Page 36 - Modern physical chemistry

2.3 Momentum and Energy Relationships 25

When the acute angle between rj and rn is 180°, dot multiplying both sides of equa-
tion (2.1) by rj and setting the other acute angles involving rj equal to 8j yields

[2.10]

whence

[2.11 ]
A similar result follows from dot multiplying both sides of (2.1) by rw But dot multiply-
ing both sides of equation (2.1) by r2 yields

[2.12]
whence

[2.13]

Here (}2 is the other acute angle.
When n = 4, formula (2.13) yields

82 = cos- (-1) = 180°. [2.14]
1
The configuration is now a square. When n = 5, we have

82 = cos- [ -~ J = 120°. [2.15]
1
The configuration is a trigonal bipyramid.
For n = 6, the most symmetry prevails when the acute angles between rj and r6, r2
and r4 , r3 and r5 all equal 180°. Dot multiplying equation (2.1) by any r j . then yields equa-
tion (2.10). As a consequence, the other acute angle is 90° and the configuration is that
of an octahedron.
Substituting other atoms or radicals for one or more of the B's attached to a given A
would perturb the structure. But since the perturbations are generally small, the sym-
metric structure serves as a reference structure.

2.3 Momentum and Energy Relationships
To measure bond distances and angles in a molecule, one may employ diffraction
methods. Approximately homogeneous beams of particles are generally employed. Elec-
tromagnetic radiation, as we have in X rays, is too penetrating.
Innumerable experiments show that a beam in which the momentum of each parti-
cle is p exhibits a wavevector k that is proportional:
h
p=-k=/ik. [2.16]
2n-
This relationship was introduced by Louis de Broglie. The wavevector has the magnitude

k = 2n- [2.17]
;.,'