J . Phys. Chem. 1990, 94, 6512-6514
6512
there is a one-to-one correspondence between the energy levels of a rigid C,, ammonia molecule which is localized in one well and an inverting ammonia molecule (in which the hydrogens are completely scrambled). Thus, there is no simple factor of R. In 2 in the entropy of inverting NH3 relative to that of rigid NH3. It would indeed be awkward if scrambling processes were to lead to simple entropy increases based just on the number of minima available to a molecule since whether we state that the identical nuclei are scrambled or not scrambled in a molecule is mostly a matter of rates; so long as the temperature is above 0 K, all possible scrambling of identical nuclei will occur given an infinite amount of time. It is recognized that Kramer et al. will not accept the above argument on ammonia as a criticism of their postulate since the three hydrogens in ammonia are chemically equivalent so that they also would not apply an extra entropy factor to inverting ammonia. However, the arguments for ammonia apply equally well to identical nuclei exchanging between chemically nonequivalent positions as in formic acid or even in the classical norbornyl cation, as shown in the following argument. For the purposes of the following argument, consider a molecular system that contains ni identical nuclei of type i; the extension to more than one kind of identical nuclei is trivial. The identical nuclei need not be associated with ni chemically equivalent positions in the molecule. If the identical nuclei are numbered, then the potential energy surface for rovibrational motion of the molecular system contains n,! minima of equal depth and equal shape. If the barriers separating these minima are very high compared to kT, it is now shown that the energy levels of the molecule localized in one well are identical and in a one-to-one correspondence with the energy levels obtained by using the nonlocalized wave functions in which the ni identical nuclei are scrambled and the molecule has equal probability of being in each of the ni! wells. Consider the wave function +&) corresponding to a state k for a molecule "localized" in the well j ; indeed, a molecule under observation would correspond to such a nonstationary state for long periods of time if the barrier height between the wells is very high compared to kT; the stationary state wave function must be nonlocalized and have correct symmetry properties with respect to permutation of identical nuclei. Depending on whether the nucleus i is a fermion or a boson, one obtains with use of the group theoretical projection operator3 the following nonlocalized + k functions as the only possibilities
.
n'
I
n;!
for bosons: where the operator P, permutes the numbering of the ni nuclei corresponding to the well j to that numbering of the nuclei corresponding to one of the other wells and (-1)' equals + I or -1 depending on whether the permutation is even or odd. The argument here is the same one that leads to the Slater determinant just mentioned for the ground-state wave function of neon, and indeed the terminology of eq l a is similar to that which is used in constructing the properly antisymmetrized functions (Slater determinants) in orbital theories. If the functions +&) are a complete set for a given well, then the functions d)k are expected to form a complete set for the nonlocalized problem. Thus, for each energy level of the localized well, there exists one delocalized energy level. As long as the barrier between wells is high in energy compared to the energy level being considered, the function C#k will yield exactly the same energy as the energy calculated in the localized problem. As the barrier between the wells is lowered, there will be interactions between the localized wave functions, and the diagonal elements of the Hamiltonian matrix based on the set d)k will change and also there will be off-diagonal matrix elements (which are zero for the infinitely ( 3 ) Bunker, P. R. Molecular Symmetry and Spectroscopy; Academic Press: Orlando, FL, 1979.
0022-365419012094-65 I2$02.50/0
high barrier). Thus, indeed, the values of the energies will change gradually as the barrier is lowered, giving rise to changes in partition functions and thermodynamic properties, but there is no change in the number of energy levels leading to an R In n,! factor in the molar e n t r ~ p y . ~ If one were to accept the postulate in the publication by Kramer et al., one would have to give up quantum mechanical postulates which explain chemical phenomena. Hence, one must reject the postulate of ref 1. Acknowledgment. It is a pleasure to acknowledge stimulating discussions with Professor Martin Saunders. (4) I t is, of course, well-known in quantum mechanics that perturbations (such as a gradual lowering of a barrier) change the energies of levels and can lead to a breakup of degeneracy, but perturbations do not lead to an increase in the number of states in a system. The authors suggest that, if perturbation could change the number of states of a system, then one could generally use that system to build a device that violates the second law of thermodynamics; i t is possible that this conjecture has already been proven.
Department of Chemistry University of California, Irvine Irvine, California 9271 7
Eckhard Spohr Max Wolfsberg*
Received: October 24, I989
Conflguratlonal Entropy of Fluxional Molecules Sir: In response to our recent paper1 on the norbornyl cation, Saunders2 and Spohr and Wolfsberg, have provided critical comments which are contained in this Journal. It should be noted that their comments do not address the major issue of the preceding paper, namely, that the enthalpy change for the hydride-transfer equilibrium between the rert-butyl cation and norbornane was found to be 9.6 kcal/mol, a value which was internally checked by a traditional thermochemical cycle and which indicates that the norbornyl cation being observed was not stabilized by any unusual means, Le., bridging. Instead, their critique is directed primarily at the theory used to rationalize the data, and this depends in large part on an understanding of what is distinguishable and what is not. We have no quarrel with Saunder's leading quote, that "a permutation of indistinguishable objects cannot be regarded as a permutation at a11",4 but differ in how we treat the permutations of atoms that are spectrally distinguishable. We provided important references to the subject, namely, the papers of Denbigh,s Feynman,6 and Guggenheim,' and believe it is appropriate to say a little more at this time. The configurational entropy at issue is the change in entropy of an ion once formed in an equilibrium reaction to that of the single ion which has time to reach a number of degenerate isomers before it is returned to a stable product molecule. For this entropy change to be other than zero we assert that the permutation must be of atoms that are in nonequivalent positions in the ion. We would define equilibrium for the fluxional component as the state ( I ) Kramer, G. M.; Scouten, C. G.; Kastrup, R. V.; Ernst, E. R.; Pictrowski, C. F. J . Phys. Chem. 1989, 93, 6257. (2) Saunders, M. J . Phys. Chem., first of three Comments in this series. (3) Spohr, E.; Wolfsberg, M. J . Phys. Chem., second of three Comments in this series. (4) Golden, S. In Introduction to Theoretical Physical Chemistry; Addison-Wesley: Reading, MA, 196 1 ; Chapter on the Maxwell-Boltzmann Method, p 99. (5) Denbigh, K. F. R. S. The Principles of Chemical Equilibrium, 4th ed.; Cambridge University Press: Cambridge, UK; 1981; Chapters 1 1 , 12, pp 1-17. ( 6 ) Feynman, R. P.; Leighton, R. B.; Sands, M. The Feynman Lectures on Physics; Addison-Wesley: Reading, MA, 1963; p 46-5. (7) Guggenheim, E. A. J . Chem. Phys. 1939, 7. 103.
0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 16, 1990 6513
Comments in which it has a lifetime sufficient for it to reach all of its degenerate isomers and is conventionally taken as the state in which its entropy and that of the system have been maximized. An outline of the theoretical treatment of such a system follows from a very brief review of Denbigh’s discussion of the partition function of a perfect gas. (We use his nomenclature in what follows and refer to Chapter 12 as being particularly pertinent to our paper.) The entropy of a gas containing N molecules of the same kind may be written as R In Q where Q is the partition function of the system. Q itself may be expressed as the product of the partition functions of the N molecules
Q = (I/N!)Y”
( 1 2.8)
f = xwie-dkT
(1 2.9)
where The symbol f denotes the molecular partition function, and wi is the molecular degeneracy. In ( 1 2.8) division by N! allows for the indistinguishability of the N degenerate molecules. One notes that the molecular partition function may be factored to a product of the translational and internal quantum states of the molecule, f =p’P(12.25). Letting w y and wkintbe the degeneracies of thejth translational level and the kth internal level, the energy, t i , and degeneracy, wi, of any energy state may be written as ti
=
t.lr i
+
k
int
and wi = w.“wpt J
(12.23)
The degeneracy follows from the fact that any of the w y states of energy ti” may be combined with any of the wkintquantum states to give a state of a given energy t i . The partition function of the collection of degenerate molecules may be written as
This expression may be factored into the product of its translational and internal components. The configurational entropy of a system of degenerate molecules follows directly as essentially the R X In Wkin‘ component of R In Q. We identify the wkintterm with the number of distinguishable isomers present in our treatment of the norbornyl ion. Our procedure may perhaps be better appreciated by considering the way a small series of fluxional molecules containing varying numbers of “identical” and “nonidentical” but permuting groups are to be treated. We note at the outset that if we did not treat atoms in chemically different environments as distinguishable and followed otherwise accepted procedures for estimating the entropies of the fluxional entities, we would be led to the unacceptable thermodynamic consequence of predicting the entropy of a system containing fluxional molecules to be less than it would be if the molecules were static. This problem disappears once these atoms are accepted as distinguishable. To aid discussion Table I is provided. It illustrates the entropy changes expected for a series of fluxional molecules. They are ozone (hypothetically assuming that the terminal oxygens can be interchanged), PFS, bullvalene, and the norbornyl cation. The second column lists the entropy change expected for the permutation of identical atoms or groups within the molecule, and the third lists the entropy change expected for the formation of degenerate configurational isomers (Le., the number of formally identical molecules in which however the connectivity or ordering of the atoms has been changed). The last column contains the sum of both contributions. Note that in molecules like PF, and bullvalene, whenever they are present in a bimolecular equilibrium reaction, their entropic contribution to the system must depend on their lifetime. The entropic terms listed in Table I are corrections to the entropy of the listed molecules to provide a proper counting of the number of their distinguishable configurations. The second
TABLE I: Entropy Changes of Fluxional Molecules molecule 03‘
PF5 bullvalened, C,oH,o norbornyl+
PSI0 -R In 2! -R In 2! + -R In 3! R In 3! + -R In 3! + -R In 3! + other (-) terms 0
PS,b
+ R In 2! + R In 5 ! + R In IO! + R In (7!1 I!)
PSI
+ hs, 0 4.6
