In the Classroom
Desymmetrization of the Tetrahedron: Stereogenic Centers Paul Lloyd-Williams* and Ernest Giralt** Department of Organic Chemistry, University of Barcelona, Barcelona 08028, Spain; *
[email protected]; **
[email protected] Undergraduate students are taught that a molecule Cabcd, formed by an sp3 hybridized carbon atom bonded to four substituents, a, b, c, and d, is chiral only if all four substituents are different, that is a ≠ b ≠ c ≠ d. If all are identical, a = b = c = d, then the molecule is achiral. Furthermore, if any two substituents are identical then the molecule is again achiral. This is often demonstrated using drawings or molecular models that represent the molecules in question as regular tetrahedra with the substituents at the vertices and by inspecting whether or not the resulting structures are superimposable. Further consideration, however, leads to the realization that the situation is rather more subtle, and perceptive students may well grasp this. The molecule Cabcd can only have the geometry of a regular tetrahedron if all substituents are identical, as represented by the general case CL4. Molecules such as methane CH4 and tetrachloromethane CCl4 can be modeled as regular tetrahedra and are achiral. Other combinations of substituents however, such as a ≠ b ≠ c ≠ d but also a ≠ b ≠ c = d, or a = b ≠ c = d or a = b = c ≠ d give rise to structures that cannot, strictly speaking, be modeled as regular tetrahedra since the magnitudes of the bond lengths and bond angles depend on the different substituents. The first of these cases is that of the stereogenic center but the others all represent achiral molecules. How, then, does the fact that these structures cannot be modeled as regular tetrahedra correspond with the principles of stereochemistry that are commonly taught (1–4) at the undergraduate level? We present the following analysis for the instructors of those students who may have doubts as to the validity of the stereochemical model normally used. We also look at why stereochemistry in molecules of the type Cabcd is almost always analyzed (5) using the regular tetrahedron as model, even though an irregular tetrahedron might be physically more realistic. The Stereogenic Center In the 1870s van’t Hoff recognized (6) that a carbon atom bound to four substituents (Cabcd) might best be modeled by a tetrahedron in which the differences between the substituents were expressed in the form of the tetrahedron itself. The lowered symmetry of this tetrahedron would reflect the substitution pattern of the central carbon atom. van’t Hoff regarded such a model as being more realistic than the model that considered the molecule to have the geometry of a regular tetrahedron with the different substituents at its vertices (or at the centers of the faces, which is equivalent). Mislow and Siegel pointed out (7) that when the angles subtended by the central carbon atom are constrained to remain tetrahedral so that changes in the shape of the tetrahedron are solely the result of changes in bond length, the two models become equivalent. Each of the models can only give rise to structures that belong to the same five symmetry point groups, Td , C3v , C2v , Cs , and C1. This means that for mol1178
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Figure 1. Regular tetrahedral models and the corresponding symmetry point groups.
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Figure 2. Models with tetrahedral geometry around the central carbon atom, but with varying C–substituent bond lengths and the corresponding symmetry point groups.
Journal of Chemical Education • Vol. 80 No. 10 October 2003 • JChemEd.chem.wisc.edu
In the Classroom
ecules of the type Cabcd, modeling them as regular tetrahedra with the substituents at the vertices or modeling them as different irregular tetrahedra having nonequivalent bond lengths to the different substituents gives the same result and adequately explains their stereochemistry. The equivalency can be understood by considering graphical representations or, better, three-dimensional molecular models of the different possibilities for the molecule Cabcd. First, consider modeling it as a regular tetrahedron 1 (Figure 1), which serves as a framework for permutation of the substituents and in which all four vertices are topologically equivalent. All bond lengths are equal, all bond angles have the regular tetrahedral value of 109.47⬚, and the central carbon atom is always located at the center of the tetrahedron. Of the five different general types 2 (where a = b = c = d) through 6 (where a ≠ b ≠ c ≠ d) that are possible, it is only 6, which incorporates a stereogenic center, that belongs to a chiral symmetry point group C1. The others, 2–5, belong to the achiral symmetry point groups Td , C3v , C2v , and Cs respectively. If the bond angles are constrained to remain at 109.47 ⬚ but the bond lengths are allowed to take different (arbitrarily assigned) values depending upon substituent, then new representations can be produced, as shown in Figure 2. Differences in bond lengths have been exaggerated for clarity. When a = b = c = d the achiral representation 2 belonging to the Td symmetry point group again results. Of the representations 7–10, the only one that can be assigned to a chiral symmetry point group C1 is 10, which, like 6 above, incorporates a stereogenic center. The achiral representations 7–9, on the other hand, still belong to the same achiral point groups C3v , C2v , and Cs, respectively, as representations 3–5. Although 2 can be modeled by the regular tetrahedron 1, representations 7–10, must be modeled by the irregular
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tetrahedra 11–14 shown in Figure 3. Each of these tetrahedra belongs to the same symmetry point group as the representation 7–10 that it models: tetrahedron 14 to the chiral symmetry point group C1 and tetrahedra 11–13 to the achiral C3v , C2v , and Cs point groups, respectively. The location of the central carbon atom within each tetrahedron is a function of the bond lengths and bond angles, the latter being fixed at 109.47⬚ in the cases under consideration. If the tetrahedron possesses planes of symmetry then the central carbon atom either lies along the line formed by their intersection or, if only one plane is present, within the plane itself. Removal of the constraint upon the value of the bond angles causes the position of the central carbon atom within tetrahedra 11–14 to change. For those tetrahedra possessing planes of symmetry this atom will still lie along the line of their intersection or within the plane itself if there is only one. However, for the bond angles permissible in organic compounds, the symmetry point groups to which each of these tetrahedra belongs remain unchanged. As a result only five different types of structure belonging to five symmetry point groups, Td , C3v , C2v , Cs , and C1, are possible, irrespective of whether Cabcd is modeled as the regular tetrahedron 1, as tetrahedra that are desymmetrized as a function of bond lengths alone, or as tetrahedra that are desymmetrized as a function of both bond lengths and bond angles (7). Moreover, when substituents are nonidentical it is only when all are different (a ≠ b ≠ c ≠ d) that the molecule is chiral. All other combinations give rise to achiral molecules even though the tetrahedra that model them are irregular. Conclusions Stereochemical issues in the molecules Cabcd, where chirality may be analyzed as being the result of the presence of a stereogenic center, can be fully explained using a model based on the regular tetrahedron 1, even though this model does not take into account different substituents causing the molecule to have different bond lengths and bond angles. While the desymmetrized model may be physically more realistic it leads to the same result as the simpler regular tetrahedron. This makes it preferable to use the regular tetrahedron as the basis for rationalizing stereochemistry at the undergraduate level for organic molecules containing stereogenic centers. However, it is important that students understand that this is a simplification and that a molecule such as chloroform cannot, strictly speaking, be represented by a regular tetrahedron. Literature Cited
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Figure 3. Irregular tetrahedral models and the corresponding symmetry point groups.
1. 2. 3. 4. 5.
Eliel, E. L. J. Chem. Educ. 1964, 41, 73–76. Beauchamp, P. S. J. Chem. Educ. 1984, 61, 666–667. Barta, N. S.; Stille, J. R. J. Chem. Educ. 1994, 71, 20–23. Luján-Upton, H. J. Chem. Educ. 2001, 78, 475–477. Prelog, V.; Helmchen, G. Angew. Chem., Int. Ed. Engl. 1982, 21, 567–583. 6. Ramsay, O. B. In van’t Hoff-Le Bel Centennial; Ramsay, O. B., Ed.; American Chemical Society: Washington, DC, 1975; Vol. 12. 7. Mislow, K.; Siegel, J. J. Am. Chem. Soc. 1984, 106, 3319–3328.
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