J. Phys. Chem. 1993,97, 13326-13329
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The Cage Effect and Environmental Isomers in Liquids with Special Emphasis on Water1,* Ernest Crunwald’ and Colin Steel Department of Chemistry, Brandeis University, Waltham, Massachusetts 02254 Received: September 7 , 1993; In Final Form: September 28, 1993@
Environmental isomers are molecules of the same molecular species which are distinguishable because they exist in different liquid cages. A possible example are the two “states” of water which account for many of the water anomalies. This paper derives the conditions for distinguishability when the cages exist in dynamic equilibrium. The approach involves first a Fourier transform and then is analogous to interpreting the coalescence of spectral lines in dynamic magnetic resonance. The analogy exists because cage switching causes a frequency switching of the electronic-state Schroedinger wave of the caged molecule. The result is that the distinguishability of two cage environments (a and b) depends both on the potential energy difference (Ea- Ed of the caged molecule in the two environments and on the mean time 7 for cage switching. Fluctuations in Ea(b) due to random Brownian motions of molecules in the cage wall produce intrinsic widths that are similar to the natural line widths in NMR. Practical criteria for distinguishability are: (i) IEa 4 kJ/mol and (ii) 2 ~ l E a Eb)r/hL > 5 . The two states of water meet these criteria and thus can be environmental isomers.
Introduction The cage effect in liquids plays an essential role in many of the free-radical and CIDNP phenomena that were studied so memorably by Gerhard Clossa3 In this paper we extend the cage concept to include a kind of isomerism that will be called environmental isomerism. Environmental isomers are defined as molecules that belong to the same molecular species but are distinguishable because they exist in different liquid cages, that is, in cages with distinct potential minima separated by a barrier. For liquids with nonspherical molecules, the existence of two or more minima is often predicted by theory. For example, according to a detailed model study,’ the pair potential for two benzene molecules has several minima, the two lowest of which (shown in Figure 1) differ by 0.9 kJ/mol. Ordinarily, the molecules occupying different liquid cages would differ not only in potential energy but also In entropy, because different cages offer different hindrances to libration, and thus might be isomers. However, two necessary conditions must be fulfilled. Because cage configurations interconvert, the rate of exchange must be slow enough to preclude exchange averaging of the thermodynamic differences, and we shall discuss the relevant time scales below. Also, owing to the random Brownian motions of molecules in the cage wall, the potential energy of the caged molecules fluctuates. This “noise” must not be so large as to obscure the differences between the potential energies of cage isomers. In fact, for liquid benzene there is no compelling evidence that environmental isomers exist, and the development given in this paper will rationalize that. Water, by contrast, presents quite a different picture. There is longstanding and consistent evidence, derived from infrared absorption,s7ultrasonics,8the heat capacity,9JOand the dielectric constant,” that water is an equilibrium mixture of two species or “states”. Thisevidenceis now definitive, as a result of accurate isosbestic points seen not only in infrared absorptionsbut also in polarized Raman scattering.12 The enthalpy difference LW between the two states of water is 10 1 kJ/mo1.5-*2 The two states are not conformational or structural isomers-the water molecule is too well characterized for such a theory to be tenable. The states could be distinct hydrogen-bonded complexes or, as we shall show, environmental isomers. These descriptions are not equivalent. In a molecular complex the ligands translate and
*
*Abstract published in Advance ACS Abslrucrs, November 15, 1993.
0 (a) (b) Figure 1. An example of distinct potential minima separated by a barrier: the two lowest pairwise potential minima of benzene, according to Jorgensen and Severance.‘ (a) One molecule is tilted with respect to the ring of the other; E. = -9.7 kJ/mol. (b) One molecule is displaced parallel with respect to the other ring; Eb = -8.8 kl/mol. When the pairwise potential shows discrete minima, it is almost certain that the cage potential will also.
rotate as a single unit, while in a cage, on the time scale of (say) the rotational relaxation time, the substrate molecule and its neighbors move as separate entities. The theory that the two states are two different hydrogen-bonded complexes, or uncomplexed water molecules in equilibrium with a single complex, has long been in dispute because, in the case of hydrogen bonding, one would expect to find many states rather than merely two.I3 Theview that the twostatesareenvironmentalisomers is permitted (although not required) by molecular dynamics simulations. At least one authoritative study14 permits a grouping of the water environments into two categories, with a mean AH of about 10 Idlmol. The two water environments, according to one of us,I1 differ in the number of neighbors to the caged water molecule: four nominally tetrahedral neighbors in one state (\4w) and five neighbors in the other (\Sw). Distinguishability of Environments
It has been stated that liquid cages interconvert and that the cage potential has a finite width due to Brownian motions of the molecules in the cage wall. The questions of distinguishability which arise are at least formally similar to those concerning exchange averaging and natural line width (1/T2) in magnetic resonance, and it will pay us to review some features of the latter. For simplicity, we shall consider two interconverting states (A and B) of equal population and assume, at first, that 1/T2 = 0. Figure 2 then shows a series of NMR absorption lines as a function of the distinguishability index [, defined in eq 1, as calculated from the McConnell-Bloch equations.1s In eq 1, Y. and % are
0022-365419312097- 13326%04.00/0 0 1993 American Chemical Society
Cage Effect and Environmental Isomers in Liquids (b) FFT
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The Journal of Physical Chemistry, Vol. 97, No. 50, 1993 13327
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Figure 2. A triptych view of exchange averaging. (a) Steady-state solutions of the McConnell-Blwh equations in NMR, u, = 40 Hz,6 = 50 Hz, and 1/ 2' = 0. (b) FFT power spectra of linearly polarized wave trainswith stochasticfrequency switching accordingto exponential lifetime distributions. Y, = 40 Hz and 6 = 50 Hz. Values of the mean lifetimes fare thesameas in (a). (c) Sameas (b),but thewave trains arecircularly polarized.
F
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Time Figure 3. Small samples of the kind of wave trains subjected to FFT. The upper wave train yields a power spectrum consistingof two well-resolved lines. The lower wave train, which is on a shorter time scale, yields a single collapsed line. The FFT's of these wave trains are not included in Figure 2b. the resonance frequencies, and 1/ r = 1/ r . + 1/?b, r g and ?b being
the mean first-order (or pseudo-first-order) kinetic lifetimes of the respective states. distinguishability index: 5 = 2?rlvb- vel?
(1)
In Figure 2a, un = 40 Hz and Yb = 50 Hz. As the switching time r decreases, the spectral lines lose distinguishability as they first broaden, then coalesce, and finally collapse into a single sharp line. Exchange averaging of magnetic resonance lines is a wellknown phenomenon. What is less well-known is that exchange averaging is not peculiar to magnetic resonance; it is a general property of waves. For example, Figure 3 (top) shows a continuous plane-polarized wave train in which the frequency switches stochastically between vn and Yb, with an exponential lifetime distribution. The mean switching time is relatively long, and the segments at ve and 3 are well-defined. Figure 3 (bottom) shows a similar wave train but on an expanded time scale, with the same Y'S but with a short T , so that on averagethere are several frequency switches during one period. This wave train resembles a noisy sinusoid at a frequency of (va + Yb)/2. Figure 2b shows the power spectra resulting from fast Fourier transformations (FFT) of such wave trains,161* using the parameters identical to those in Figure 2a. The wave trains used in these calculationsencompass 104-105 frequency switcheswith exponential lifetimedistributions,
and the resulting FFT's are nearly stochastic and smooth. It is clear that the lineshapes in Figure 2a,bare essentiallyequivalent. Essentially identical stochastic FFT power spectra areobtained when the original wave trains are circularly rather than plane polarized. Thus, for a given value of [,the normalized line shapes in Figure 2a-c are all the same. In Figure 2b,c the heights were normalized by applying standard FFT theorems, and the spectra are presented at positive frequencies. The heights of the NMR absorption spectra in Figure 2a were normalized by equating the peak heights at = 4.94 to the mean of the corresponding peak heights in the FFT spectra. Figure 2c will be important to us because Schroedinger stationary quantum waves are circularly polarized. The technique of formulating the problem of frequency switching first in the time domain and then transforming it into the frequency domain provides a comfortable link to chemical kinetics.*g It also permits certain properties of the spectra to be derived from basic Fourier transform theorems.16J7 It thus follows from the modulation theorem16 that the absolute value of Y. and Yb, Le., IYb + Val/& determines the centroid of the power spectrum, while the difference, IYb - u.1, determines its shape. Thus, distinguishabilitydependsonIvb-~~l, asstated byeq 1;theabsolute frequency, 1% + va1/2, is irrelevant. To define conditionsfor distinguishabilityof cage environments, we shall track the electronic energy and the electronic wave function of the caged molecule, at this point neglecting stochastic noise due to molecular motions in the cage wall. The spacing of the librational levels in the cage is dense enough so that the energy per mode is near RT per mole, the classical limit for an oscillator, regardless of the cage environment. The caged molecule remains in its electronic ground state throughout. We shall treat the interaction with the cage as a perturbation and follow the perturbed ground-state energy as the cage environment switches back and forth between \a and \ba2O The problem is simplified because the transit timesZ1in cage switching are long enough so that Ehrenfest's adiabatic principle applies.22 The transit times are on the time scale of atomic displacements or vibrations ( s), while the electronic energy of the electronic ground state responds on the much shorter time scale of electronic motions (leLs s). Accordingly, a caged molecule, on cage switching, arrives in its new cage with the electronic wave function and associated electronic energy essentially in a stationary state. Moreover, the transit time for surmounting the intercage barrier is short compared to the mean time 7 for exchange and thus is short compared to the mean residence times ra and 7b. Our tentative model for the latter is the mean lifetime of an encounter pair, u2/6D, or s at room temperature for a benzene-like liquid; u is the encounter diameter and D the diffusion coefficient. Cage switching may thus be viewed as a sudden transit from one stationary (albeit noisy) wave function to another. The analogy to NMR is imperfect, however, because in NMR the A F? B transit time is muchsmaller than 111%- u.1, while in cage switchingthis inequality is less marked. Instead, cage switching more nearly resembles exchange averaging in the infrared,23 in the sense that the phase of the wave motion is only approximately continuous during frequency switching. The time-dependent equation for a stationary Schroedinger wave is given in eq 2,where e denotes the state energy per molecule
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\ k ( x , t ) = 4(x) exp[-i2a(e/h)t]
(2)
(E per mole), and 4(x) the wave amplitude. Recalling that e x p [-io] = cos 8 - i sin 8, we note that the wave is circularly polarized, with frequency Y = e/h. When the cage switches from state a to state b, the perturbed electronic energy switches from Cn to Cbr and the wave frequency switches from .Y = s,/h (or EJLh) to Vb = cb/h (Or &/Lh). The cage switching occurs with a mean time r and a stochastic, exponential lifetime distribution in each
Grunwald and Steel
13328 The Journal of Physical Chemistry, Vol. 97, No. 50, 1993
state. Neglecting possible differences in wave amplitude, the power spectrum associated with the cage switching is therefore depicted in Figure 2c, with the distinguishability index, in terms of molar energies, given in eq 3. Note that eq 3 involves only the energy difference (Eb- E,l. Absolute values of E, and Eb are not needed.
4 = 27rlEb- E,(r/hL = 1.575 X 10'olE, - EAT
(3)
where E is in joules per mole and T is in seconds. Thus, within this framework, stochastic sudden switching between two cage environments produces concomitant energy switching of the electronic stationary state of the substrate molecule, and the power spectrum of the Schroedinger wave has virtually the same shape, as a function of 4, as does the absorption mode in infrared or magnetic resonance: At high 4 the two state waves are well resolved, at low they are exchange averaged, and coalescence occurs when 4 is of order 1. Adopting a test value of 10-l1sfor~,andt>l,eq3 indicatesthat twocageenvironments will be distinguishable if IE, - Ed > 1/(1.575 X 1Olo X lo--'-'), i.e., >6 J/mol. This is a small energy quantity, whose magnitude is exceeded by most qualitative changes in the nearest-neighbor shell and which therefore gives the impression that environmental isomerism must be common. This impression is false, however, because electronic energies of caged molecules in liquids have "intrinsic widths" owing to Brownian motions of molecules in the cage wall, and these widths greatly exceed 6 J/mol. Distinguishability is retained only if the two peaks in the power spectrum are far enough apart to remain separate in spite of these intrinsic energy widths.
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Energy (kJ/mol) Figure 4. Simulated power spectrum for a hypothetical liquid with two equally populated cage environments whose potential minima differ by 0.9 kJ/mol. (a,b) Without thermal "noise" there are two sharp lines. (c) With thermal "noise" each line broadens with qc) = 2.6 kJ/mol, and the power spectrum coalescts into a single unresolved band. For the sake of realism, the 0.9 kJ/mol value was borrowed from Figure 1, and the 2.6 kJ/mol value is comparable to the thermal "noise" predicted for the cage potential in liquid benzene.
Intrinsic Energy Widths The calculation of the potential energy "noise" due to the Brownian motion of the molecules in the cage wall requires a knowledge of the relevant pair potential and of the effective number, s,of molecules in the cage wall. A paper describing our approach is in preparation. At this point we shall give only an outline. We assume that the cage is spherical and that the s molecules of the cage wall move independently. Only one direction of motion, that normal to the cage wall, is relevant. We adopt a suitable pair potential (such as the Lennard-Jones 6-12 potential and, if appropriate, augment it by electrostatic interactions4) and use it to construct a cage potential by superposition of pairwise interactions. The minimum in the cage potential (which is normally at a greater intermolecular distance than the minimum in the pair potential) defines the cage radius. Using the cage potential, we calculate the amplitude of libration normal to the cage wall when the mechanical excitation energy (potential + kinetic) in that mode equals the mean thermal energy of RT per mole. Given this amplitude, as well as the original pair potential, a straightforward statistical calculation then yields the standard deviation u(2)in the pairwise interaction energy for the molecule in the cage wall and the substrate molecule at the cage center. The total standard deviation U(C) in the potential energy at the cage center, due to all s neighbors, then is U(C) = dsu(z), The number, s,is large enough for the central limit theorem16to work, and the distribution of the potential energy about its mean value at the cage center is therefore approximately Gaussian. In short, the intrinsic line width is q c ) , and the intrinsic line shape is nearly Gaussian. Our estimates of U(C) are approximations, but their accuracy is better than order of magnitude. When the cage radius, and hence s, are relatively small, the cage potential is approximately harmonic at distances of interest, and U(C) = (4/5s)1l2(RT),or about 1 kJ/mol. This result is rather insensitive to the nature of the pairwise potential. As the cage radius (and s) increases, U(C) becomes greater, and at values that might reasonably apply to liquid benzene, up) = 2.6 kJ/mol.
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Energy (kJ/mol) Figure 5. Power spectrum for model water molecules in \4w and \5w environments,as described in the text. (a,b) Without thermal noise there are two sharp lines. (c) With thermal noise each line broadens substantially, but the two lines remain resolved.
Distinguishabilityfor Noisy Cage Environments We found that in the absence of thermal "noise" two cage environments, \a and \b, interconverting at room temperature at typical rates ( l / ~ )are , distinguishable, with 4 > 1, if p a - Ed 1 6 J/mol. However, with intrinsic line widths U(C) of order 1-3 kJ/mol, this condition is much too weak. A reasonable pair of round-number criteria to use for distinguishability is that paEd 1 4 kJ/mol and that 4 must be >5. If this is granted, environmental isomers will be observed only in special cases, such as water, where values obtained for the parameter -Ed are consistently>-'*around 10 kJ/mol and 4 >> 5 . For example, Figure 4 shows the power spectrum for a liquid with two environmental isomers of equal population whose parameters resemble those for benzene; (E, -Ed = 0.9 kJ/mol, 4 = 400, and U(C) = 2.6 kJ/mol. In the absence of noise the power spectrum would consist of the clearly resolved sharp lines (a) and (b). Butowing to thepresenceofnoise, (a) and(b) areintrinsically quite broad and overlap unresolvably. Figure 5 shows a similar power spectrum for water. Here the two states of water are environmental isomers in the ratio 0.7/ 0.3,9J1with IE, - Ebl = 10 kJ/mol. (Since IE, - Ed = A P ,the ratioof0.7/0.3 shownin Figure5 ofcoursealsoreflectsanentropy difference between the two states.) Dielectric relaxation times*' which yield 7 then suggest that 4 = 1000. Cage (a) is the familiar icelike cage \4w (sa = 4), and cage (b) is a less polar cage \5w (sb = 5).l-' To calculate qc), we used the HzO-dipole/cage-
Cage Effect and Environmental Isomers in Liquids dipole potential tested by Grunwaldll and added a LennardJones interaction with UW = 2.125 A and cw = 6.0 kJ/mol, to allow for dispersion forces and repulsion. The results for U(C) were 1.1 kJ/mol for (a) and 1.0 kJ/mol for (b). The sharp lines (a) and (b) represent the power spectrum in the absence of noise. The broadened but, nonetheless, separate lines are in the presence of noise. We may infer that the two states described by this model are distinguishable and that a theory in which the two water states are environmental isomers is tenable.
W e r e These Concepts Can Be Useful Much of Gerhard Closs's scientific work has been very basic in nature. We shall begin by stating, therefore, that the present concepts can add to the precision of basic discussions because they clarify the nature of molecular species in liquids. For instance, they enter discussionsof solvation phenomena by setting criteria for when the molecules in solvation shells should be regarded as distinct from those in the bulk solvent. Thus, in continuum models of liquid environments, environmental isomerism which is imposed by the local cage structure is impossible. The quasi-continua which represent the dense high-energy vibrational levels of most polyatomic moleculesZ5 may be contrasted with the liquid continua that result from the averaging of environmental distinctions. The caged molecule in our model exists in a stationary electronic quantum state whose energy is perturbed by interaction with the surrounding medium. While the perturbation energy switches between two (or several) discrete values, the quantum descriptors of the electronic state of the caged molecule remain fixed. Thus, if thedifferencein theenergy values becomes unresolvable, owing to exchange averaging or thermal %oisen, all distinguishability disappears. By contrast, in vibrational quasi-continua the state energies may overlap, but other quantum descriptors, such as the set of vibrational quantum numbers, retain their distinguishability. The concepts of environmental isomers are also relevant in chemical thermodynamics. As is well-known, when a thermodynamic component is in fact a mixture of distinguishable molecular species, its partial free energy and derived properties includes a term for the entropy of mixing of these species. It is less well-known that when the solvent consists of two (or several) discrete species, the addition of a solute can shift the equilibrium among the solvent speciesand thereby modify the partial enthalpy, entropy, heat capacity, etc., of the solute component.26 In the case of solvent environmental isomers, these equilibria proceed with 1:1 stoichiometry,and the effects of the solute-inducedshifts can be highly significant. For example, entropies and heat capacities of solvation of nonpolar solutes in water deviate dramatically from regular models.26b The entropies can be substantially predicted, however, and the heat capacities rationalized, if the thermodynamic treatment assumes that water is a mixture of two environmental isomers with properties as given by the parameters of the two-state mode1.Z6b While water may be the most remarkable case, environmental isomers may play a significant role also in the solution chemistry of other hydrogenbonding liquids, such as methanol, formic acid, or N-methylacetamide.
The Journal of Physical Chemistry, Vol. 97, No. 50, 1993 13329 Environmental isomers differ from other isomers in chemistry in that for the latter the differentiating constraint is a nuclear motion associated with the molecule, while in the former the constraint is variation in the structure of the surrounding cage. Also, this cage is peculiarly sensitive to conditions. Rabinowitch and Wo0dz7showed that the cage effect only becomes significant when the packing fraction exceeds a threshold value. Thus, even when environmental isomerism is manifest at high fluid densities, it will almost certainly disappear at low densities. One of the arguments that has been used against the two-state model of water is that the phenomena which justify it tend to disappear as water approaches the critical point.28 If the two states were environmental isomers, however, such behavior is to be expected.
Acknowledgment. C.S. thanks the Research Corporation for financial support. References and Notes (1) Dedicated to the memory of Gerhard L. Closs. (2) Presented at the 23rd IUPAC Conference on Solution Chemistry, Leicester University, U.K., Aug 1993. Pure Appl. Chem., in press. (3) Closs, G. L., Ed. In Chemically Induced Magnetic Polarization; D. Reidel: Urbino, 1977. Lepley, R., Closs, G. L., Eds. Chemically Induced Magnetic Polarization; Wiley: New York, 1973. Closs, G. L.; Calcaterra, L. T.;Green, N. J.; Penfield, K. W.; Miller, J. R. J. Phys. Chem. 1986,90, 3673. (4) Jorgensen, W. L.; Severance, D. L. J . Am. Chem. Soc. 1990,112, 4768-74. ( 5 ) Worley, J. D.; Klotz, I. M. J. Chem. Phys. 1966, 45, 2868. (6) Senior, W. A,; Verrall, R. E. J. Phys. Chem. 1969, 73,4242. (7) Angell, C. A. J. Phys. Chem. 1971, 75, 3698. (8) Davis, C. M.; Litovitz, T. A. J . Chem. Phys. 1965,42, 2563. (9) Benson, S.W. J . Am. Chem. Soc. 1978, 200, 5640. (10) Benson, S.W.; Siebert, E. D. J . Am. Chem. Soc. 1992,114,4269. (11) Grunwald, E. J. Am. Chem. Soc. 1986, 108,5719-26. (12) Walrafen, G. E.;Hokmabadi, M. S.;Yang, W.-H. J. Chem. Phys. 1986, 85,6964-69. Walrafen, G. E.; Hokmabadi, M. S.;Fischer, M. R.; Yang, W.-H. Ibid. 1986.85, 6970-82. (13) Nemethy, G.; Scheraga, H. A. J. Chem. Phys. 1%2,36,3382; 1964, 42,680. Falk, M.; Ford, T. A. Can. J . Chem. 1966,44,1699. Wall, T. T.; Hornig, D. F. J. Chem. Phys. 1965,43, 2079. Luck, W. A. P.; Ditter, W. J. Phys. Chem. 1970, 74, 3687. (14) Stillinger, F. H.; Rahman, A. J . Chem. Phys. 1972,57, 1281. (15) McConnell, H. M. J. Chem. Phys. 1958,28,430-31. (16) Bracewell, R. N. The Fourier Transform and its Applications, 2nd ed.; McGraw-Hill: New York 1978;Chapters 6 and 8, p 168. (17) Champeney, D. C. Fourier Transforms and their Physical Applications; Academic Press: London, 1973. (18) Ramirez, R. W. The FFT: Fundamentals and Concepts; Prentice Hall: Englewood Cliffs, NJ, 1985. (19) Grunwald, E.; Herzog, J.; Steel, C. J . Chem. Educ., submitted. (20) Throughout this paper, the symbol 'V denotes 'cage". For example, A(\x) denotes an A molecule in an x cage. (21) The switching time for A s B is the mean time T for exchange, with l / = ~ 1/~. 1/7b. The transit time is the mean time a system takes in surmounting the barrier between state A and state B. (22) Tolman, R. C. The Principles of Statistical Mechanics; Oxford University Press: London, 1938;p 414. (23) Wood, K.A.; Strauss, H. L. J. Phys. Chem. 1990, 94, 5677-84. (24) Hasted, J. B. In Water, A Comprehenriue Treatise; Franks, F.,Ed.; Plenum Press: New York, 1972; Vol. 1, Chapter 7. (25) Gilbert, R. G.; Smith, S.C. Theory of Unimolecular and Recombination Reactionr; Blackwell Scientific Publications: London, 1990;p 309. (26) (a) Grunwald, E. J. Am. Chem. Soc. 1984,106,5414 (correction 1986,108, 1361). (b) Ibid. 1986, 108, 5726-31. (27) Rabinowitch, E.;Wood, W. C. Trans. FaradaySoc. 1936,32,1381. (28) For example, the isosbestic relationship seen in Raman scattering from water12 below 80 OC disappear as the temperature approaches 300 O C . Ratcliffe, C. I.; Irish, D. E. J . Phys. Chem. 1982,86,4897.
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