Self-Association of 2-Pyrrolidinone
The Journal of Physical Chemistry, Vol. 82, No. 18, 1978 2031
Self-Association of 2-Pyrrolidinone. 2. Spectral and Dielectric Polarization Studies of Benzene Solutions Judlth A. Walmsley Department of Chemistry, University of Toledo, Toledo, Ohio 43606 (Received April 17, 1978) Publication costs assisted by the University of Toledo
The self-associationof 2-pyrrolidinonein benzene has been investigated using infrared and Raman spectroscopy and dielectric polarization measurements. For solutions up to 0.5 M this system is best described as an equilibrium between monomer and cyclic dimer solute species. No evidence for the presence of trimers or other higher n-mers was found. Using a computer fit of the dielectric polarization data a dimerization constant of 33 5 M-l and dipole moments of 3.91 f 0.20 D for the monomer and 2.06 f 0.23 D for the dimer were determined. The Raman spectra confirm that the major associated species is a cyclic dimer. The presence of acyclic dimers cannot be ruled out, but on the basis of all the experimental measurements it is concluded that the concentration must be small, if it exists at all in this solvent.
*
Introduction The structure and hydrogen bonding behavior of lactams have attracted the interest of scientists for many years. Lactams have been used as models of the amide group in peptides and the self-association of the cis-lactams serves as a model for the hydrogen bonding of the bases in nucleic acids. A generalized formula for lactams (a), where n = O
H
reinvestigate the dielectric polarization of this lactam in benzene and, in addition, to examine the infrared and Raman spectra for the presence of oligomers higher than the dimer. Huisgen and Walz have studied the dielectric polarization of a series of cis-lactams in benzene, determining the dimerization constant and dipole moments of the species with the assumption of a monomer-dimer models4 Hopmann did not find evidence for trimerization in a dielectric relaxation study of t-caprolactam in benzene, interpreting the data as indicating the presence of monomers and acyclic dimers.8 Dimerization constants for cis-lactams in benzene have also been determined by using infrared,8J1 ebullioscopic,lland vapor phase osmometric12 techniques. The results of this study indicate that up to a concentration of at least 0.5 M the system is best described as an equilibrium mixture of monomer and cyclic dimer. Within the limits of their sensitivity, none of the experimental techniques used gave any indication of the presence of higher polymeric species. The dimerization constant and the dipole moments of the monomer and dimer have been determined using a computer fit to the dielectric polarization data.
ring size, the formula for 2-pyrrolidinone (b), and the structure of its cyclic dimer (c) are shown. For smallmembered ring lactams, n < 9, the cis conformation per~istsl-~ and these lactams undergo self-association in nonpolar solvents to form cyclic dimer^.^ When n > 9 the Experimental Section trans conformation is adoptedlV3and chain oligomers are MateriaZs. 2-Pyrrolidinone was purified by a double formed in nonpolar solvent^.^ A review of hydrogen fractional distillation. The second distillation was carried bonding in lactams has been written by Kedziora and out using a 30-cm heated fractionating column containing co-~orkers.~ Although in 1960 Lord and Porro6 noted that the dia tantalum spiral and the fraction boiling at 86 "C at 1.3 Torr was used for the measurements. The 2-pyrrolidinone merization constant of t-caprolactam in CC14 was not constant for solutions with a concentration greater than was stored in a desiccator and all transfers were done in 0.2 M, it has only been since 1974 that investigators have a glove bag under an N2 atmosphere. considered the possible existence of higher polymeric Reagent grade thiophene-free benzene was further species in these systems. Results of a dielectric polarization purified by drying over sodium ribbon and fractionally and spectrophotometric examination of 2-pyrrolidinone distilling through a 1.2-m column packed with stainless in CC12 and the results of dielectric relaxation s t u d i e ~ ~ ? ~steel curlings. It was stored either over sodium ribbon or of several lactams indicated that, at least in some solvents, the solvent bottle was stored in a plastic bag containing the self-association of cis-lactams does not obey a simple Drierite. monomer-cyclic dimer model. This has been found to be All chemicals and their solutions were exposed to air as true of the self-association behavior of carboxylic acids as little as possible. The procedure for the preparation of the well, which can also form hydrogen bonded cyclic dimers.l0 solutions has been described previ~usly.~ Three individual A study of a lactam in a variety of nonpolar solvents should sets of solutions were used to obtain dielectric polarization indicate if the tendency to form trimers, tetramers, or other data. Each set contained ten solutions and two solvent higher oligomers is a general phenomenon of these commeasurements with solution concentrations ranging from pounds or if their behavior is strongly dependent upon 3 x 10-3 to 5 x 10-1 M. specific solute-solvent interactions. In view of our finding Apparatus. The heterodyne beat method13J4using the that the system of 2-pyrrolidinone in CC14is best described apparatus, glass cell, and procedure described previously' by a monomer-dimer-trimer model7 it was decided to was employed to measure the dielectric constants of the 0022-3654i7ai2oa2-203i$oi .oo/o
0 1978 American Chemical Society
2032
The Journal of Physical Chemistry, Vol. 82, No. 18, 1978
Judith A. Walmsley
TABLE I: Dimerization Constant and Dipole Moments of 2-Pyrrolidinonein Benzene at 25 "C K = 3 3 i 5M-' P,,D
3.91 * 0.20
P2>
6
D
2.06 t 0.23
solutions. The infrared and Raman spectra were obtained as previously described7 except that the 488.0-nm line of an argon ion laser was used as the excitation line for the Raman spectra.
Data Treatment The derivation of the equations and the details of the data fitting procedures are the same as those used for 2-pyrrolidinone in carbon tetrachlorides7 The method of Guggenheim15JGawas employed in the calculation of the dipole moments, assuming that a plot of (t - n2),/(t+ 2)(n2 + 2) vs. concentration could be represented as the summation of contributions from each type of species. For a system containing monomers and dimers in equilibrium the Guggenheim equation becomes to - no2 t - n2 + ( t + 2)(n2+ 2) (€0 + 2)(no2 + 2) where t is the dielectric constant of the solution, n is the refractive index of the solution, p is the dipole moment in esu cm, c is the concentration in moles per mL, N is Avogadro's number, k is the Boltzmann constant, T i s the absolute temperature, and the subscripts 0, 1, 2 refer to the pure solvent, monomer, and dimer, respectively. No correction was made for the atomic p~larization.~ The concentrations of monomer and dimer can be obtained by choosing a value for the association constant, K, and utilizing the relationships: K = c2/ci2 (2) c = ~1 + 2 ~ = 2 ~1 + 2Kci2 (3) The analysis of the dielectric polarization data was done using a linear least-squares fit of the experimental data to that calculated from eq 1. The value of K was varied systematically until a minimum in the sum of the residuals squared was obtained. Only experimental points whose concentrations were less than 0.1 M were used in the calculations because of indications that above 0.1 M the Debye theory, upon which the Guggenheim expression is based, does not hold for this system. Inclusion of data in the concentration range 0.1-0.5 M resulted in the inability to fit the data to any model system or resulted in a physically unrealistic fit (with respect to spectral data and to the dipole moment of 2-pyrrolidinone in CC1: and dioxane7J7). In this case it is thought that the deviations are due to solute-solute interactions which are causing the dielectric polarization to be greater than that predicted by the Guggenheim expression.
Results A plot of the dielectric polarization data of 2-pyrrolidinone in benzene is shown in Figure 1. The solid line represents the curve calculated on the basis of a monomer-dimer model. The values of the dimerization constant and the dipole moments given in Table I were determined by combining the three sets of data (22 points). The deviation accompanying each value was obtained by treating each individual set of data separately and comparing the results with those from the combined data. The
0
2
5
X h
N
+
N
r
v
h
N +
4
(P
C l
0
u 3
2
0 03
0 06
0 09
CONC. ( h l ) Flgure 1. Plot of the dielectric polarization data for 2-pyrrolldlnone In benzene at 25 'C: (0) experimental;(-) calculated.
magnitude of the deviations of the monomer and dimer moments results largely from imprecision in determining the best value of the equilibrium constant rather than from large deviations in the experimental measurements. Since the values of K and the p's obtained from the calculations are dependent upon the shape of the ( E - n 2 ) / ( t 2)(n2 + 2) vs. c plot, should two or more adjacent experimental points have relatively large errors in the same direction, the shape of the curve will be changed and the values obtained for K and the p's can be appreciably different. It is desirable to use a large number of experimental points in order to reduce the probability of this occurring. To obtain an indication of the magnitude of the deviations in the experimental measurements, K was set at 33 M-l for each set of data and the dipole moments were calculated. The deviations for the moments were found to be k0.07 D for the monomer moment and f O . l l D for the dimer moment. Attempts were made to fit the experimental data to a monomer-dimer-trimer and a monomer-dimer-tetramer model. Neither of these models gave a minimum in the sum of the residuals squared and both models were trying to fit a situation in which there were smaller and smaller amounts of the third species present. Thus the dielectric polarization results clearly indicate that up to concentrations of 0.1 M, benzene solutions of 2-pyrrolidinone are best described as an equilibrium between monomer and dimer. Because a planar cyclic dimer possesses C2h symmetry and has no coincidences in its infrared and Raman spectra, a spectral investigation of this system can confirm the presence or absence of a cyclic dimeric species. The infrared spectra of solutions of 2-pyrrolidinone in benzene have been obtained in the carbonyl and N-H stretching
+
The Journal of Physical Chemistry, Vol. 82, No. 18, 1978 2033
Self-Association of 2-Pyrrolidinone
TABLE 11: Infrared Freauencies of 2-Pvrrolidinonein the N-H and C= 0 Stretching Regions
benzene
0.45 3432 3210 3110 1699 0.10 3435 3208 3105c 1710 sh 1701 0.05 3434 3212 -3113c 1714 1700 0.01 3434 3209 1715 1701 0.001 3436 3200 1715 1700 3108 1701 0.29 3450 3210 CS, 0.11 3452 3207 3110 1700 0.011 3453 3201 3108 1715 1701 0.0011 3450 3208 1715 CC1,d 0.27 3461 3213 3110 1698 0.01 3460 3210 3110 1716 1701 cyclohexaned 0.10 3464 3205 3117 1711 0.01 3462 3205 3115 1732 1712 Combination of C=O stretching and N-H in-plane deformation vibrations. Reference 21. Contains ontributions from dimer and oligomer. Solvent interference makes the frequency uncertain. Reference 7.
-
-
-
TABLE 111: Raman Frequencies of Benzene Solutions of 2-Pyrrolidinonein the C=O Stretching Region concn, M
v (C= 0
v (C=0 !&mer,
cm-
cm0.49 1715 1676 0.28 1716 1675 0.14 1712 1670 a This appears as a shoulder on a benzene band at 1701 cm-'.
-
regions and are shown in Figure 2. In both regions the higher frequency band has been assigned to the stretching vibration of the nonhydrogen bonded group and the lower frequency band has been assigned to the stretching vibration of the hydrogen bonded group in the dimeric s p e c i e ~ . ~ *The ~ J ~infrared - ~ ~ frequencies at various concentrations of 2-pyrrolidinone in benzene are listed in Table I1 and the frequencies in CS2,CC14,and cyclohexane at representative concentrations are included for comparison purposes. The v(C=O) region of the benzene solutions shows two absorptions, a band at 1715 cm-l due to the free (nonhydrogen bonded) carbonyl of the monomer and a band at 1700 cm-l which is assigned to the asymmetric stretching vibration, vas(C=O), of the cyclic dimer. The relative intensities of these bands change with changing concentration in the expected manner, but their frequencies remain invariant (within experimental deviation) over the whole concentration range. In contrast, in CC14 solutions above 0.05 M, the v,(C=O) vibration at 1701 cm-l is shifted to 1698 cm-l. This has been interpreted as the result of an increasing concentration of a trimeric s p e ~ i e s . ~ The Raman frequencies of benzene solutions at several concentrations are listed in Table 111. The band at 1715 cm-l is assigned to the carbonyl stretching vibration of the monomeric species. The band at 1676 cm-l is assigned to the symmetric stretching vibration of the carbonyl group, v,(C=O), in the cyclic dimer in accordance with assignments in CC14solution7and in the liquid and solid states.lg This band is polarized in agreement with expectation for a vibration of A, symmetry and it gives conclusive evidence that the major associated species in benzene solution is the cyclic dimer. Models of a cyclic trimer and cyclic tetramer have been made and they have a low degree of symmetry. Thus it seems most unlikely that the observed IR and Raman behavior could result from a cyclic species higher than a dimer. In CCll solutions, three bands are observed in the 1850-1600-~m-~ region of the Raman spectrum. They are a t 1712, 1696, and 1672 cm-l and have been assigned to the monomer, trimer, and dimer species, respectively.'
W 0
z
c I-
t
fz U p:
c
3800
3400
3000
1800
1600
F R E Q U E N C Y (cm-')
Flgure 2. Infrared spectra of 2-pyrrolidinone in benzene: (A) 0.01 M (1.0-mm cell), (8) 0.05 M (1.0-mm cell in N-H region and 0.1-mm cell in C=O region), (C) 0.10 M (0.1-mm cell), (D) 0.45 M (0.1-mm cell in N-H region and 0.025-mm cell in C=O region).
Unfortunately, there is a band at 1701 cm-l in the Raman spectrum of benzene which, although weak, results in solvent interference in the dilute 2-pyrrolidinone solutions. Therefore it is impossible to reach any further conclusions from these data about the possibility of a third species, either an acyclic dimer or a higher oligomer. Discussion Although dielectric polarization and spectrophotometric studies of 2-pyrrolidinonein CC4show that species higher than the dimer are present, and dielectric relaxation
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The Journal of Physical Chemistry, Vol. 82, No. 18, 1978
investigation^^^^ of this and other cis-lactams in cyclohexane suggest similar behavior, we have found that in benzene solutions up to a concentration of 0.5 M, 2pyrrolidinone exists as monomers and dimers, mostly if not completely in cyclic form. It is certainly possible that at higher concentrations higher polymeric species may also exist, but it is not possible to use dielectric polarization techniques at these concentrations without having to consider solute-solute interactions. A comparison of the dimerization constants of 2pyrrolidinone in cyclohexane (1600 M-1),8carbon tetrachloride (118 M-9,' and benzene (33 M-l) gives evidence that specific solute+olvent interactions are very important, even when the solvents being compared are normally considered to be incapable of forming hydrogen bonds. Nozari and Drago have reported that oxygen-containing donor molecules such as amides tend to form aggregates in cyclohexane.22Perhaps cyclohexane can be considered an example of a solvent for 2-pyrrolidinone in which there is little or no solute-solvent interaction. However, in benzene the dimerization constant is very much lower and this is attributed to the interaction of the 7~ electrons with the lactam. lH NMR studies of l-methylla~tams~~ and of 1,3-dimethylpyrimidine~~~ have been interpreted to indicate the solvation of these molecules by benzene or toluene with the formation of a complex in which the solute and solvent rings are in parallel arrangement. Furthermore it is known that aromatic compounds are capable of forming hydrogen bonds with molecules which contain suitable acceptor protons.26 Evidence for a specific interaction between monomeric 2-pyrrolidinone and benzene comes from the frequency of the N-H stretching vibration, which is shifted 15-30 cm-' lower than the frequency in the other solvents listed in Table 11. Klemperer and co-workers observed the same type of effect in benzene solutions of N-ethylacetamide and ascribed it to the interaction of the amide hydrogen with benzene.20 This band in benzene is also noticeably broadened as compared to its width in CS2 and CC4. A slight broadening has also been observed in cyclohexane. An alternative explanation for the broadening is that it could be caused by the presence of acyclic dimers or chain oligomers. At least in benzene the infrared and Raman data from the carbonyl stretching region seem to rule out the presence of chain oligomers and indicate the amount of acyclic dimers would have to be small. In contrast to the monomer frequencies, the infrared and Raman frequencies of the cyclic dimer in both the v(N-H) and v(C=O) regions are virtually identical in benzene, CC4, and CS2, indicating that dimer-solvent interactions in these solvents are either very similar or nonexistent. Three previous determinations of the dipole moment of monomeric 2-pyrrolidinone in benzene gave values of 3-55: 3.7,26and 3.1 D.27 All of these values were determined by a graphical procedure which requires the evaluation of the slope of the tangent t o either the dielectric constant increment or molar polarization curves at infinite dilution and, therefore, contain large uncertainties. The monomer moment of 3.91 D determined in this study using a computer treatment of the data is higher than these and perhaps a little higher than might be expected considering its moment of 3.74 D in CC14 and 3.80 D in d i ~ x a n e . ~ However, it is difficult to predict what effect, if any, interaction with benzene will have on the monomer dipole moment. For 1-methyl-2-pyrrolidinonethe dipole moments in benzene and dioxane are nearly identical, being 4.0926and 4.06 D,17respectively. However, this molecule does not have a hydrogen attached to the nitrogen atom
Judith A. Walmsley
TABLE IV: Summary of Dimerization Constants of 2-Pyrrolidinone in Benzene
K, M-' t, "C experimental method 25.4 f 7.6a 33 f 5 46i: 2 31b 81.7 f 8.6
ref
25 25 25 25 37
dielectric polarization 4 dielectric polarization this work infrared 11 ebullioscopic 11 vapor phase 12 osmometry Converted from mole fraction units. Calculated at 25 "C from the thermodynamic parameters given in the reference.
which could interact with benzene. Devoto28reported the first dipole moment value for 2-pyrrolidinone as 2.3 D in benzene. This is close to the value of 2.2 D determined for the dimer by Huisgen and Walz4 and clearly was obtained from measurements on solutions containing predominantly the dimeric species. The dimer moment of 2.06 D reported here agrees within the experimental deviation with these values. It is considerably higher than the dimer moment of 1.64 D obtained in CC14,but still within the experimental deviations of the two values. A cyclic dimer having the structure as shown in (c) would be expected to have a dipole moment of zero. As discussed in some detail previously,7 there are several possible explanations for the magnitude of the observed dimer moment. One possibility is that the observed moment arises from a large atomic polarization contribution which is uncorrected in the data treatment. This is certainly reasonable16bbecause hydrogen bonds are not rigid and the cyclic dimer will be a large, flexible molecule. A second possibility is that the moment arises from the presence of acyclic dimers which would not be expected to have a zero moment. The presence of appreciable amounts of acyclic dimer can be largely discredited on the basis of the Raman spectra and by the fact that the dimer moments are similar in benzene and carbon tetrachloride. If both acyclic and cyclic dimers are present, the measured dipole moment will be the square root of the weighted average of the two moments squared, and in order for the measured dimer moments to be the same in different solvents, it is necessary that the equilibrium constant, K', for the reaction acyclic dimer cyclic dimer be the same in all solvents. In view of the large differences observed in the overall dimerization constants in various solvents (vide supra) it seems highly unlikely that the values of K' would be unchanged from one solvent to another. The value of the dimerization constant agrees well with most other values which are reported in the literature. A summary of the values and the methods by which they were obtained is given in Table IV. Although one value is very different from the others, it is noteworthy that several different experimental techniques result in similar values of the dimerization constant. This agreement lends added strength to the validity of a monomer-dimer model choice for 2-pyrrolidinone in benzene. Acknowledgment. The author thanks Dr. H. Bradford Thompson for helpful discussions, Ronald P. Hohmann for running the Raman spectra, and Lorri M. Sills for assistance with the computations. Supplementary Material Available: Tables of the concentration,dielectric constant, and refractive index data are available (3 pages). Ordering information is available on any current masthead page.
Calculation of Decomposition Rate of Polyatomic Molecules
References and Notes (1) H. E. Hallam and C. M. Jones, J . Mol. Struct., 1, 413 (1968). (2) C. Y. S.Chen and C. A. Swenson, J. Phys. Chem., 73,2999 (1969). (3) K. L. Williamson and J. D. Roberts, J . Am. Chem. SOC.,98, 5082 (1976). (4) R. Huisgen and H. Walz, Chem. Ber., 89, 2616 (1956). (5) P. Kedziora, J. Jadzyn, and J. Malecki, Wiad. Chem., 29, 347 (1975). (6) R. C. Lord and T. J. Porro, Z. Necfrochem., 64, 672 (1960). (7) J. A. Walmsley, E. J. Jacob, and H. B. Thompson, J. Phys. Chem., 80, 2745 (1976). (8) R. F. W. Hopmann, J . fhys. Chem., 78, 2341 (1974). (9) L. Hellemans and L. De Maeyer, J. Chem. fhys., 83, 3490 (1975). (10) M. A. Goldman and M. T. Emerson, J. fhys. Chem., 77,2295 (1973), and references therein. (1 1) K. Wagner, G. Rudakoff, and P. Frolich, 2.Chem., 15, 272 (1975). (12) G. Montaudo, S.Caccamese, and A. Recca, J. fhys. Chem., 79, 1554 (1975). (13) N. E. Hill, W. E. Vaughn, A. H. Price, and M. Davies, “Dielectric Properties and Molecular Behavior”, Van Nostrand-Reinhold, London, 1969, p 124.
The Journal of Physical Chemistry, Vol. 82, No. 18, 1978 2035
(14) H. B. Thompson, J. Chem. Educ., 43, 66 (1966). (15) E. A. Guaaenheim. Trans. Faraday SOC..45. 714 (1949). (16) J. W. Sm%, “Electric Dipole Mom&ts”, Bt&erworths,‘London, 1955: (a) p 58, (b) p 260-264. (17) C. M. Lee and W. D. Kumler, J. Am. Chem. SOC..83,4593 (1961). (18) H. E. Affsprung, S. D. Christian, and J. D. Worley, Specfrochim: Acta, 20, 1415 (1964). (19) M. Rey-Lafon, M.-T. Forel, and J. Lascombe, J . Chlm. Phys., 64, 1435 (1967). (20) W. Klemperer, M. W. Cronyn, A. H. Maki, and G. C. Pimentel, J. Am. Chem. SOC.,76, 5846 (1954). (21) T. Miyazawa, J . Mol. Spectrosc., 4, 168 (1960). (22) M. Nozari and R. S.Drago, J . Am. Chem. SOC.,94, 6877 (1972). (23) R. M. Morlarty and J. M. Kliegman, J. Org. Chem., 31, 3007 (1966). (24) I. Rosenthal, Tetrahedron Lett., 3333 (1969). (25) G. C. Pimentel and A. L. McClellan, “The Hydrogen Bond”, W. H. Freeman, San Francisco, Calif., 1960, p 202. (26) E. Fischer, J. Chem. SOC., 1382 (1955). (27) G. F. Longster and E. E. Walker, Trans. Faraday Soc., 49, 228 (1953). (28) G. Devoto, Gazz. Chim. Ita/., 63, 495 (1933).
Pseudocanonical Description of Non-Boltzmann Excited Ensemble of Oscillators I. Oref” and N. Gordon Department of Chemisfty, Technion-Israel Institute of Technology, Haifa, Israel (Received May 2, 1978)
The rate coefficient for decomposition of polyatomic molecules is calculated by assuming a canonical ensemble of harmonic oscillators. The dependence of the rate coefficient on the number of modes of a polyatomic molecule in an homologous series is given and an apparent temperature is calculated for each member of the series.
Introduction Some important aspects of unimolecular decomposition such as ergodic and nonergodic behavior of decomposing molecules’-3 are still not settled. Arguments pro and con ergodicity flared anew with the advent of C02laser induced unimolecular decomposition. However, there are other methods of producing nonequilibrium distributions of excited molecules14 which can be used to study inter- and intramolecular energy transfer. The concept which is involved here is the production of an ensemble of excited molecules with a delta function distribution of energy. Stated differently, there is a distribution D(E) such that
D(E)=l/o E - ? < E < E + ?
2 2 where u is the width of the distribution determined by the method of excitation. The methods most often used are chemical activation and photoactivation. In the first method an excited species is formed by an insertion of an H or a CH2 into a double bond forming a species with excess energy obtained by the thermochemistry of the system. In a photochemical excitation, the vibrationally excited molecules in an electronic or ground state (following internal conversion) are formed by absorption of a photon. The distribution function D(E) is of course not a pure delta function since there is a ground state distribution of thermal energies in the parent molecules prior to excitation. An additional small distribution exists in the excitation source, be it in the radical in chemical activation or the spectral distribution in the light source in photoactivation. The two distributions which formally must be convoluted into P(E)can be ignored whenever the mean value of the excitation energy is much larger than the average value of E given by the two distributions. If it is assumed that the excited molecule with an adequate number of oscillators samples all phase space prior 0022-365417812082-2035$0 1.OO/O
to decomposition; that is to say, the molecules is ergodic on the time scale of the experiment, one can assign a “distribution” and a “temperature” to this ensemble of oscillators. These will be used later to derive pragmatic quantities such as the rate coefficient of unimolecular decomposition. Theory The excitation of a molecule by a delta function distribution of energy $(E), as described before, takes place on a time scale short compared to the decomposition of the excited molecule. In addition, the later event is much slower than the intramolecular energy relaxation following the excitation event. In other words the molecule is ergodic in less1Y2than 50 ps while decomposition takes place in the nanosecond range.’l4 The (ergodic) molecule is considered as an ensemble of oscillators with partition function5t6 z = Jmg(E)e-fl*’E dE
g(E) is the density of states at energy E and ,L?* is related to the “temperature” of the ensemble by the expression (derived, for example, by the steepest descent method6) (a In z/afl)p,8* = - ( E ) (2) where ( E ) is the average energy of the ensemble given by the delta function distribution P ( ( E ) )and (P*)-’ = RT*. The probability for an oscillator, in thermal contact with the molecular heat bath, to have an energy above the threshold energy for decomposition is given by (3)
Once the energy is in the reaction coordinate on the potential hypersurface the frequency of the crossings to 0 1978 American Chemical Society
