Employing Vector Algebra To Obtain the Tetrahedral Bond Angle George H. Duffey Department of Physics, South Dakota State University, Brookings, SD 57007 Recently, Kawa' used the law of cosines in calculating the tetrahedral angle. A useful alternate procedure employs simple vector algebra.= Consider vector A, of length IAl = A, and vector B, of length I B ~ = B, drawn from the same point, with 0 the angle between, as Figure 1shows. The lengthof the projection of A on B times the length of B is called the scalar product A . B. Using the definition of the cosine, we have where Figure 1. Proiecting vector A on vector B to get A m d
By convention, the vectors of unit length lying along the x , y, and z axes are designated i, j, k, respectively, as Figure 2 shows. Applying eq 1 to these yields i.i=j.j=k.k=l
Similarly,
(3)
Multiplying these out using eqs 3 and 4 gives us Any vedor may he represented as the sum of its projections on the three axes. From Figure 2 we have
' Kawa, C. J. J. Chem Edw. 1988, 85,884-885.
Duffey. G. H. Physical Chemistry; McGraw-Hill: New York, 1962: pp 21-28.
Now, construct a tetrahedron within a cube as Figure 3 illustrates. Take the center as the origin of a Cartesian coordinate system. Let the x axis pass through the center of the front face; they axis, through the center of the right face; the z axis, through the center of the top face. Choose the unit of distance as one-half the length of an edge. Vertices of the
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F i w e 2. Unn vectas I. I. and k. Also, vecta A as Uw sum of projection Ad. projection AJ, and projection A*.
Figre 4. Vectas drawn hm me wmer of an inscribed tetrahedmn to two opposite comers, along two adjacent edges.
The other combinations yield the same value. For the square of the length of A, we have Each of the other vectors has the same so A itself eouals 3. length. The angle between any two adjacent bonds equals the angle between the corresponding vectors. Substituting the values of A. B, A, and B into eq 2 gives us
whence
The remainingcombinations of vectors yield the same angle. Other angles can be obtained similarly. For the angle hetween two edges of the tetrahedron, consider the vector from (-1, -1,l) to (1, -1, -I), Figure 3. Vectas ham hm U-e centsr of a tetrahehon to its vertices at anemate caners of a cube.
tetrahedron then appear at points (1, 1, I), (-1, -1, 1). (1, -1, -I), and (-1, 1,-1). Consider CH4 with the carbon atom at the center of the cube and hydrogen atoms a t the designated comers. Place a vector on each bond. The initial point of each vector is then (0,0,0); the end points are those listed above. Inserting the resulting components into eqs 5,6,.. .then yields A=i+j+k
Applying eq 7 to vectors A and B leads to 36
Journal of Chemical Education
(8)
and the vector from (-1, -1,l) to (1,1, I), as in F i m e 4. We find that
whence The centers of the faces of a cube, together with two opposite comers, form the vertices of a rhombohedron. From the first cube corner, the edges travel along the edges of one of the inscribed tetrahedra. So the rhombohedra1 angle is here 60'.
