Page 35 - Modern physical chemistry

24 Structure in Molecules and Atoms

negative to the more positive atom. The different dipole moments add vectorially to give
the dipole moment of the molecule.

2.2 Symmetric Coordination Structures

The bond angles in small symmetric coordination structures are determined by the
equivalence of the attached atoms.
Consider a molecule or ion of the type AB n , where n B atoms are bonded to a single
A atom. Let rj be the vector drawn from the center of A to the center of the jth B.
N ow, the dipole moment of the jth A- B bond lies along r j . Furthermore, we suppose
that its magnitude varies with the bond length r j • So when the ligands are arranged sym-
metrically around the central atom, the net dipole moment is zero and
[2.1]

The dot product of two vectors equals the projection of the magnitude of the first one
on the second one times the magnitude of the second one. If Ojk is the acute angle between
rj and r k , we have
[2.2]

When all the bonds are equivalent, they have the same length and
1j = r. [2.3]

When all the acute angles are the same, we also have
0jk = O. [2.4]

Let us now dot multiply both sides of equation (2.1) by vector rj and introduce con-
ditions (2.3) and (2.4) to construct

r2 + ( n -1 )r2 cos () = 0, [2.5]

whence

o = cos (1-n) [2.6]
.ll j .
1
For n = 2, we have

[2.7]

The corresponding structure is linear. When n = 3, we find that

() = cos-{ - ~) = 120°. [2.8]

The ligands now lie at the comers of an equilateral triangle. When n =4, we have

() =cos- 1(-i) = 109.47°. [2.9]

These ligands are arranged tetrahedrally about the central atom.