On the Validity of Wallach’s Rule: On the Density and Stability of Racemic Crystals Compared with Their Chiral Counterparts

Journal of the American Chemical Society

Volumn 113, Issue 26, 1991, Pages 9811-9820

  Brock, Carolyn Pratt ^a   Schweizer, W Bernd ^b   Dunitz, Jack D ^b  

^a UNIVERSITY OF KENTUCKY   (United States)

^b ETH ZURICH   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000602940     PISSN: 00027863     EISSN: 15205126     Source Type: Journal    
DOI: 10.1021/ja00026a015     Document Type: Article

References (61)

1. Wallach, O. Liebigs Ann. Chem. 1895, 286, 90–143. “Hieraus ist ersichtlich, dass nur bei dem zuletzt gennanten Körper keine Aenderung des Volumes zu bemerken ist. In alien übrigen Fällen findet die Vereinigung der optisch isomeren Körper zu einer krystallisierten racemischen Verbindung unter Kontraction statt.”
2. See especially: Jacques, J.; Collet, A.; Wilen, S. H. Enantiomers, Racemates and Resolutions; Wiley: New York, 1981; pp 23–31.
3. Statements of Wallach’s rule appear in several classic textbooks. Consider the following excerpt from p 574 of Basic Principles of Organic Chemistry by J. D. Roberts and M. C. Caserio (Benjamin: New York, 1964).
4. “The racemic tartaric acid has a noticeably higher melting point and lower solubility than the separate component enantiomers, which means that the racemic acid has the more stable crystal structure. In other words, the 1:1 mixture of enantiomers gives a stronger packing than either enantiomer separately. This is analogous to the observation that right- and left-handed objects usually can be packed in a box better than all right- or all left-handed objects.” In the 1977 edition of the same text, however, the extrapolation from the specific example of tartaric acid to a general rule is absent. Consider also several statements from pp 44-45 of Stereochemistry of Carbon Compounds by E. L. Eliel (McGraw-Hill: New York, 1962). “It may happen that in a crystal each enantiomer has a greater affinity for molecules of the same kind than for molecules of the other enantiomer. … A rather more common situation than that described above is that the molecules of one enantiomer have a greater affinity for those of the opposite enantiomer than for their own kind. … Racemic compounds have lower enthalpies than pure enantiomers.”
5. In this paper we use the term “racemic crystal” to describe any crystal whose space group includes improper symmetry elements (symmetry elements of the second kind), i.e., centers of inversion, mirror planes, glide planes, and alternating (rotation-inversion) axes. Thus we apply the term not only to most racemates—or “racemic compounds” as they have been dubbed (ref 2, p 4)—but also to crystals built from achiral molecules. [The qualifier “most” is needed because a racemate or racemic compound is usually defined as a crystalline addition compound containing both enantiomers in equal amounts. Since a racemate or racemic compound is occasionally found to crystallize in a chiral space group (see ref 14), it is not necessarily a racemic crystal in our parlance. Indeed, such a racemate can occur in chiral and achiral polymorphic forms, as it does in the case of methylsuccinic acid (DLMSUC). According to the definition used here, only one of the two polymorphs of methylsuccinic acid would be a racemic crystal.] In the latter context we would imply the existence of another polymorphic form with a chiral space group. Racemic mixtures of enantiomeric chiral crystals are here referred to as “conglomerates”.
6. Jacques, J.; Collet, A.; Wilen, S. H. Enantiomers, Racemates and Resolutions; Wiley: New York, 1981; p 28.“In other words, the free energy for the ‘reaction’ {Formula Ommited} is almost always negative”.
7. Allen, F. H.; Kennard, O.; Taylor, R. Acc. Chem. Res. 1983, 16, 146–153.
8. Jacques, J.; Collet, A.; Wilen, S. H. Enantiomers, Racemates and Resolutions; Wiley: New York, 1981; pp 94–95.
9. Jacques, J.; Collet, A.; Wilen, S. H. Enantiomers, Racemates and Resolutions; Wiley: New York, 1981; p 29.
10. Δ(%) = 100(ρR − ρA)/[0.5(ρR + ρA)] or, what is equivalent, Δ(%) = 100[(V/Z)A − (V/Z)R]/{0.5[(V/Z)A + (V/Z)R]}, where R refers to the racemate and A to the enantiomer (i.e., the “antipode”), and where ρ is the density, Vis the volume, and Z is the number of formula units in the unit cell. The value of Δ(%) should be significantly positive if Wallach’s rule holds.
11. The numbers in parentheses are the estimated standard deviations of the means.
12. Mason, S. F. Molecular Optical Activity and the Chiral Discriminations; Cambridge University Press: Cambridge, 1982; p 171.
13. For the pair designated Thioph6, the chiral structure has been described briefly (Kwiatkowski, S.; Syed, A.; Brock, C. P.; Watt, D. S. Synthesis 1989, 818–820).
14. a full description of both structures is in press (Acta Crystallogr., Sect. B). Reports of the structures of the P3121 and C2/c forms of ZZZKPE are likewise in preparation (Brock, C. P.; Simpson, G. H.; Fu, Y., 1991).
15. Too late for inclusion in this study was the structure analysis of the elusive anhydrous D, L-glutamic acid (Schweizer, W. B. Unpublished results), which has V/Z = 152.4 Å3 compared with 159.5 Å for the α form (Lehmann, M. S.; Nunes, A. C. Acta Crystallogr. 1980, B36, 1621–1625).
16. and 155.0 Å3 for the β form of L-glutamic acid (Lehmann, M. S.; Koetzle, T. F.; Hamilton, W. C. Cryst. Mol. Struct. 1972, 2, 225–233).
17. The Δ(%) values are then +0.046 (α form) and +0.017 β form). Depending on the conditions, crys tallization of racemic solutions of glutamic acid can yield crystals of the enantiomers, of the racemate as a monohydrate, of the anhydrous racemate, or, often, as mixtures of several forms (Dunn, M. S.; Stoddard, M. P. J. Biol. Chem. 1937, 121, 521–529).
18. This dependence on conditions of crystallization suggests that kinetic, as well as thermodynamic, factors are operative. The thermodynamic transition point between the monohydrate and the anhydrous crystal forms of the racemate are 20.3 °C (Ogawa, T. J. Chem. Soc. Jpn., lnd. Chem. Sect. 1949, 52, 71—72;).
19. (Ogawa, T. Chem. Abstr. 1951, 45, 1860g).
20. Although crystallization in a chiral space group of a racemic com pound formed from resolvable enantiomers is thought to be rare (ref 2, p 17), we came across the following examples (names as given in the CSD): BIJVEV (rac-3, 3′-biindan-l-one, P2121, 21)); CEHBUM (rac-3-methoxy-18-methylestra-l, 3, 5(10)-trien-17-one, P21, 21, 21, ); CUMFEV ((+-)-rel-(lR, 3S, 3′S)-3, 3‣-di-re-butyl-l, l′-spirobiindan, P21, 21, 21, ); DCPENT (D, L-2, 4-dicyanopentane, P41); DLMSUC (D, L-methylsuccinic acid, P21); FAXMIA (charge-trans-2-hydroxy-2-methyl-4-(p-methoxyphenyl)-3, 4-dihydro-2H, 5H-pyrano-[3, 2-c][l]benzopyran-5-one, P21, 21, 21, ); SSESOX (D, L-l, 4-diphenyl-l, 4-dithiabutane 1, 4-dioxide, P212121).
21. McCrone, W. C. In Physics and Chemistry of the Organic Solid State; Fox, D., Labes, M. M., Weissberger, A., Eds.; Interscience: New York, 1965; Vol. II, pp 725—767. According to McCrone, polymorphs are different solids that melt to give the same liquid.
22. The term “molecule” as used here should also be understood to include ions [e.g., dichlorobis(2, 2′-bipyridine)iron(III) (CAVDOS)], sets of ions [e.g., piperidinium 1-piperidinecarbodithiolate (PIPPTC)] and sets of molecules [e.g., D, L-methylsuccinic acid (DLMSUC)], and repeating units in network solids (e.g., the SiO2 unit in quartz).
23. In chiral structures (e.g., quartz, NaC103) built from essentially achiral units, the optical activity is a consequence of the packing arrangement only. Individual crystals containing such building units may be optically active, but once the crystal is dissolved, melted, or sublimed, the optical activity is lost. The sense of the optical rotation of any one crystal depends on accidents of growth. [It has recently been reported (Kondepudi, D. K.; Kaufman, R. J.; Singh, N. Science 1990, 9757—976) that crystallization of stirred NaC103 (aq) solutions produces crops of nearly homochiral crystals.]
24. In phase-rule terminology, two species in chemical equilibrium are considered to form a single component. For example, see: Berry, R. S.; Rice, S. A.; Ross, J. Physical Chemistry; Wiley: New York, 1980; p 869.
25. Strictly speaking, solvated crystals of achiral molecules contain two components, and solvated crystals of resolvable enantiomers contain three. This complication does not, however, affect the phase diagrams in any way that is important to the argument presented below.
26. Only two assignments (BPACUA, a pseudotetrahedral Cu(II) com plex, and CBPACU/BENJAF, a dinuclear complex containing two five-co ordinate Cu(II) ions) presented problems. Both were assigned to group I because Cu(II) complexes usually cannot be resolved (see: Hathaway, B. J. In Comprehensive Coordination Chemistry; Wilkinson, G., Gillard, R. D., McCleverty, J. A., Eds.; Pergamon: Oxford, 1987; Vol. 5, pp 596—619).
27. Although DACWUZ [(η6-mesitylene)(2, 3, 5-η4-3a, 4, 6, 7, 7a-pentahydro-4, 7-methano-lH-inden-5-yl)osmium(II) hexafiuorophosphate; Bennett, M. A.; McMahon, I. J.; Pelling, S.; Robertson, G. B.; Wickramasinghe, W. A. Organometallics 1985, 4, 754—761] is reported to occur in chiral and racemic space groups (space groups P21 and P2/a), we decided to omit it from our tabulations. Both crystal structures have been determined, but, as the authors state, the structure of the chiral form is defined only poorly by the available data; the esd’s of some of the atomic coordinates are greater than 0.05 Å. The reported positions of most of the atoms correspond closely to the centrosymmetric space group P21/m so that there is a possibility that both enantiomers are present, perhaps in different regions of the crystal. Moreover, we do not know whether the enantiomers are isolable (although the molecule does appear to be rigid on the NMR time scale), so it is not clear whether the pair should be assigned to group I or group II.
28. The statistical distributions for both group I and group II samples are so irregular and dependent on accidents of choice (why a particular compound was chosen for detailed crystal structure analysis) that it would be dangerous to draw too dogmatic conclusions from them. In particular, our use of probabilities derived for normal distributions in discussing distributions that deviate very far from normal may be open to criticism, so that our deductions from them need to be viewed with some reservation. Note that in Figures 3 and 4 the distributions for groups I and II are both strongly skewed—and in opposite directions. Thus, although the mean Δ(%) is smaller for group I than for group II, the four largest positive values (8.29%, 7.62%, 7.36%, 5.76%) are all in group I, and the largest negative value (−5.22%) is in group II.
29. Kinetic effects play a role (see, for example, ref 13), but they are unlikely to favor a phase that is thermodynamically disfavored to a large extent. If crystals of the two members of the pair were grown at very different temperatures but studied at similar temperatures, one is almost certainly metastable with respect to the other, but there are only a few such pairs in the list.
30. The second sample (group II) contains many pairs for which the chiral crystals were derived from natural products and the racemic crystals from synthetic material.
31. The melting point curves for the enantiomers were calculated from the simplified Schröder–van Laar equation (see Jacques, J.; Collet, A.; Wilen, S. H. Enantiomers, Racemates and Resolutions; Wiley: New York, 1981; pp 46–47), {Formula Ommited} and the melting point curves for the racemic crystals were calculated from the Prigogine-Defay equation
32. (see Jacques, J.; Collet, A.; Wilen, S. H. Enantiomers, Racemates and Resolutions; Wiley: New York, 1981; pp 374– 375): {Formula Ommited}
33. For polymorphic materials at sufficiently low temperatures, where the entropic term becomes unimportant, the relative free energies of polymorphs must be in the same order as the internal crystal energies. The form that is stable at low temperature tends to have the highest density (Richardson, M. F.; Yang, Q.-C.; Novotny-Bregger, E.; Dunitz, J. D. Acta Crystallogr. 1990, B46, 653–660).
34. Since the entropy of all ordered crystals is zero at 0 K (a statement of the third law of thermodynamics), it follows that the configurational entropy of ordered crystals is zero at all temperatures.
35. Reference 2, p 30. Essentially the same argument is given by Collet, A. In Problems and Wonders of Chiral Molecules; Simonyi, M., Ed.; Akademiai Kiadó: Budapest, 1990; pp 91–109.
36. If the racemic and chiral crystals are equally stable, the melting point of the former is close to the eutectic temperature T of the enantiomers (see Figure 5d), which can be calculated, assuming ideality, from the simplified Schröder-van Laar equation (see ref 25) by setting x = 0.5 so that {Formula Ommited} For crystals having T ≅ 400 K and ΔH ≅ 7 kcal mol, the eutectic temperature 7 is about 30 K lower than T, the melting point of the homochiral crystals.
37. Two of the sets of values given in the table on pp 94-95 of ref 2 are inconsistent and were therefore omitted from the regression calculation. For the racemate of N, N′-bis(α-methylbenzyl)thiourea |ΔH − TΔSfusR| = 0.51 kcal mol, and for the racemate of (trans-exo-1, 5-dichloro-1 l, 12-di(hydroxymethyl)-9, 10-dihydrc-9, 10-ethanoanthracene |ΔfusR − TΔ5fusR| = 0.24 kcal mol. For all other entries in the table |Hfus − TΔSfus| < 0.12 kcal mol.
38. Thermal expansion coefficients of molecular crystals typically correspond to a ca. 1% increase in the crystal volume for a ca. 100 °C increase in the temperature.
39. The estimated standard deviation of the distribution exceeds 2% so that in any given pair the more stable crystal may be the less dense. Certain kinds of interactions, such as hydrogen bonds, may systematically favor more open structures.
40. Few direct comparisons of packing energy and density are available. For the group II pair BAGMOL/BAGMUR, the racemic product, obtained by irradiation of the chiral starting material (Ohashi, Y.; Yanagi, K.; Kurihara, T.; Sasada, Y.; Ohgo, Y. J. Am. Chem. Soc. 1982, 104, 6353–6359).
41. is estimated to have a PPE (packing potential energy) (Gavezzotti, A. J. Am. Chem. Soc. 1983, 105, 5220–5225) of–90 kcal mol compared with −83 kcal mol for the chiral crystal.
42. (Uchida, A.; Dunitz, J. D. Acta Crystallogr. 1990, B46, 45–54). The Δ(%) value is +1.14 (see Table I).
43. Coquerel, G.; Bouaziz, R.; Brienne, M.-J. Chem. Lett. 1988, 1081–1084.
44. Coquerel, G.; Bouaziz, R.; Brienne, M.-J. Tetrahedron Lett. 1990, 31, 2143–2144.
45. Jacques, J.; Collet, A.; Wilen, S. H. Enantiomers, Racemates and Resolutions; Wiley: New York, 1981; p 81.
46. Mighell, A. D.; Himes, V. L.; Rodgers, J. R. Acta Crystallogr. 1983, A39, 737–740.
47. Donohue, J. Acta Crystallogr. 1985, A41, 203–204.
48. Wilson, A. J. C. Acta Crystallogr. 1988, A44, 715–724.
49. Most of these chiral structures are in space group P212121 (11.9%) or space group P2X (6.6%).
50. While the members of a pair of enantiomorphous space groups (e.g., P31, and P32) are nonequivalent for homochiral substrates, they lead to essentially equivalent, isometric structures for an enantiomeric pair of substrates. Thus when crystals are grown from a racemic solution or melt, the number of chiral space groups available is effectively reduced from 65 to 54.
51. Kitaigorodskii, A. I. Organic Chemical Crystallography; Consultants Bureau: New York, 1961; pp 65–112.
52. The commonly held idea (e.g., Nicoud, J. F.; Twieg, R. J. In Non-Linear Optical Properties of Organic Molecules and Crystals; Chemla, D. S., Zyss, J., Eds.; Academic Press: New York, 1987; Vol. I, p 253) that large dipole moments are an important factor leading to centrosymmetry in molecular crystals is refuted by a recent study based on data from the CSD
53. (Whitesell, J. K.; Davis, R. E.; Saunders, L. L.; Wilson, R. J.; Feagins, J. P. J. Am. Chem. Soc. 1991, 113, 3267–3270) and is in any case contrary to elementary physics. When the potential or the electric field at distance r from a molecule is expressed as a series in r, r, r, etc., the series expansion is only valid when r is large compared with the distances within the molecule. This is not the case for interactions between neighboring molecules in crystals.
54. Addadi, L.; Berkovitch-Yellin, Z.; Weissbuch, I.; Lahav, M.; Leiserowitz, L. Top. Stereochem. 1986, 16, 1–85.
55. And certainly the oldest! See Pasteur, L. Ann. Chim. Phys. 1848, 24, 442–459.
56. For a detailed description of Pasteur’s achievement with instructions for repeating his crystallization experiments with sodium ammonium tartrate, see: Kauffman, G. B.; Myers, R. D. J. Chem. Educ. 1975, 52, 777–781.
57. A useful survey of methods of optical resolution by direct crystallization is found in Chapter 4 of ref 2. The method of resolution by entrainment is discussed in detail.
58. The most common chiral space group, P212121, is unambiguously determined by Laue symmetry and systematic absences; the same is almost true of the second most common group, P21, because P21/m is relatively rare as long as the molecules do not contain a mirror plane. The main problem concerns the members of the space group pair P1/Pī, which are not distinguishable by systematic absences alone. The presence of only a single (noncentrosymmetric) molecule in the unit cell would, however, strongly suggest P1.
59. The individual crystals of a conglomerate, although formally in a chiral space group, are not necessarily chirally pure (see: Davey, R. J.; Black, S. N.; Williams, L. J.; McEwan, D.; Sadler, D. E. J. Cryst. Growth 1990, 102, 97–102). The crystals may contain the wrong enantiomer in solid solution, or they may be twinned at the macroscopic or microscopic level.
60. For example, see: Eliel, E. L. Stereochemistry of Carbon Compounds; McGraw-Hill: New York, 1962; p 46.
61. Collet, A.; Brienne, M.-J.; Jacques, J. Chem. Rev. 1980, 80, 215–230.