J. Phys. Chem. 1989, 93, 6257-6260 quencher, excited triplets are formed via intersystem crossing directly from the singlet excited pyrene. Smaller triplet yields were observed in polar solvents such as acetonitrile than in cyclohexane (b(CH3CN)/(b(C6H12) = 0.58 due to the competitive double photon photoionization. In the presence of I m M PA, the photodissociation process of excited singlet (D-A) radical pair in polar solvent, k i , is the dominant decay process of the excited singlet complex, and a higher relative yield ratio of cationic radical to triplet of (b+/(bT was observed in acetonitrile (2.03). On the contrary, in nonpolar solvents, ki is greatly depressed, and the triplet radical pairs formed by intersystem crossing generate excited triplets by fast back electron transfer reaction. Therefore, the absorption or yield of the triplet is not reduced but increased from 0.078 to 0.106 on adding 1 m M PA in cyclohexane, and a lower (b+/(bT is observed (0.09). The above data support the proposed mechanism. Further studies are still needed for back electron transfer mechanism in the formation of triplet. Conclusion The quenching of excited pyrene by phthalic anhydride in bulk
6257
poly(dimethy1siloxanes) shows that these reactants diffuse rapidly in the studied polymer. The rapidity of the motion is not in agreement with a Stokes-Einstein model of the system. It is suggested that the reactants move rapidly because of their low affinity for the silicone polymer and due to the rapid motion of the C H 3 group “spaces” between the polymer chains. Larger side groups, e.g., phenyl and cetyl, which do not rotate rapidly in the polymer network, lead to much lower reaction rates and diffusion of reactants. Temperature affects the reaction rate and the polymer matrix viscosity in a direct inverse way. Temperature changes both macroviscosity and microviscosity to the same extent, but increasing the degree of polymerization above a certain value increases the macroviscosity rather than the microviscosity. From the view of free volume theory, temperature changes affect the free volume fraction, but changes in molecular weight have little effect above the critical transition molecular weight.
Acknowledgment. W e thank the NSF and IBM Corp. for support of this work. We also thank Dr. J. Kuczynsk of IBM for support and helpful discussions. Registry No. PA, 85-44-9; pyrene, 129-00-0; squalene, I 1 1-01-3.
Ludwig Boltzmann and the Norbornyl Cation G . M. Kramer,* C. G . Scouten, R. V. Kastrup, E. R. Ernst, and C. F. Pictroski Corporate Research Laboratories of Exxon Research & Engineering Company, Clinton Township, US.22 East, Annandale, New Jersey 08801 (Received: July 1 , 1988; In Final Form: December 13, 1988)
The norbornyl cation is a fluxional species capable of rearranging to as many as 7! X 1 I! degenerate configurations. The process has an associated entropy, depending on how many distinguishable structures [those not involving the interchange of identical atoms or groups] are attainable under experimental conditions and this markedly affects the position of hydride-transfer equilibria observable in solution. The equilibration of norbornane with the rert-butyl cation has been observed to be endothermic by 9.6 kcal/mol and be driven toward the formation of the fluxional ion by an entropy change of 34.4 gibbs/mol in the 50-deg span from 193 to 243 K.
Introduction The norbornyl cation, whose structure has been a subject of debate for many years, is known to be fluxionally degenerate] and appears to form with remarkable ease.* The effect of fluxional degeneracy upon its stability has not hitherto been considered, yet this factor appears to be crucial in affecting the thermochemistry of the ion. A theoretical estimate of the entropy associated with the existing degeneracy and an experimental measure of the entropy change in an equilibrium involving the norbornyl ion lead to the conclusion that the hydride affinity of norbornane is about 10 kcal/mol greater than that of isobutane. This difference is comparable to the hydride affinity differences of 9-1 2 kcal/mol, which are typically observed in equilibria between secondary and tertiary carbonium ions from simple alkane^.^,^ Clearly, the hydride affinity of the norbornyl cation is not un( I ) (a) Saunders, M. L.; Schleyer, P. v. R.; Olah, G. A. J . Am. Chem. SOC. 1964, 86, 5680. (b) Holmes, J. L.; McGillivray, D.; Isaacs, N. S. Can. J . Chem. 1970, 48, 2791. (c) Org. Mass Specfrosc. 1974, 9, 510. (2) (a) Mirda, D.; Rapp, D.; Kramer, G. M. J . Org. Chem. 1979,44,2619. (b) Redetermination of this value under improved experimental conditions in the course of the present work led to a minor revision of the value but confirmed the essential conclusions of the earlier work. (3) Arnett, E. M.; Petro, C. J. J . Am. Chem. SOC.1979, 100, 5402, 5408. (4) Brouwer, D. M.; Hogeveen, H. Prog. Phys. Org. Chem. 1972, 9 , 179.
0022-3654/89/2093-6257$0l.50/0
usually low, its ease of formation as inferred by equilibrium measurements being strongly affected by the large entropic changes. A theoretical estimate of the configurational entropy of the ion under conditions where it is undergoing rapid 2,3- and 2,6-hydride and Wagner-Meerwein shifts is readily made by noting that these reactions lead to the total permutation of its 7 carbons and of its 11 hydrogens.lb The number, W,of degenerate and distinguishable configurations a t equilibrium for a long-lived ion (easily realized in superacids about 170 K) is 7! X l l ! = 2 X IO” if it has a classical structure5 as all of its constituent atoms would be expected to be spectrally distinct. This very large number of structures can be thought of as the ratio of the number of microstates of the fluxional entity to those of the static component and represents the thermodynamic probability of the system. It has an associated entropy, S = R In W = 51.7 gibbs/mol, estimable from the inscription on Ludwig Boltzmann’s gravestone.6 [The nonclassical (5) Johnson, C. K.; Collins, C. J. J . Am. Chem. SOC.1974, 96, 2514. We believe that the complete permutations of carbons and hydrogens carried out by Johnson and Collins will explicitly enumerate all of the possible enantiomeric forms of the ion without the need for the mirror operator, G 3 ,which doubles the number of possible configurations. The higher symmetry of the bridged ion would lead to a -R In 2 = -1.38 gibbs/mol reduction of the calculated entropy. Consequently, in the case of the norbornyl ion, W = 7! X 1 I!, rather than the 7 ! X 1 I ! X 2! given by Johnson and Collins.
0 1989 American Chemical Society
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The Journal of Physical Chemistry, Vol. 93, No. 16, 1989
SCHEME I: Degeneracy of the Norbornyl Cation Strongly Affects Hydride-Transfer Equilibria
Kramer et al. t-Bu-
+ 2NbH
I-Bu'
+ 2-NbH 2 I-BuH + 2-Nb'
+ 2-Nb'
2-NCH
Conditions: H(-) transfer is fast, but s13w on the N M R time scale Experimental value: AGO = 2.1 kcal/mol (at -55
t-BUH
(1) OC)'
Theoretical values when all configurational isomers are attainable: A S o = 55.3 gibbs/molb
AHo = 14.2 kcal/mol
"Reference 2b. bThis value is obtained after accounting for the fluxionality and symmetry or all components in the equilibrium. ion would have configurational entropy of 40.7 gibbs/mol, the reduction being caused in part by the symmetry of the ion and in part by the exchange of seven sets of two indistinguishable atoms during the fluxional process.] This unusually large configurational entropy could have an important effect on processes such as hydride-transfer equilibria involving the norbornyl ion (reaction 1). For these reactions the magnitude of the effect is expected to vary with the lifetime of the ion and hence the rates of the reversible hydride-transfer process. Under conditions where the back reaction is relatively slow while the fluxional process i y fast [which obtains in the experiments to be described], the entropy should rise from 0 toward 5 1.7 gibbs/mol as the number of experimentally attainable configurations increases. The large entropy change expected for a relatively long-lived ion would therefore be expected to shift the equilibrium constant to the right. This must reduce the observed free energy change from that normally expected from comparisons of hydride affinities of alkanes yielding less fluxional secondary and tertiary cations. A n alternative way of considering the origin of this entropy component in an N M R experiment is by thinking of a single norbornane molecule which is converted to an ion. Before reverting to norbornane the ion isomerizes to a number of degenerate isomers. The number of distinguishable isomers formed during the round trip, W, then yields the configurational entropy increment. These considerations apply to the present experiments. It is of interest to note that if rates of the reversible hydride-transfer process were to approach or exceed those of the fluxional process, norbornane as well as the ion could be rapidly scrambled and the net configurational entropy change would be expected to diminish toward 0, but this situation is not relevant to the present study. Scheme I summarizes the conditions applicable to this work. At -55 OC in the AIBr,/CH,CI, system, reaction 1 is but slightly disfavored; AGO = 2.1 kcal/mol.2b If the estimated entropy change of reaction 1 is appropriate, the enthalpy change would be 14.2 kcal/mol. This value is about that normally expected for the difference in each of forming secondary and tertiary ions in the gas phase7a while a difference of 10 kcal/mol is usually found in s ~ p e r a c i d s .Thus, ~ ~ ~ theory indicates that the apparent ease of forming the norbornyl ion, inferred from AG, arises not from some structural feature which imparts special stability, but rather from the high entropy of the ion conferred by its exceptionally high fluxional degeneracj. Experimental Section The propriety of this interpretation was probed by determining the reaction entropy and enthalpy in an N M R study of the hydride-transfer equilibrium. A typical 0.5-mL solution containing tert-butyl chloride (0.2 M) and norbornane (0.2 M) and AIBr, (6) (a) In the central cemetery of Vienna. (b) Glasstone, S. Thermodynamics for Chemists, 5th printing; Van Nostrand: Toronto, 1950: p 187. (e) Toda, M.; Kubo, R.: Saito, N. Statistical Physics 1 . Equilibrium Statical Mechanics: Springer-Verlag: Berlin, 1983: p 30. (7) (a) Solomon, J. J . : Field, F. J . Am. Chem. Soc. 1975, 97, 2624. (b) Saunders. M.; Hagen, E. L. J . Am. Chem. Soc. 1968, 90, 2436. (e) Brouwer, D. M.: Mackor, E. L. Proc. Chem. Soc. 1964, 147. (d) Brouwer, D. M. Red. Trar. Chim. 1968. 87, 210. (e) Brouwer, D. M.: Hogeveen. H. Prog. Phys. Org. C'hern. 1972. 9. 179-240.
1111l
'
Figure 1. Hydride-transfer equilibrium as a function of temperature, probed by 'H NMR. As the temperature was raised from -80 to -30 ' C , the production of isobutane caused upfield shifts in the peak due to all the methyl protons in the butyl system (rert-butyl ion + isobutane averaged by hydride exchange). At the same time, intramolecular rearrangements within the norbornyl ion increased, leading to a coalescence of its resonances. Upon cooling from -30 OC back to -80 O C , the peak positions and relative intensities were restored, thereby establishing the observation of an equilibrium which occurs rapidly, yet slowly on the N M R time scale.
(1 .O M) was prepared in CH2Cl2by adding the components to an N M R tube which was cooled to -80 "C. All additions were done in a nitrogen atmosphere. [Caution: These solutions must be kept below -20 "C to avoid potentially explosive halide exchange and oxidative chemistry!] The mixture of components was maintained frozen till minutes before spectra were to be obtained. At this time the samples were warmed slowly and when the solvent had liquefied shaken rapidly, recooled, and inserted into the N M R probe which had been preset a t -80 OC. The samples appear to remain fluid throughout the measurements, although small amounts of solids are occasionally observed on the walls of the N M R tube. A similar sample, containing isobutane and exo-2-bromonorbornane and a series of reference samples containing isobutane, norbornane, tert-butyl cation (from tert-butyl chloride) norbornyl cation (from exo-2-bromonorbornane) were likewise prepared in the AIBr,/CH,CI, solvent system. The resulting solutions were examined at temperatures between -80 and -30 OC by 'H N M R , using a 400-MHz JEOL GX-400 spectrometer.
Results and Discussion The spectra (Figure 1 ) show separate bands for norbornane, the norbornyl cation, and a large temperature-dependent shift of a single resonance representing all the methyl protons of the butyl system. The distinct bands of the norbornyl components means that intermolecular hydride transfer between them or with either isobutane or the tert-butyl cation is slow on the N M R time scale. As temperature is raised there is also a distinct increase in the ratio of norbornyl cation to norbornane indicating that intermolecular hydride transfer is occurring fast enough to provide the equilibrium of reaction 1 although the rates are slow on the N M R scale.
Ludwig Boltzmann and the Norbornyl Cation
"."
Reactants 0
iC,H,,+exo-2.BrNb
2.0 -
z
1.0
-
AGO= 9630-T(34.4)
The Journal of Physical Chemistry, Vol. 93, No. 16, 1989 6259 equilibrium constant and AGO on the temperature. That this is not the case is apparent from this study, which provides an unusual example of Boltzmann's principle operating in a chemical equilibrium containing a very large degeneracy in one of the components. The same approach was used to study the hydride-transfer equilibrium involving tert-butyl and tert-amyl (t-Am) ions (reaction 2). For this reaction, one expects no configurational t-Bu+ + t-AmH e t-BuH
AG*= 90-T(4) -1.0
A 5 'A --
+ t-Am+
(2)
entropy when methyl scrambling is not occurring in the ?-Am+ ion. [Permuting the three methyl groups, two of which are identical, between two distinguishable carbon sites would yield an entropic contribution of 3.6 gibbs/mol but this should not be occurring at the temperatures used.] Experimental results over the range of -80 to -30 OC yielded ASo = 4 gibbs/mol (Figure 2), as expected a much smaller value than that obtained for the tert-butyllnorbornyl equilibrium, and a barely endothermic AH". The validity of the AHo and AS" values obtained in the isobutane/norbornyl ion equilibrium can be tested by the use of a thermochemical cycle. Thus, the enthalpy of the hydride-transfer equilibrium (reaction 1) may be combined with the analogous heats of ionization of tert-butyl chloride and em-2-norbornyl chloride (the latter in SbF5/CH2Cl2a t -55 O C 3 (reaction 4) to estimate the heat of reaction of a hydrogen-chlorine exchange between tert-butyl chloride and norbornane. The enthalpy derived from the solution measurements can be compared with the energetics of this reaction in the gas phase which can be estimated independently (reaction 3). The comparison provides a stringent estimated gas-phase heat at reaction (298 K): t-BuH -32.1 (ref 12)
AHf, kcal/mol:
+
2-CINb -21.8 (ref 13)
-
t-BuC1 -43.8 (ref 14)
+
2-NbH -14.7 (ref 15) (3)
AHo = -4.6 f 2.3 kcal/mol (ref 16) estimated heat of reaction from liquid-phase studies: 2-CINb
+ t-Bu+
-
t-BUCI
+ Nb+
(4)
AHo = 4.43 kcal/mol 2-NbH + f-Bu+ -* t-BuH + Nb+ AHo = 9.6 kcal/mol (at 218 K ) (4 - I ) :
-
2-CINb + t-BuH t-BuCI+ 2-NbH AH" = -5.2 kcal/mol
(1)
(5)
test of the validity of the measured enthalpy change and necessarily of the measured AS" as well. The agreement between the vapor-phase and solution-phase estimates for the hydrogen/chloride interchange [-4.6 f 2.3 vs -5.21 is excellent. The error limits on equilibria measured by N M R are normally less than 1 kcal/mol and there is little reason to believe they are different for determinations of hydride-transfer equilibria. Still, measurements involving highly fluxional components such as the norbornyl cation may be suspected of having larger uncertainties than normal. Determination of AHo and So from a AGO vs T plot in itself is independent of the processes occurring, but if a major factor affecting the entropy change in the system is molecular degeneracy, then this factor must remain sensibly constant for a linear relationship to be found. This appears to be the case in both the tert-butyl/norbornyl and tert-butyl/ tert-amyl systems involved in this study. We do not propose that either the theoretical or experimental entropy value, per se, provides direct evidence of the structure of the 2-norbornyl ion. Indeed, under complete scrambling conditions the entropies of the classical racemic and nonclassical achiral ions should both be very high, differing by only 11 gibbs/mol. However, we find it difficult to reconcile the normal hydride affinity of the 2-norbornyl carbonium ion (vs secondary carbonium ions
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The Journal of Physical Chemistry, Vol. 93, No. 16, 1989
from simple alkanes) with proposals that the unusual stability of the 2-norbornyl ion requires that it have an achiral, resonancestabilized, nonclassical structure. On the contrary, the present results clearly indicate that there is no resonance stabilization. Therefore, we conclude that these results are most consistent with the classical 2-norbornyl carbonium ion. General Treatment of the Entropy of Fluxional Molecules The theory just applied to the norbornyl cation has not to our knowledge been previously used in discussions of the thermodynamic properties of fluxional molecules. This leads to questions about the treatment which deserve further comment. That a Boltzmann type of analysis must be applied to such species is however anticipated by the writings of many authorities including Feynman,’ Denbigh,” Guggenheim“ and Glasstone.” Feynman has succinctly summarized the situation by noting “We measure “disorder” by the number of ways the insides can be arranged, so that from the outside it looks the same. The logarithm of that number of ways is the entropy”, a definition perfectly applicable to the norbornyl ion and all other fluxional molecules, as long as groups which are distinguishable from one another in the static configuration are being interchanged. A conceptual issue which has been raised in discussions of these molecules which deserves consideration is the question of why, if an equilibrium is obtained wherein a fluxional component is (9) Feynman, R. P.; Leighton, R. B.; Sands, Matthew The Feynaman Lectures on Physics; Addison-Wesley: Reading, MA, 1963; p 46-5. ( I O ) Denbigh, K. F. R. S . The Principles of Chemical Equilibrium, 4th ed.; Cambridge University Press: Cambridge, U.K., 1981; 1-17. and Chapters I I and 12. (1 I ) Guggenheim, E. A. J . Chem. Phys. 1939, 7 , 103. (12) Rosenstock, H . M.; Draxl, K.; Steiner, B. W.; Herron, J . T. “Energetics of Gaseous Ions”; National Bureau of Standards, 1977. (13) Sen Sharma, R. B.; Sen Sharma, D. K.; Hiraoka, K.; Kebarle, P. J . Am. Chem. SOC.1985, 107, 3747; especially the reference to a private communication from P. v. R. Schleyer, N. L. Allinger, and Y. Yuh, indicating MM2 force field calculations were made. The error limit for such calculations is typically f 2 kcal/mol. (14) Stull, D. R.; Westrum, E. F.; Sinke, G. C. The Thermodynamics of Organic Compounds; Wiley: New York, 1969. 7 1 5 ) Steel;, W. V . J . Chem. Thermodyn. 1978, I O , 919. (16) A rough estimate of the error in AHo is f2.3 kcal/mol. The major source of error is probably in AH, of 2-CINb. Steele estimates that AHrof N b is known within f0.8 kcal/mol. If the errors in the AH;s of i-C,H,, and f-C,H,CI are each about 0.5 kcal/mol, the probable error in AH should be P = [(0.5)2 (0.5)* (0.8)2 (2)2]05= f 2 . 3 kcal/mol. (17) Glasstone. S. Theoretical Chemistry; Van Nostrand: New York, 1944; p 373.
+
+
+
Kramer et al. present on one side of the expression, are not the properties of this component transmitted to the entities on the other side of the expression so as to cancel the configurational contribution? That they are not so transmitted may be readily seen by considering the possible equilibria of methane with three different Clo hydrocarbons [and hydrogen], n-decane, I-decene, and bullvalene. IOCH, z C,oH22+ 9H2 (6) IOCH, s C I O H 2 + 0 10H2 (7)
+
10CH.q s C,oH,o 15H2 (8) bullvalene Only in equilibrium 8 is there a fluxional component in the system and there is just no way for its configurational entropy to affect in any way the properties of methane or hydrogen. The example emphasizes the fact that the properties of a fluxional moiety are not to be transmitted to the “static” components of an equilibrium. How to estimate the configurational entropy of bullvalene is the next question. The molecule has I O C H units which completely scramble yielding lo! configurational isomers. Of these a small number involve the permutation of identical C H units and should not be counted as distinguishable entities. Thus, the static molecule contains three carbons in a cyclopropyl ring, three equivalent a-carbons, and three equivalent &carbons as well as a C3 axis. The permutation process then requires a correction of -3R In 3! and the symmetry axis leads to further -R In 3 [Le., -R In g ] correction to the molar entropy of the compound which might be estimated by a group equivalent procedure. Note that if bullvalene was to be formed in an equilibrium process, its configurational entropy would in general be time dependent and would only be maximized under conditions where its lifetime was long enough for an average bullvalene molecule to attain all IO! configurations. Lastly we turn to the question of how fluxionality is manifested in equilibrium constants when they are expressed in terms of partition functions of the products and reactants. The partition function of each component always contains a coefficient sometimes called the statistical weight or the molecular degeneracy of the entity.I0 This term is equivalent to the number of distinguishable isomers of each fluxional component. The logarithm of the quotient of the molecular degeneracy of the equilibrium components [in each case corrected for the applicable symmetry of the species] provides the configurational entropy of the system. Registry No. Norbornyl cation, 12169-78-7; norbornane, 279-23-2; t-butyl cation, 14804-25-2.
Editor’s Note. This is a controversial paper. Several distinguished reviewers think that some basic aspects of the interpretation are wrong; but several other distinguished reviewers think the treatment of these same aspects is right. The Editors hope that publication of this paper, and the discussion it will prompt, may prove unusually instructive.
