Biomedical Engineering Reference
In-Depth Information
The integral has to be done over all possible values of the angles, keeping
R
fixed. After
some standard integration, we find
2
p
A
p
B
3
k
B
T
(4πε
0
)
2
1
R
6
U
AB
dip...dip
=−
(2.10)
The overall value is therefore negative, and the term is inversely dependent on the
temperature. It also falls off as 1/
R
6
.
2.6
Induction Energy
The next step is the case of two interacting molecules, one of which has a permanent dipole
moment and one of which is polarizable but does not have a permanent electric dipole
moment.
Figure 2.5 shows molecule A with a permanent dipole moment
p
A.
I have indicated the
direction of
p
A
in the diagram, and an arbitrary point P in molecule B. The dipole
p
A
is
distant
R
from point P, and makes an angle θ as shown. The molecules are sufficiently far
apart for the precise location of the point P inside the second molecule to be irrelevant.
Molecule A
Dipole p
A
p
A
θ
R
P
Molecule B
No permanent dipole
polarizability
α
B
Figure 2.5
Dipole-induced dipole
The basic physical idea is that the electric dipole
p
A
induces a dipole in molecule B,
since B is polarizable. We evaluate the potential energy involved and finally average over
all possible geometrical arrangements, for a fixed value of the intermolecular separation.
The steps involved are as follows. The electrostatic potential due to the small dipole
p
A
is
p
A
R
R
3
This is related to the electrostatic field by the general formula
1
4πε
0
φ
A
(
R
)
=
E
(
R
)
=−
grad φ (
R
)
