BRUCE H.MAHAN
100
Vol. 62
THE NATURE OF COLLISIONAL PROCESSES I N UNIMOLECULAR REACTIONS BY BRUCEH. MAHAN Department of Chemistry, Universily of California, Berkeley, Calif. Received September 3,1967
Application of the Herzfeld theory of vibrational energy transfer to unimolecular reactions indicates that the predominant collisional processes involve small changes in the vibrational energy of a potentially reactive molecule and are not “activation or deactivation” events as in previous interpretations of the Lindemann mechanism. The relatively slight variation in the efficiencies of inert gases in maintaining the first-order ratte is a consequence of these small collisional energy changes. The diqrepancy between the low efficiencies for collisional energy exchange measured in sonic experiments and the high efficiencies usually assumed in unimolecular reactions can be rationalized in terms of this description of the collisional processes in unimolecular reactions.
The Lindemann mechanism for unimolecular reactions is often written as A+M+Ai*+M Ai* M - - - t A M Ai* +products
+
+
Bi*=---
ai
A
bi ci
where A is the active or decomposing species, Ai* is a molecule with vibrational energy greater than the critical activation energy EO,and M is either A or some added inert gas. Most theoretical investigations of unimolecular reactions have been concerned with calculating the specific dissociation probability ci of an excited m0lecule.~-4 In applications of these theories it is often assumed that deactivation of the excited molecule occurs upon every collision and bi is set equal to 2, the bimolecular collision frequency. Sound velocity measurements show,Showever, that as many as lo4 to 106 collisions may be necessary to deactivate a molecule from its first excited vibrational level to its ground state. If a one quantum deactivation process (between two excited vibrational levels) is a true description of the deactivation step in unimolecular reactions, then the assumption of unit collision efficiency obviously is not valid. I t is the purpose of this note to show that this apparent discrepancy can be explained in terms of a current theory of vibrational energy transfer and to indicate the nature of the physical processes which correspond to the rate constants ai and bi. The unimolecular rate constant
can be written in terms of the above mechanism as K = E - aiciM biM
of the collisional processes which are responsible for maintaining the steady state fractional concentration of a species Ai* of energy Ei > EO. It has been common to picture the collisional processes as statistical redistributions of energy between two molecules. That is, the active species Ai* is produced in an event in which a molecule with vibrational energy less than Eo receives additional vibrational energy amounting to as much as several kT from translational energy relative to its collision partner and the possible internal degrees of freedom of the latter. An expression given by Schwartz and Herzfelda shows that the probability P of such a translational-vibration energy exchange decreases sharply as the amount of energy transferred increases.
where In this equation AE is the net energy converted from vibration to translation of the two molecules and p is the reduced mass of the collision pair. Po(a) and Po(b) are steric factors for the two molecules a and b involved, with values in the range ‘/3 to 1. The constant a is related to the u of the Lennard-Jones expression for the intermolecular potential energy
+ ci
where if (i) is taken as specifying the total energy of a molecule, and if the individual energy states are close together, the sum may be replaced by an integral over all energies greater than the critical energy EO. The expression shows that in order to calculate the unimolecular rate constant a t all concentrations we must first find a detailed description (1) L. S. Kassel, “The Kinetics of Homogeneous Gas Reactions,” The Chemical Catalog Co., New York, N. Y., 1932, p. 95. (2) N. E. Slater, Phil. Trans.,A846, 57 (1953). (3) J. C. Giddings and H. Eyring, J . Chem. Phys., 22, 538 (1954). (4) R. A. Marcus, ibid., ao, 352 (1952). (5) F. D. Rossini, Editor, “The Thermodynamics and Physics of kiatter.“ Princeton University Press, Princeton, N. J., 1955, Section H.
aiM biM -I-ci
v
= -4E
-
(y]
by the approximate expression LY = 17.5/u, and E is the Lennard-Jones attractive energy for the collision pair. The vibrational factor V 2 is defined as
where inis the initial quantum number of normal mode n of frequency vn for the molecule possessing N , surface atoms. The factor Asnz/ms is related to the transformation coefficients from normal coordinate n to the Cartesian coordinate of surface (6) R. N. Schwartz and K. F. Herzfeld, J . Chem. Phys., 2a, 767 (1954).
r
Jan., 1958
atom s of mass m,. The calculation of this term has been discussed by Tanczos.' 5Equation 1 successfully predicts the small transition probabilities measured in sonic velocity ex4periments with pure gases where the important transitions are those between the first excited state of the normal mode of lowest frequency and the L.H.S. 3ground state. These molecules in the lowest ex- Eq*3* cited vibrational level have no choice but to lose 2their energy to translation in the relatively large increments corresponding t o one vibrational quantum. In the case of the highly excited mole1cules involved in unimolecular reactions it is not necessary for these large translation-vibration en0ergy exchanges to occur, however. The unique difference between molecules involved in sonic experiments and those involved in unimolecular re-1 actions is that the latter highly excited molecules possess a near continuum of vibrational energy levels.8 The possibility thus exists for a molecule -2OSF, to leave a state with energy Ei and go to another I energy state Ej without converting as much as one -3-
probable on this basis than a simple one quantum deactivation. Rice later showed'o that since less energy is converted to translation, the over-all probability of such a process is relatively high. The following example makes use of the Herzfeld theory to demonstrate that even in a molecule with quite low vibrational frequencies, the complex deactivation (or activation) process is more probable than a one quantum process. The non-linear molecule F20 has vibrational frequencies v1 = 926, v2 = 460 and v3 = 830 cm.-I. If one assumes the Lennard-Jones constants of fluorine to apply here, one can calculate 2 X lod6 for the collisional probability of a simple 1 -c 0 deactivation of the VI mode, while the probability of a collision inducing a simultaneous deactivation of VI and excitation of v3 is 1 X loq2. This complex deactivation has a higher probability than the deactivation of the lowest frequency mode, which is 3X One may thus expect that complex two quantum deactivation processes will be more probable than simple deactivation of one normal mode for polyatomic molecules which have frequencies which differ by less than 200 cm.-l and have no frequencies lower than 400 cm.-'. In simpler molecules with widely separated vibration frequencies three quantum intramolecular processes may become important. Complex processes will become relatively more important as the total energy of an excited molecule increases. Since the harmonic oscillator probabilities ( V 2in equation 1) are proportional to the quantum number i, the oscillator probabilities for an m quantum process will increase with 2, the average vibrational quanF.I. Tanczos. J . Chem. Phys., 26, 439 (1956). (8) See reference 1, pp. 44-46. (9) 0. K. Rice, J . Am. Chsm. SOC.,49, 1617 (1927), (10) 0.K. Rice, {bid., 64, 4558 (1932).
(7)
101
NATURE OF COLLISIONAL PROCESSES IN UNIMOLECULAR REACTIONS
H2 O
0
0
Nzo
0 Q$02
GI,
He
Ne
N2
A
0
'Kr
oXe I
I
1
I
I
I
tum number, as (2)". For the average excited molecule involved in a unimolecular reaction, the probability of a two quantum process will be one order of magnitude greater than indicated by the above calculation which involved the i = 0, 1 levels only. Since all the possible collisional processes which change the energy of a molecule Ai" are parallel, competitive reactions, the rate a t which such molecules are created or destroyed will be regulated by the fastest of these processes. We may therefore discard processes in which large amounts of energy are delivered or removed from an excited molecule as being unimportant in maintaining its steadystate concentration and consider the rate constants ai, bi as representing the fastest processes by which the energy of an excited molecule is changed. The fact that these processes do not correspond to deactivation of an excited molecule in the sense of lowering its energy to less than EO is immaterial, since any change in its energy an excited molecule will have a renewed spontaneous dissociation probability (which may be greater or less than the previous value) and will be effectively starting life afresh as a potentially reactive molecule. The un: certainty involved in choosing an average vibrational quantum number for an excited molecule and the approximate nature of equation 1 for small energy changes prevent exact calculation of the collisional transition probabilities for excited molecules. If P is plotted against AE (equation I ) , and a smooth extrapolation to AE = 0 (resonant case) is made, then by choosing AE = 10 cm.-' (vide infra) and the average vibrational quantum number of an activated molecule equal to five, one can estimate that values for the deactivation rate constant are in the neighborhood of 0.1 of the
102
BRUCEH. MAHAN
bimolecular collision frequency. If these complex multi-quant,um processes are accepted as rate determining collisional processes, certain generalizations can be made concerning the efficiency of various M gases, The efficiency of an M gas will not be markedly sensitive to its vibrational structure because even if an M gas has one vibrational frequency which is identical to a frequency of the excited A molecule, “resonant” transfers of vibrational energy between the excited molecule and the internal degrees of freedom of the M gas are only slightly more probable than the intramolecular two quantum process discussed above. The noble gas atoms will then be as efficacious in energy transfer as complex molecules of similar mass and van der Waals forces. The parent molecule A will be a good M gas due to its ability to effect many different resonant or near resonant intermolecular energy transfers as well as to induce complex internal transitions in the excited gas, but it is possible for molecules with more favorable mass and intermolecular force characteristics to be more efficient M gases than A. The data on relative efficiencies of M gases so far collected seem to agree with these generalizations. 11,12 As pointed out by Rice,ls the small energy change mechanism must lead to a concentrationdependence of the rate constant which is different from the one derived under the assumption that statistical redistribution of energy occurs on each collision. In the former case active molecules must be at least in part produced by collisions involving other active molecules having higher or lower energies, whose concentrations will be more or less depleted by reaction. Difficulties involved in studying unimolecular reactions over large concentration ranges prevent a satisfactory proof of the small energy change hypothesis by analysis of rate constant-concentration behavior. Some evidence can be presented to support the small energy change mechanism, however. Because of the approximations involved in application of the Schwartz-Herzfeld equation to processes in which the vibrational energy change is small one must only expect order of magnitude accuracy in predicting collisional transition probabilities. One might expect to be able to predict relative efficiencies of various M gases, however. Reference to equation 1 shows that if the efficiency of an M (11) A. F. Trotman-Dickenson, “Gas Kinetics,” Butterworths, London, 1955, p. 84. (12) M. Volpe and H. 8. Johnston, J . A m . Chenz. Soc., 78, 3903 (1856).
(13) 0.K.Rice, J . Chem. Phys., 4,242 (1936).
Vol. 62
gas is taken as being proportional to the probability of a vibration-translation energy exchange P , the only significant terms which represent the variation in efficiency for two gases are the exponential and p z in the pre-exponential factor. This approximation is particularly good if the predominating collisional processes are two quantum internal transitions. If Pa is the probability of M gas (a) inducing a transition, and likewise for M gas (b) P. = In -
- gb)
pb
+ ‘E +b 2 In e + const. kT Pb
(2)
If Q is taken to represent the relative efficiencies of two M gases E - Eb In Q - 2 In e = 3 (gb - & + const. kT
(3)
Pb
.
If, as is generally done experimentally, efficiencies are measured relative to the parent molecule A, a plot of the left-hand side of equation 3 vs. the difference between the quantity (pu2)‘/a for the parent molecule and the various M gases should be a straight line of slope 2 X 4 ( ~ ~ ) 2
‘A
(t~~kT(17.6)~)
Figure 1 shows such a plot of the data of Volpe and Johnston12 for the un’imolecular decomposition of nitryl chloride. The approximations involved in such a plot are most extreme for Hr and He. The slope of the line drawn through the points for the heavier gases leads to AE = 12 crn.-I. Although this value has little quantitative significance it does indicate qualitatively that the energy change in an effective collision is small compared to one vibrational quantum. This phenomenon seems to be quite general, since in the cases which have been investigated’ 1,12 the variation of the efficiencies of M gases lie within a factor of 20 for molecules which have non-chemical interactions with the decomposing molecule. This is to be expected if A E is always small. The opposite behavior is often observed in sonic experiments6 where AE is large and certain added gases change vibrational relaxation times by several factors of ten. The result of this analysis of energy transfer data indicates that the theory of unimolecular reactions should be reconstructed after discarding the idea that the rate of production of active molecule? is unaffected by the occurrence of the reaction. In order to do so a theory of energy transfer specifically designed to handle small vibrational energy changes is required.
1
