Tetrahedral Geometry Made Simple A. A.Woolf University of the West of England, Frenchay Campus, Coldharbour Lane, Bristol, England BS16 1QY Glaister has added two rather complex derivations of the (27anele to the orevious eieht recorded in this tetrahedral ~~~-~~~~~ Journal (1).kealikcally if student c-ot manage to remember the 109.5' anele the shorter and more concrete a derivation the better.yhus, if alternate corners A, B, Q and Y of a unit cube are sliced perpendicular to body diagonals until the Tedges marked in Figure 1 emerge (other edges are omitted for clarity) the Tangle is 2 tan-' *from insoection of the shaded triangle. All derivations, of ~
a
ing an expl&ation of why four-fold coordination should be almost always tetrahedral rather than square or some inbetween geometry. Deriving this preference is likely to be more instructive for a student than just obtaining an angle from a pre-existing symmetry. The shape itself does not arise from sp3or d3s hybridization (2)as some believe, because Tsymmetry is used to generate the requisite combination of orbitals. However, the shape appears from the energy minimization of four points, or electron pairs in Gillespie's VSEPR model (3), repelled by some inverse power law over the surface of a circumscribing sphere. Because there is no simple algorithm for minimization, it has to be found by systematic movement of the points over the surface (4). A less mathematically demanding explanation can be based on close packing or maximum space filling. It starts with four spheres of unit radius in a square array leaving a central space for a sphere of radius 0.225. Skewing the array as in Figure 2 leaves two trigonal holes to fit two spheres of radii 0.155 and a smaller total volume. Finally, when one sphere is placed on top, the spare volume is
Figure 1 . Short derivation of the tetrahedral angle
J?
Figure 2. Tetrahedron formed by close packing
Figure 3. Tetrahedron formed by ring closure.
Figure 4. X-section of tetrahedron containing C, and C3axes. Edge along AB, centroid at C, FC= CE, BC = 3 x CD. halved to a minimum value and a tetrahedron lying on a face results. Rolling about the axis shown, re-orients it to the position in Figure 1. Tetrahedral shape generation from mono- and di-cyclo shapes is nicely illustrated in Maier's work on tetrahedrane hydrocarbons (5). Only the t-butyl derivative has been isolated with a central Tskeleton of four carbons each subtending 60' angles instead of the normal 109.6. The skeleton is maximally strained a s expected fmm the old Baeyer strain theom Interestindv. the more stable rectaneular cvclobutadieLe and the t&khedrane fonns can be ikerco&erted. O ~ w s i t edees e need a 90'twist and bonds need miss-crossinr! (fig. 3).The'josure ofwing-shapeddi-cycle (110)butanes has yet to be achieved. F o l k to a dihedral angle of 70.5 is required to make all faces equilateral. The tetrahedron's geometry can be enumerated using the close-packing approach of Figure 2. The center of mass (centroid) in orientation A is obviously mid-way between opposite edges formed by the 2 + 2 sphere array; whereas, in orientation B with a 3 + 1array, the centroid is nearer the set of three than the single sphere in the ratio 1:3. Thus, the B centroid lies a t 1/(1 + 3) of the height. This corresponds with the 114 levels of T-holes between closepacked layers (6).Examination of more of the shaded plane in Figure 1enables other characteristic angles to be evaluated. This plane contains one Czand two Cp axes (Fig.4 ) and is a ~ s bthe bisector plane inthe wing structure of R ~ a r 3. e The sections oftwo of the four cube truncations are shown by the dotted areas. These fragments make up Volume 72 Number 1 January 1995
19
two tetrahedra and hence the cut-out tetrahedron is one third of the cube volume. The dihedral angle between faces is given by AEB the angle subtended by an edge because Cz and S4axes are coincident. The external 144.7C is the angle the substituents make with three edges of a tetrahedrane. One angle is sufficient to derive the others; i.e., (180-109.47)and 0.5(180 f 109.47).In Figure 4 there are five similar triangles to the shaded one in Figure 1.Three are indicated by shading the half-tetrahedral angle. The heights of tetrahedra inA and B orientations also can be extraded from F i 4.In Athe height to edge length
AE - 21 1 FE
20
Journal of Chemical Education
is cot 54.74;in B it is sin 54.74 (BDIAB).The height ratio BIA by division i s m . The tangent ratio tan (54.7435.26)han54.74ofll4confirmsthecentroidpositionalong theC~ximbtainedintuitivelyahove. Thus,allthe relevant geometry ofthe tetrahedronis evaluatedsimplyfromright-angledtrianglescontained withinasingleZsection. Literature Cited 1. Maister, P. J C k m . Educ 1993,351, 546. 2. Kimbal1.G. E. J C k m Phys. 1940,8, 188. 3. G3lesoie.. R.J.Molecular h m t m Van Noatrand hihold: New York. 1972 4. Edmundson. J. R.Acto Cvst l992,A48,60. 5. Maier, G A w C k m . (Int. Ed.) 198827. 309. 6. Ho, S. M.; Douglas,B. E. J Chem. Educ 1868.45.47;.
.
