J . Phys. Chem. 1990, 94, 6509-651 1 "
."
( a ) 1000 rDm
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(b) 400 rpm
Time (Seconds)
5 25
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Time (Seconds) Figure 6. Effect of rotation rate (0.05 M H202. 0.10 M H2S04, 10 mA): (a) 1000 rpm, (b) 400 rpm.
small and one large. Note that the small peaks occur at low values of the potential, in contrast to those shown in Figure 1. The oscillations of the type shown in Figure 5 have been seen by many investigators in several types of systems and thus will not be discussed further here. See, for example, the recent extensive study of Schell et al. and the references therein.*O
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The effect of rotation rate is shown in Figure 6. The top portion (a) shows the potential measured with fl = lo00 rpm. The rotation rate was then decreased to 400 rpm, resulting in the behavior shown in (b). (There is a short transient between parts a and b of Figures 6.) The period has obviously increased although the general shape is unaltered. An increase in rotation rate to 1000 rpm returns the behavior to the oscillations shown in Figure 6a. In general, over the conditions of these experiments the frequency of oscillations increased with increasing rotation rate. This indicates that mass transfer is one of the resistances that control the rate of the electrocatalytic reaction under the conditions of the experiments described in this note.
Concluding Remarks We have shown that oscillatory behavior can occur for the reduction of hydrogen peroxide under galvanostatic conditions on a platinum electrode. Further investigation of this oscillating system appears to be warranted since it involves a relatively simple electrocatalytic reaction which yields interesting dynamic behavior. It is also likely, judging from the complexity of the oscillations reported in this paper, that low-order chaos can be found for some parameter values. Also note that a rich variety of oscillatory behavior has now been found in several types of electrochemical processes; electrochemistry is a fertile field for the applications of theories of nonlinear dynamics. Acknowledgment. This work was supported in part by grants from the National Science Foundation (CBT-8713070) and the Center for Innovative Technology, Commonwealth of Virginia. Registry No. H202,7722-84- 1; H2S04,7664-93-9; platinum, 744006-4.
COMMENTS Response to "Ludwig Boltzmann and the Norbornyi Cation" Sir: Recently, Kramer et al.' have made the novel proposal that molecules, such as the norbornyl cation, which undergo very rapid rearrangements returning to the same chemical structure so as to scramble their atoms (degenerate rearrangement processes), have much greater entropy and therefore more favorable free energy than similar molecules which are not rapidly scrambling. They contend that a molecule with a group of N interchanging atoms would be favored at equilibrium by a factor of N! over a similar but nonscrambling molecule since this is the number of permutations which would be distinct if the atoms could be individually labelled. The purpose of this paper is to show that the above idea is incorrect. The following statements by well-known theoreticians are relevant: "a permutation of indistinguishable objects cannot be regarded as a permutation at all."2a "Proceeding now to consider an assembly of N non-localized identical systems, we may suppose first that these are hypothetically labelled, 1, 2, ..., N . Then any particular microscopic state of the assembly is just one ( 1 ) Kramer, G. M.; Scouten, C. G.; Kastrup, R. V.; Ernst, E. R.; Pictroski, C. F. Ludwig Boltzmann and the Norbornyl Cation. J . Phys. Chem. 1989, 93. 6257. (2) (a) Golden, S. In Introduction to Theoretical Physical Chemistry; Addison-Wesley: Reading, MA, 1961; Chapter on the Maxwell-Boltzmann Method, p 99. (b) Rushbrooke, G. S. In Statistical Mechanics; Oxford University Press: London, 1949; p 38. (c) Gibbs, J. W. In Elementary Principles in Starisrical Mechanics; Yale University Press: New Haven, CT, 1902; Chapter XV, p 187.
0022-3654190 /2094-6509$02.50/0
member of a set of N! such states obtained from this one by permuting the labels 1, 2, ..., N: and these N ! states of the assembly are mutually distinguishable providing that labels are attached. Thus all the distinguishable microscopic states of the hypothetical assembly of labelled systems fall into sets of N! states, the members of any set being mutually distinguishable only on account of the labels artifically attached to the systems. Consequently, if we enumerate the complexions of N identical nonlocalized systems by the expedient of attaching hypothetical labels to the systems, then we count all the truly distinguishable states of the assembly N! times."2b "The essence of statistical equilibrium is the permanence of the number of systems which fall within any given limits with respect to phase. We have therefore to define how the term "phase" is to be understood in such cases. If two phases differ only in that certain entirely similar particles have changed places with one another, are they to be regarded as identical or different phases? If the particles are to be regarded as indistinguishable, it seems in accordance with the spirit of the statistical method to regard the phases as identical."2c The above opinions seem conclusive; however, resort to authority is not the most satisfactory form of scientific argument. It would be better to construct a proof from a commonly agreed on set of first principles; but, the Kramer paper effectively does not accept the above statements which might well be considered as stemming from a fundamental postulate by others. In other words, these authors seem to be starting from different basic ideas then have others before them. Although the authors do not state this directly, the entropies that they predict for various symmetrical and unsymmetrical systems seem to indicate that they do not regard 0 1990 American Chemical Society
6510 The Journal of Physical Chemistry, Vol. 94, No. 16, 1990
protons or carbons as indistinguishable unless there is a symmetry element which carries one into another. Thus, they seem to be saying that interchange of two protons with different N M R chemical shifts, for example, gives a separately countable arrangement, while interchange of two protons with the same chemical shift does not. Even with this disagreement on fundamentals, one can nevertheless construct thought experiments and compare the results predicted using the Kramer conjecture with common universal experience. Such "gedanken" experiments were the subject of numerous enlightening discussions by physicists in the early days of quantum mechanics when no set of postulates were universally accepted. In the case of the Kramer idea, it can be shown that there are grave problems in applying it to a number of other cases. One does not need to consider exotic cases such as the rearranging norbornyl cation or the bullvalene molecule. Any simple gas will do. In a given volume, the molecules are interchanging rapidly so that (according to Kramer's idea), for any geometric arrangement of the molecules, there should be a redundancy factor of N ! . The phase space usually considered for the different positions and momenta of the molecules must be duplicated N! times, if one counts all the permutations of the identical molecules as denumerably different. Putting an identical volume of the same gas alongside the first one gives us an equal situation with a second Kramer permutation factor of N!. If we now remove a partition between the two volumes so that the molecules can pass from one to the other, we would now have 2N molecules which are rapidly interchanging. The Kramer factor for the joint volume is now ( Z N ) ! , which is enormously greater than the ( N ) *factor which we should expect with the partition in place. Application of the Kramer idea would tell us that the arrangement with the partition removed would be very strongly favored (by entropy?). It therefore should be hard to put the partition back. However, common experience tells us that there is no free energy change on either removing or inserting the partition, in strong contrast with the prediction derived from the Kramer conjecture. (Where would the force on the partition come from?) The molecules cannot be regarded as indistinguishable according to the Kramer definition, since no symmetry element allows us to relate the different molecules in the gas and they would instantaneously have different NMR shifts. If the continuous degrees of freedom in a gas are of concern, one can substitute a case where the scrambling atoms are in a crystal. An ice crystal can be appropriately thought of as one giant molecule. It is well-known that the protons in an ice crystal are extremely mobile. Through the extremely rapid chain transfers of ions or through migration of Bjerrum faults, quite rapid scrambling of the protons occurs. One can thus interconvert among all Kramer's N! permutamers rapidly. If one now compares two crystals of a certain size with a single one of twice this size, the situation is exactly the same as with the gas. In the case of the crystals, the sum of the free energies of the smaller crystals is not actually precisely equal to the free energy of the larger crystal equal to them in total weight because the surface area is not the same and there is surface energy. However, the surface energy difference is tiny compared with the enormous predicted Kramer effect. Kramer et al. imply that their entropy effect is a function of rate of scrambling but do not tell us what this function is. If there were a Kramer effect, we would need to know its rate dependence to know when it did and did not apply. If it were independent of the rate, it would always apply to all sets of similar particles in the entire universe giving us monster N! redundancy factors but producing no observable consequences since these factors would always be the same. In the two cases described above, the rate of interchange of molecules of the gas between the two, formerly separate, compartments is a function of the volumes, the gas density, and the temperature and can be made as fast or as slow as we want it to be. In the ice crystal, it is a function of the temperature and whether the crystal has been doped with a tiny amount of acid or not. Again, experience tells us that we do not see any Kramer effect under any conditions.
Comments In the case of any molecule with more than one atom of a given type, there are finite (sometimes very high) barriers for processes which interchange atoms. In other cases, simple group rotation will do it. Isobutane, one of the molecules mentioned in the paper, has nine methyl protons which are scrambling extremely rapidly (but not every permutation can be obtained). There is no single structure which allows all three hydrogens on any methyl of isobutane to be the same. However, rotation of each methyl interchanges nonequivalent hydrogens. (There would be 27 Kramer arrangements.) Rotation of each methyl is a scrambling reaction just as much as any process in norbornyl. Other hydrocarbons scramble to different extents depending on the number of rotations which interconvert different atoms. Are there any experimental entropy differences between isomeric hydrocarbons which can be assigned to this origin? (Experience tells us no.) An interesting situation arises if we have a case where a scrambling process can be catalyzed. (Acid-induced exchange of hydroxyl protons in a polyhydroxy compound is an obvious instance of this kind.) If the process is slow in the absence of the catalyst and fast in its presence, we have the unusual (and very unphysical) result that the entropy and free energy are altered considerably by the addition of a trace of catalyst. If one considers using a solid catalyst which could be readily added or removed, one could construct a heat engine that violates the second law, if the Kramer conjecture were correct. If a Kramer effect occurred for interchanging protons or carbons, it would follow that there should also be such an entropy effect for electrons. It is harder to know whether electrons within a molecule are interchanging and at what rate, since we have no isotopes of electrons. However, we can detect such processes via hyperfine interactions in molecules with unpaired electrons. They occur at a variety of rates. Since there are many more electrons than protons, Kramer effects should be incredibly large. Since these effects would all be in the entropy, they could be distinguished from the usual energy effects involved with electronic structure. No such enormous entropy effects have been seen. From the reduced value given for the "configurational entropy" for the symmetrical nonclassical norbornyl cation compared with that given for the classical cation, as well as from the treatment of bullvalene in the last section of the paper, it is clear that the IV term for exchanging N atoms is modified by Kramer, if there is a group of atoms which are related by symmetry. This leads to a problem. Suppose we have a molecule which is rapidly passing over a very low barrier (as the presumed classical norbornyl would be). If we could alter this barrier, perhaps by changing the solvent, to make it smaller and smaller until it disappeared, the Kramer conjecture would tell us that there would be a sudden (discontinuous) change in the entropy and free energy just at the moment when the barrier went to zero because of the gain of symmetry. This is not in accord with the continuous nature that we expect of thermodynamic functions. Phase changes do involve discontinuities in thermodynamic functions in response to changes in external variables; however, the existence of varying amounts of the phases involved provides a mechanism for this. There is nothing equivalent here. The above discussion has been purely classical, but quantum mechanics does predict the tiniest hint of a Kramer effect. Just as there are always processes that interchange atoms by passage over a barrier in all molecules, we also always have to consider tunneling (often with rates so low that tunneling is unlikely in time periods comparable with the age of the universe). When we do consider tunneling, the time-independent solutions of the wave equation are no longer the structures of the molecule at minima on the classical energy surface, but an equal number of symmetric and antisymmetric superpositions of the wave functions of these minima. These new states differ in energy but only by microscopic amounts in most cases. Even for the few cases with clearly observable very rapid tunneling, like ammonia, the energetic consequences of replacing the classical structures with the correct quantum mechanics combinations are still completely negligible. In the particular case of norbomyl, the two scrambling processes with measured barriers are the 6,2-hydride shift and the 3,2-
J . Phys. Chem. 1990, 94, 651 1-6512 hydride shift. (If the ion were classical, there would also be a Wagner-Meerwein rearrangement with a very much lower barrier.) A series of 6,2-shifts returns to the starting arrangement of the nonclassical ion in a cycle of four steps. (In the classical ion a similar cycle would involve intervening Wagner-Meerweins and would have a length of eight). So, starting with Kramer’s 201 180 672 000 arrangements for the classical ion, we have 25 147 584000 very rapidly interchanging groups of eight arrangements. Going from one group of eight to another requires passage over the 10.8 kcal/mol barrier for the 3,2-shift. A t the highest temperature where Kramer shows a spectrum, -30 ‘C the rate of this process has been measured to be 700 s-I. At this rate it would take 66 years to go to each of Kramer’s arrangements if one visited each one just once. Of course, since the rate process is actually a random one, the motion on the lattice of permuted structures is similar to diffusion and therefore arrangements in the lattice close to the starting point will be visited many, many times before we get to all of them. It therefore will require truly astronomical times to visit them all. If the Kramer conjecture were true, is this fast enough to get the entropy bonus? The simplest way of avoiding all of the problems discussed above is to assume that interchanging particles such as protons leads to no change in entropy at any rate of scrambling. It must also be pointed out that the N M R spectroscopic evidence presented in the Kramer paper is not consistent wiih the authors’s interpretation. The changes in peak areas required as a consequence of the proposed change in the equilibrium constant are not evident from the spectra shown. Even if there had been an observable entropy effect, we would have to consider other possible sources before concluding that there must be a Kramer effect. Equilibria involving ions often show large entropy differences because of differences in solvation of the ions. This possibility was not considered by Kramer et al. However, the novel N! conjecture presented in this paper is of the greatest moment, since it would so fundamentally alter our thinking in so many areas if it were true. Since changing ones way of thinking is an extremely difficult thing to do, it is fortunate that the Kramer conjecture can be demonstrated to be untrue. Comments on the Response of Scouten and Kramer.3 Scouten and Kramer’s reference to the discussion of configurational entropy4 deserves discussion. The examples cited by Denbigh are ones where different groups (A and B) are distributed in various ways among identical sites. This leads to separately countable arrangements. The cases which Kramer et al. discuss (norbornyl, etc.) are, in contrast, ones where identical atoms are distributed among chemically different sites. Another book by DenbighS has a very informative chapter on the history of the concept of indistinguishability. (Boltzmann had it wrong and was corrected by Gibbs.) The suggestion by Scouten and Kramer that another way of interchanging the terminal oxygens in ozone produces a result different from simple rotation is incorrect. Finally, it must be pointed out that Scouten and Kramer do not respond to the examples described above where application of their idea leads to predictions which are contrary to experience. Acknowledgment. I would like to acknowledge many stimulating discussions on this topic with Max Wolfsberg and Eckhard Spohr. Registry No. Norbornyl, 12169-78-7. (3) Kramer, G . M.; Scouten, C. G . J . Phys. Chem., companion paper in this issue. (4) Denbigh, K. G . In The Principles of Chemical Equilibrium, 4th ed.; Cambridge University Press: Cambridge, UK. 1981; pp 49-56. (5) Denbigh, K. G.; Denbigh, J. S.In Entropy in Relation IO Incomplete Knowledge; Cambridge University Press: Cambridge, UK, 1985; Chapter 4.
Department of Chemistry Yale University New Haven, Connecticut 0651 I
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Entropy and Degenerate Rearrangements Sir: In a recent paper in this journal Kramer et al.’ state that molecules which are undergoing very rapid rearrangements which return to the same chemical structure so as to scramble their nuclei (degenerate rearrangement processes) have a greater entropy and therefore a more favorable free energy than similar molecules which are not rapidly scrambling. A closer reading of the paper shows that indeed only scrambling of atoms in nonidentical positions in the molecule is considered and that the authors postulate that the scrambling of N identical nuclei (atoms)among N nonequivalent positions contributes a factor of R In h! to the molar entropy of the molecule. The classical norbornyl carbonium ion C7HII+has seven nonequivalent carbon positions and 11 nonequivalent hydrogen positions. Since the carbons and the hydrogens in this fluxional ion are known to be involved in degenerate rearrangement processes, Kramer et al. then postulate that the classical norbornyl carbonium ion has an extra entropy of R In 7!1 I! by virtue of the scrambling processes and rationalize the known stability of the norbornyl carbonium ion in terms of a scrambling classical structure of the ion without recourse to the nonclassical structure, which is invoked by most workers today to explain the stability of the species. It is the purpose of this paper to show that the postulate of Kramer et al. is wrong, namely, that it contradicts the postulates of quantum mechanicsfstatistics and that the consequences of the postulate are contradictory to experience. Gibbs already recognized that in counting states of a system in order to compute its entropy one could not count states that arise from other states by the exchange of indistinguishable particles; Gibbs was led to this postulate because in the absence of such a postulate the entropy of a system would not be proportional to the size of the system and the entropy of mixing of identical molecules would be nonzero. With the advent of quantum mechanics, it was found that Gibbs’ postulate follows directly from the postulates of quantum mechanics which include the concepts of the BoseEinstein and the Fermi-Dirac statistics (vide infra). Before proceeding further, it is of value to emphasize where the postulate made by Kramer et al. is in conflict with the postulate of Gibbs. The conflict revolves about the term indistinguishable particles. Apparently, Kramer et al. have decided that nuclei at chemically different positions in a molecule, such as the hydrogens and carbons in a classical norbornyl ion, are not indistinguishable while at the same time they would agree that the two hydrogens in H 2 0 and the three hydrogens in NH, are indistinguishable. In fact, the hydrogens in the classical norbornyl cation are indistinguishable as are the carbons. An example of this concept is very familiar to chemists. Take a IO-electron neon atom for which the ground state is described in the orbital approximation by ( 1 ~ ) ~ ( 2 ~ ) ~ (There 2 p ) ~ are . 10 different atomic spin orbitals here (Le., lsa, Is& 2sa, etc., where a and p refer to spin functions). While one can write IO! products of these spin functions which correspond to the IO! permutations of the 10 electrons among the 10 spin orbitals, only one wave function is permitted by the Fermi-Dirac statistics, namely, the antisymmetric combination of the lo! different spin orbital products which is usually written as a so-called Slater determinant. In the first edition of their classic text, Mayer and Mayer2 already called attention to the fact that the partition function of NH3 is calculated to be the same whether it is regarded as a noninverting molecule of symmetry C,, or an inverting molecule of symmetry D3,,except for the effect of energy shifts which result from the direct interaction between wave functions ”localized” in the two possible minima of the molecule. A comforting confirmation of the conclusion of Mayer and Mayer can be derived by using the methodology for calculating rovibrational energy levels of nonrigid molecules3 from which it is easy to show that
Martin Saunders
Received: October 23, 1989: In Final Form: February 8. I990 0022-3654/90/2094-65l1$02.50/0
( 1 ) Kramer, G . M.; Scouten, S. G.; Kastrup, R. V.; Ernst, E. R.; Pictroski, C. F. J . Phys. Chem. 1989, 93, 6251. (2) Mayer, J. E.; Mayer, M. G. Statisticul Mechunics;Wiley: New York, 1940.
0 1990 American Chemical Society
