Biomedical Engineering Reference
In-Depth Information
Comparison of the macroscopic and microscopic equations gives
1
1/2
N
α
ε
0
V
n
=
+
which can be expanded by the binomial theorem to give
N
α
2ε
0
V
n
=
1
+
and if the gas is a perfect gas we have finally
p
α
2ε
0
k
B
T
n
=
1
+
(23.7)
which gives an experimental route to the polarizability α.
In condensed phases, the separation between molecules is of the order of the molecu-
lar dimensions and the result is that each molecule is polarized not by just the ordinary
applied electric field
E
but by a
local field
F
comprising
E
plus the field of all the other
dipoles. Once the local field is known, we can use Equation (23.6) to find
P
simply
by substituting α
F
for α
E
. The calculation of
F
is in fact difficult because the dipoles
that contribute to
F
are themselves determined by
F
and a self-consistent treatment is
needed.
A little manipulation yields the well-known
Clausius-Mossotti
equation
ε
r
−
1
N
α
3ε
0
V
2
=
(23.8)
ε
r
+
which can be used to calculate α from an experimental value of ε
r
.
So far we have assumed that the molecules in the system have no permanent electric
dipole moments. When this restriction is relaxed we have to allow for the
orientation
polarization
; the interaction energy of a dipole moment vector with an electric field is
=−
U
p
0
.
E
so that in a fluid the molecules will tend to orient themselves parallel to the field. This
tendency will be opposed by random thermal agitation, and a calculation by the methods of
statistical mechanics yields Debye's (1912) correction to the Clausius-Mossotti equation
α
ε
r
−
1
N
3ε
0
V
p
0
3
k
B
T
2
=
+
(23.9)
ε
r
+
Debye's dipole theory accounts quantitatively for the dielectric properties of gases and
qualitatively for those of condensed phases. A more thorough analysis of the problem
shows that one should not expect the dipole moment to remain constant because real
molecules have a nonzero polarizability. The polarization of the dielectric in the electric
field of the molecule itself gives rise to a
reaction field
. The classic literature reference is
Onsager (1936).
Continuum models of the solvent have their origin in these physical considerations; one
component of the system (the solute
M
) is treated microscopically and the remainder of
the system (the solvent) is treated macroscopically.
