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 decompositiontakes 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
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The Journal of Physical Chemistty, Vol. 82, No. 18, 1978
the product zone is given by a Slater-type treatment as the mean frequency P which lies between the high and low frequencies which contribute to decomp~sition.~ The final expression for the average rate coefficient ha(/?*) is
ha@*) = 3 Pr(P*)
(4)
which yields upon integration for nondegenerate oscillators (for the case ET >> Eo)
k,(fi*) = ije-Ed* (5) The meaning of eq 5 is as follows: if we were to find a highly excited molecular system where E(excitation) >> E(therma1) then, one can expect the graph of In k,(P*) vs. P* to be a straight line. It should be mentioned that data for homologous series were reported empirically as In ha vs. n where n is the number of carbon atoms in the m ~ l e c u l e . ~ The above discussion indicates that P is a natural parameter to use in as much that it relates to the internal energy content of the molecule.
Examination of the Validity of Pr(p*) We next examine the validity of the expression for Pr(P*) as given by eq 3 and as used in eq 5. To do so we define the average energy of the decomposing molecule ( E )
I. Oref and N. Gordon
TABLE I : Values for the Activation Energy and Frequency Factor from Experimental Results and from This Model ~~~
molecule CF,-c-C,F, D, a C,F,-C-C,F,D,~ C,F, ,-c-C,F,D,” CF,-c-C,H, CF,CH,-C-C,H, CF,(CH ) -cC,HSb2
CH,-C-C,H,~ C,H,-C-C,H,~ C,H, -c-c,H, C,H,
log a (this model)
log A
,
E , , kcal E,, kcal mol (this mol model)
14.8 -15.27‘
~
47 .5a
54
61.4b
68.5
61.4b
83
30.2d
29.5
-14.61‘ 13.1 14.39-13.7
15.6
15-49‘ 14.46 -14.62
12.3 -13’
13.48 11.2-12.1 a Reference 3. erence 4.
Reference 9.
’ Reference 10.
Ref-
Substituting eq 3 into eq 4 and differentiating with respect to P* we obtain, after trivial manipulation and with eq 2 d In h,/dfi* = - ( ( E ) - ( E ) ) (7)
(e)
The difference - ( E ) is Tolman’s definition of the activation energy.8 Therefore, the expression for Pr(p*) and ha@*) satisfy the criterion for the temperature dependence of the rate coefficient.
k,(fi*)of Non-Boltzmann Excited Homologous Series Consider a large polyatomic molecule with s normal modes which decomposes following a delta function of excitation. For example, chemically activated molecules in the alkyl radical series decompose as follows4 H + RCH&H=CH, R + H~CECHCH~
I 01 02 03 04 05 06 07 08 09 I O I I
12 13 14 15
OK”
+
The excitation takes place by the insertion of the H atom into the double bond and the molecule decomposes when sufficient energy is available in the C-C bond. The alkyl tail serves only as a heat sink. The activated complex is identicalll for all members of the homologous series. Therefore, it is possible to form a series of molecules which have practically the same Eo and P but differ by their effective “temperature”. The available experimental results are tabulated in Table I and drawn in Figure 1. It is remarkable how good an agreement is obtained by comparing the results obtained from eq 5 and those reported in the literature. The lines of In k,(P*) vs. (T*)-‘ are straight even for the case of the alkyl radicals which span the range of 33-135 degrees of freedom. The values of Eo agree in all cases, except alkylcyclopropane, to within 10% of the reported values and as good an agreement is obtained for the A factors.
Discussion and Conclusion The physics of the process which is described by the above model can be summarized by the following points: (a) The energy is statistically distributed throughout the molecule. (b) The activated complex is identical for all members of an homologous series. (c) The excited mol-
I/T*xIO-~
Flgure 1. log k, vs. (T*)-’. A. (1) CF3-c-C3H5,(2) CFCH2-c-C3H5,(3) CF3(CH2)Z+C3H,. B. (1) C3H8, (2) CHyc-CSH,, (3) C H ~ C H ~ - C - C ~(4) H~, CH3(CH2)2-C-C3H5. C. (1) CF,-C-C3F,Dz, (2) CF~(CF&-C-C~F~D~, (3) CF&CFZ),-C-C.~~DZ.D. (1) C4Hgl (2)CsH139 (3) C,H15, (4) CaHm (5) CIOHZl, (6)C1ZH253 (7) C14H29! (8) C16H33#
ecules have a “temperature” which is the one of a canonical ensemble of harmonic oscillators. Within this set of assumptions the model has predictive power in as much as one can do “back of the envelope” calculations of the value of k, of a molecule provided it is known for other members of the homologous series. That such a simple expression for k , is expected can be seen if one considers the expression for the unimolecular rate coefficient8 k(E)
k ( E ) = A ( 1 - Eo/E)”’ (8) Taking the In of both sides of eq 8 and expanding In (1 - E o / E ) in series, omitting higher terms since E > Eo, yields In k(E)= In A - (s - l ) E o / E (9) where E / ( s - 1) = RP,,, for a classical harmonic system of identical oscillators. That the A factor is identical for
Relaxation in Micellar Solutions
the series is obvious from the fact that the ratio of the partition functions of the activated complex and the molecule Z+/Z is constant for an homologous series if the assumption of identical activated complex is maintained.
Acknowledgment. This research was supported by the U.S.-Israel Science Foundation. References and Notes (1) (a) S.Chervinsky and 1. Oref, J. Phys. Chem., 79, 1050 (1975); (b) ibid., 81, 1967 (1977). (2) A. N. KO and B. S. Rabinovitch, J . Chem. Phys., 66, 3174 (1977). (3) J. F. Meagher, K. J. Chao, J. R. Barker, and B. S.Rabinovitch, J. Phys. Chem., 78, 2535 (1974).
The Journal of Physical Chemistry, Vol. 82, No. 18, 1978 2037
(4) E. A. Hardwidge, 8. S. Rabinovitch, and R. C. Ireton, J. Chem. Phys., 58, 340 (1973). (5) R. Kubo, "Statistical Mechanics", North-Holland Publishing Co., Amsterdam, 1965. (6) . . (a) M. R. Hoare and Th. W. Buiiarok, J. Chem. Phvs., 52, 113 (1970): . . (b) M. R. Hoare, ibid., 52, 5685 (1970). (7) N. B. Slater, "Theory of Unimolecular Reactions", Cornell University Press, Ithaca, N.Y., 1959, p 119. (8) W. Forst, "Theory of Unimolecular Reactions", Acedemlc Press, New York, N.Y., 1973, p 15. (9) F. H. Dorer and B. S.Rabinovitch, J. Phys. Chem., 69, 1973 (1965). (10) S. W. Benson and H. E. O'Neal, Natl. Stand. Ref. Data Ser., Natl. Bur. Stand., No. 21 (1970). (11) Some frequencies in the complex might be slightly lower due to the increase in the mass of the tail but the effect will be very small for large enough consecutive members.
Chemical Relaxation Studies in Micellar Solutions of Dodecylpyridinium Halides Tohru Inoue,* Ryolchl Tashiro, Yoko Shibuya, and Ryosuke Shimozawa Department of Chemistry, Faculty of Science, Fukuoka University, Fukuoka 8 14, Japan (Received January 16, 1978; Revlsed Manuscript Received June 23, 1978) Publication costs assisted by Fukuoka University
Chemical relaxation measurements in aqueous solutions of dodecylpyridinium halides (DPX: X = I, Br, C1) were carried out by temperature-jump and pressure-jump techniques. Under some fortunate conditions two relaxation processes could be observed by temperature-jump in DPI solutions. Among the three DPX, remarkable differences in the concentration dependence of the reciprocal of slow relaxation time, 72-1, and in the effect of added salt on 72-1 were observed. In DPI solutions T ~ increased - ~ almost linearly with the surfactant concentration, and also increased with the addition of KI. In contrast to DPI, in the case of DPBr and DPC1, 72-1 showed a concentration dependence having a maximum in the pressure-jump concentration range, and decreased with the amount of added salts at low concentrationsof added KBr or KC1, whereas by the addition of salts at high concentration 7c1showed the same behavior as that of DPI. These differences in the behavior of T ~ among - ~ the three DPX were well explained by assuming a single rate-determining step in the micellization-dissolution multiple step process and by introducing a parameter which represents the degree of counterion association on the micellar surface.
I. Introduction the linear dependence of appears when a small amount of long-chain alcohol is added to the original surfactant Since the temperature-jump technique had been applied ~ a m p l e . ~ Now, ~ , ~ Jit~ is known that the relaxation time to the surfactant solution by Kresheck et al.,l chemical associated with the micelle formation-dissolution process relaxation techniques have been extensively used to study is influenced drastically by the presence of surface active the kinetics of micelle formation, and a number of papers impurities. have been published in this field.l-12 Now it is well known Kinetics of micelle formation of dodecylpyridinium salts that in a surfactant solution there exist two relaxation (DPX) were already studied by Lang et ala8and by processes associated with micelle-monomer equilibrium. Hoffmann et aL9using various relaxation techniques. Due The faster one, designated by relaxation time T ~has , been to the facts described above it seems important to reinobserved by ultrasonic absorption or the shock-tube vestigate the relaxation in DPX systems with respect to method. On the other hand, the slower one, designated its concentration dependence. Therefore, we carried out by relaxation time r2, has been observed by temperarelaxation measurements with DPX (X = I, Br, C1) by ture-jump (T-jump) or pressure-jump (P-jump) methods. T-jump and P-jump methods using highly pure samples. Important information about the mechanism of micelle As a result, there were observed remarkable differences formation is obtainable from the concentration dependence in the effect of surfactant concentration and of added of the relaxation times. For the fast process, a linear inorganic salts on 72-1 among these DP salts. In this work dependence of T ~ on - ~the surfactant concentration has been obtained with various surfactant ~ y s t e m s , ~ ~ ~ * ' - lwe l will report the results and also propose a new model for the micelle formation mechanism which explains the whereas for the slow process, inconsistent concentration present experimental results. dependence has been reported by different workers even with the same s ~ r f a c t a n t . ~ ~ ~In* '1966, J ~ J ~Kresheck et al. 11. Experimental Section reported that T ~ obtained - ~ by the T-jump method in dodecylpyridinium iodide (DPI) solution increased linearly Materials. All the surfactants used in this study were with surfactant c0ncentration.l During the following synthesized and purified under carefully controlled conseveral years, linear dependence of 72-1 on concentration ditiohs. was reported for other surfactant system^.^,^^,^ Recently, Dodecylpyridinium Bromide (DPBr). Dodecyl alcohol however, it has become evident by P-jump measurement which was purified by successive distillation was converted with sodium alkyl sulfates that r2-l is almost independent to dodecyl bromide according to the procedure described of the concentration when a very pure sample is used, and in ref 13. After purification the product was identified as 0 1978 American Chemical Society
