaCO CHaCOCHa --t CHd CO CHzCOCHs (i.e., some of the acetyl acting as “active” methyl radicals). This might then explain the ‘‘anomaly” observed by Noyes in the activation energies for methane production a t low and high temperatures. This would not necessarily do damage to the relation which he has found for oc,,/Qi’&a, since a t the lower temperatures where the acetyl radical concentration is highest, its ratio to the concentration of methyl radicals might well be such as to permit a small contribution to the production of methane and yet maintain the form of the ratio constant.
+
+
+
M. SZWARC (University of Manchester, Manchester, England) added the following comment in hhe proofs: I t seems that Dr. Laidler misunderstood my expression “normal.” The term has therefore been explained fully in footnote 9 of my paper. FREE RADICALS I N THE PHOTOLYSIS AND PYROLYSIS OF ACETALDEHYDE’,’ PAUL D . ZEMANY AND MILTON BURTONa
Research Laboratory of the General Electric Company, Schenectady, New York, and Department o j Chemistry, New York University, University Heights, New York City Receiued August 10, 1960 INTRODUCTION
The Rice-Herzfeld theory of free-radical chain reaction (14) accounted both for the observed order and for the activation energy of the pyrolysis of acetaldehyde (11) by the folloming mechanism: CHICHO --t CH3 HCO (1) R CHSCHO -+ R H CHIC0 (2) CH3CO -+ CH3 CO (3) R R + chain end (4)
+
+
* Presented
+ + +
before the Symposium on Anomalies in Reaction Kinetics which was held under the auspices of the Division of Physical and Inorganic Chemistry and the Minneapolis Section of the American Chemical Society a t the University of Minnesota, June 19-21,1950. This paper is ahstracted from a thesis submitted by Paul D. Zemany to the Faculty of the Graduate School of Kew York University in partial fulfillment of the requirements for the degree of Doctor of Philosophy;June, 1950. a Present address: Department of Chemistry, University of Notre Dame, Notre Dame, Indiana.
’
