Edited by DAN KALLUS Midland Senior High School 906 W. Illinois Midland. TX 79705
revi~i ted Teaching VSEPR Theory The Dilemma of Five-Coordination Anna G. McKenna College of St. Benedict, St. Joseph, MN 56374 Jack F. McKenna St. Cloud State University, St. Cloud, MN 56301 In 1940, Sidgwick and Powell suggested that molecular geometry could he determined by the arrangement of bonding and nonhonding electron pairs in the valence shell ( 1 ). The resulting valence-shell electron-pair repulsion (VSEPR) theory has been developed into a useful tool to predict molecular structures? In most introductory chemistry texts VSEPR theory is explored in some detail for structural predictions, but the reasoning behind and exceptions to the theory are often overlooked. The fundamental rule of VSEPR theory is that the pairs of electrons in the valence shell adopt the arrangement which maximizes their distance apart due to the electrostatic repulsions of the electrons (3). Making the simplifying assumption that the valence electron pairs are equidistant from the nucleus, specific structures favored for various numbers of electron pairs may he derived. For molecules with two electron pairs, VSEPR theory predicts a linear arrangement. For three electron pairs, a trigoual planar structure (Fig. 1) is predicted. For four electron pairs, two possible structures must he considered: a square planar (Fig. 2a) and a tetrahedral (Fig. 2b) arrangement. T o a student encountering VSEPR theory for the first time, the square planar arrangement seems to follow the linear and trigonal planar structures for two reasons. First, the square planar structure is a logical extension of the two-dimensional trigonal planar structure, and, second, students have difficulty visualizing in three dimensions. However, since tetrahedral bond angles exceed those in a square planar structure, the tetrahedral structure is preferred by VSEPR theory. For six electron pairs, an octahedral arrangement (Fig. 3) maximizes the distance between the electron pairs. In all of the above structures, the positions of the electron pairs are equivalent and the bond angles are equal. As the coordination number increases, each predicted structure leads to the next. So, explaining VSEPR theory for coordination numbers 2, 3, 4, and 6 follows a logical process. The major stumbling block in teaching VSEPR theory for coordination numbers 2-6 is in explaining the arrangement of five electron pairs. If we logically follow the trend in structures from two t o six electron oairs. . . whv. does the ohserved trigonal bipyrilmidal strurture (Fig. 4)for five electrun pairs conrain two distinrr typrs of bonds with differenr bond An excellent reference to VSEPR theoly is by R. J. Gillespie
angles? I t would seem likely that due to the nature of electrostatic repulsions, the structure minimizing these repulsions would have equal bond angles. T o explain this dilemma we first must study polyhedra. If a closed surface consists only of portions of planes, it is called a polyhedron. The various
Figure 1. Trigonal planar structure.
la Ib) ~igure2. (a) square planar structure: (b) tetrahedral structure.
Figure 3. Octahedral struchm.
Figure 4. Trigonal bipyramidal structure.
This feature is a review of basic chemical principles and as a reappraisal Of the state of the art. Comments. suggestions fortopics, and contributions should be sent to the feature editor.
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Seotember 1984
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types of polyhedra are defined by the number of vertices, edees. and faces. If all of the faces of a volvhedron are reeular plans; regions, the polyhedron is known as a regular po&hedron. If all of the vertices of a volvhedron are eauivalwt (the same number of faces meeting a t the same angle), the pblyhedron is said to be isogonal. If all the faces of the polyhedron are dike (all squares or triangles, for example), the polyhedron is said to be isohedral. In 1758, Euler determined the relationship between the number of vertices (u), edges ( e ) ,and faces ( f ) of a simple convex polyhedron (4): u-e+f=2
By restricting the polyhedra to be isogonal and isohedral, Euler's relationship restricts the number of these regular polyhedra. In order to illustrate this, we will use Euler's formula and elementary geometry to determine the number of faces allowed for a particular face shave. Let us hegin with the simplest faces.hape: a triangle. Since the triangle must he regular, rach face angle is 60'. The face angles meet in the vertices of the polyhedron. Since a polyhedral venex mnst belong to at least three faces and the sum of the face angles must br less than 360°, therecan he3.4, or 5 face angles at each vertex. If u is the numher of vertices, for 3 face angles per vertex, there arr 31. fare angles, u faces, and 3u12 edges (hecause each face is bounded by 3 edges and each edge is an edge of 2 faces.) Substituting these expressions into Euler's formula for u, e, and I, respectively, we obtain: u
more than bonding electron pairs (2). Because nonbonding electron pairs are under the influence of only one nucleus, their orbitals are more diffuse and occupy a larger region of space than the orbitals of bonding electron pairs which are more confined in space because of the influence of two nuclei. The repulsions between electron pairs may he arranged in order of decreasing strength: nonhonding-nonhonding > nonbonding-bonding > bonding-bonding. Thus, the electron pairs will occupy positions which minimize the repulsions in the order given above. For a five-coordinate species containing one nonbonding electron pair, the nonbonding pair willoccupy one of the equatorial pasitions (in the trigonal plane). This can be exdained bv examinine the axial and euuatorial oositions: the axial position (Fig. 5ajhas three neighboring a t 90' while the equatorial ~osition(Fie. 5b) has two neiehborine pairs a t 90' a n d two more a t l20< In distributing two nonbondine electron vain in a trieonal hipyramidal strkture, three st;uctures &e possihle:both nonbondinr pairs in the axial ~ositions(Fie. 6aJ. hoth oairs in e q ~ a t o r ~ p o s i t i o(Fig. n s 61;), and onep& in axi; position with the other pair in an equatorial position (Fig. 6c). The third structure can he easily eliminated by the student because the two nonbonding electron pairs are onlv 90° aoart. Because nonbonding electron pairs re& each other more ihan bonding pairs, it would appear to the student that thecorrect structure would have thenonbonding electron pairs in the axial positions so that the angle between the nonbonding
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- 3012 + u = 2
and u=4,e=B,f=4
and the polyhedron is a regular tetrahedron. Using the same prncedure for 4 and 5 triangular faresat each vertex leads to figures with 8 faces (octahedron) and 20 facrs (icosahedron), respectively. If the faces of the polyhedron are square, the resulting polyhedron bas 8 faces (cube). If the faces are pentagonal, the polyhedron has 12 faces (dodecahedron). If the faces are allowed to have 6 or more sides, each angle in the face will be 120" or greater and the sum of the face angles in the vertex will equal or exceed 360°, which is impossible. Therefore, it follows that every regular polyhedron is one of the five types listed above: tetrahedron, octahedron, icosohedron, cube, dodecahedron (5).If the polyhedra are allowed to have faces of more than one type, 15 semi-regular polyhedra are obtained using Euler's relationship. The fact that the number of isogonal bodies is limited to the five regular solids and 15 semi-regular solids is quite important in structural prediction. Since it is assumed in VSEPR theory that the electron pairs are equidistant from the central atom, it is useful to envision placing the electron pairs on the surface of a sphere whose midpoint is the central atom. The placement of the electron pairs on the surface of the sphere is such that the distance between adjacent electron pairs is maximized. For coordination number 4, the electron pairs are arranged at the vertices of a regular tetrahedron. For coordination number 6, the electron pairs are arranged at the vertices of an octahedron. Neither ;he regular norsemi-regular polyhedra contain five vertices or facrs. Thrrefore, it is not ~ossihleto distrihuce five equivalent points uniformly on thesurface of a sphere apart from forminr a reeular uentaaon (6). . . Thus. the distrihution of five election pairs deviates from the regular polyhedra predicted by VSEPR theory for other coordination numbers. A second problem encountered in teaching VSEPR theory for five-coordinate species lies in the placement of bonding and nonbonding electron pairs in the nonequivalent positions of the trigonal bipyramid. A postulate of VSEPR theory states that nonbonding electron pairs repel adjacent electron pairs 772
Journal of Chemical Education
(.I t bl Figwe 5. Disbibutlon of one m n M i n g pair of elemom in abigonal bipynunidal struchne: (a) axial; (b) equataial.
Figure 6. Distributionof two mnbondingpairs of electrons in a trigonal bipyramidal structwe: (a) born axial: (b) both equatbrial: (c) me axiallone equatorial.
Figue 7. Distributionof mree nonbonding pairs of electrons in a trigonal bipyramidal structure.
electron pairs would be 180°.However, the two nonhonding electron pairs are found in the equatorial positions. This apparent contradiction to VSEPR theory can be explained by the aneular denendence of electronic re~ulsions.Since the repulsions between neighboring electron pairs decrease rapidlv with increasine aneles between them. it can be assumed t h i t the repulsionbe&een electron paik at 120' is quite small. Thus, we need only concern ourselves with 90' repulsions and, as stated above, the equatorial position contains two 90°repulsions while the axial position contains three such repulsions. In a five-coordinate species containing three nonhondine electron nairs. all three nonbondine.. nairs . occu~v .. equatorial positions $ig. 7). 'I'eachine VSEPR theory in introductory chemistry is only a smal~bercenta~e of lecture time i n d usually perhaps the explanations given here may seem to he in too much detail for such a small topic. However, in teaching science, the whv and how of a theow are much more important than simple roie memorization of series ot'rules. In teaching VSEI'R theory for coordination numbers 2-6, the only apparent break & the logical process is at coordination number 5. Exploring the reasons why the trigonal bipyramidal structure is favored by VSEPR theory encourages students to realize that the trigonal bipyramidal structure is not an anomaly. Mentioning the fact that a square pyramidal structure (derived from the octahedron) not favored by the theory does exist (7) also gives the student insight into the role of other factors hesides simple electron repulsions in structural ge-
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a
ometry. The placing of honding and nonbonding electron pairs in the trigonal bipyramidal structure requires more than simple rote application of rules. The student must explore not only the strength of various repulsions, hut also how these repulsions affect the placement of honding and nonbonding electron pairs. In teaching VSEPR theory, as in teaching any other simple model, the student should understand that such a simple approach neglects many other factors in honding. Its emphasis on interelectronic re~ulsionsrather than on bond formation contrasts sharply with nther models in which repulsions are considered only to refine structural details (71. VSEPR theow. like most simple models, produces mostly loirect predictio& However, such a simple theory can easily be misused or overused. It should be stressed that although VSEPR theory is an easy-tome tool, sucha simple tool-by the very nature of its simplicity-may not be as accurate as necessary. Llterature Cited (1) S i W a k , N. V., and Powll, H. M., Roc. Roy. Soc..A176,153 (1940). (2) Gillepis, R. J., J. CIIEM. EDUC.. 40,295 (1963). (3) GiUespie, R. J., "Molrmlar Geometry,"Van N m m d Reinhold Company Ltd.,landan,
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,972 ... ,nn Fr. F-l 6
(4) "Standard Mathematical Table,"21st d.,TheChemiealRubberCo..Cleveland. Ohio,
1973,~.15. (5) Ringenberg,LamnrrA.,"CollegeGwmeVy," John Wiley andSans,New York, 19% pp. 269-271. (6) Wells. A. F.. "Structural Inorganic Ch.?mistry,"4th od., Clarendon h a s . &ford, 1975, p. 64. (7) Cotton.F. Albert,and Wilkioson,Gmffrey,"AduancedInorganieChemistry,"4thd., John Wiley and Sons,NeaYork, 1 9 8 0 , ~197. .
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