Biomedical Engineering Reference
In-Depth Information
and direct differentiation gives the following formula:
p
A
R
3
R
5
R
1
4πε
0
3
p
A
R
E
A
(
R
)
=−
−
(2.11)
MoleculeA therefore generates an electrostatic field in the region of molecule B, according
to the above vector equation. The modulus of this vector at point P is
R
3
(1
1
4πε
0
p
A
E
A
=
+
3 cos
2
θ )
(2.12)
This electrostatic field induces a dipole in molecule B. For the sake of argument, I will
assume that the induced dipole is in the direction of the applied field (and so we need not
worry about the fact that the polarizability is a tensor property).
A calculation of the resulting mutual potential energy
U
AB
gives
2
3 cos
2
θ
1
1
(4πε
0
)
2
α
B
p
A
R
6
1
U
AB
=−
+
(2.13)
Polarizabilities are positive quantities and so
U
AB
is negative for all values of θ at a given
intermolecular separation. This is quite different from the dipole-dipole interaction, where
some alignments of the dipoles gave a positive contribution to the mutual potential energy
and some gave a negative one.
Finally we have to average over all possible alignments keeping the internuclear separ-
ation fixed. This averaging has again to be done using the Boltzmann weightings, and we
find eventually an expression for the
induction
contribution to the mutual potential energy
of A and B:
1
(4πε
0
)
2
p
A
α
B
R
6
U
AB
ind
=−
(2.14)
Note that the interaction falls off as 1/
R
6
just as for the dipole-dipole interaction, but
this time there is no temperature dependence. For two identical A molecules each with
permanent electric dipole
p
A
and polarizability α
A
the expression becomes
2
(4πε
0
)
2
p
A
α
A
R
6
U
AA
ind
=−
(2.15)
This of course has to be added to the dipole-dipole expression of the previous section.
2.7 Dispersion Energy
It is an experimental fact that the inert gases can be liquefied. Atoms do not have permanent
electric moments, so the dipole-dipole and induction contributions to the mutual potential
energy of an array of inert gas atoms must both be zero. There is clearly a third interaction
mechanism (referred to as
dispersion
), and this was first identified by Fritz W. London in
1930.
The two contributions to the mutual potential energy discussed in previous sections
can be described by classical electromagnetism. There is no need to invoke the concepts
of quantum mechanics. Dispersion interactions can only be correctly described using the
