Biomedical Engineering Reference
In-Depth Information
According to Gauss' electrostatic theorem, this is the same force as would be exerted
if all the charge within a sphere of radius
d
were concentrated at the centre. This charge
is
−
−
Q
times the ratio of the volumes of spheres of radius
a
and
d
and so is
Qd
3
/
a
3
. This
gives a force of magnitude
Q
2
d
3
a
3
1
4πε
0
1
d
2
Q
2
d
4πε
0
1
a
3
=
Hence the displacement
d
satisfies
Q
2
d
4πε
0
a
3
=
QE
The induced dipole is
Qd
and the polarizability is
Qd
/
E
and so
α
=
4πε
0
a
3
(6.4)
Apart from the dimensional factor, the volume of the sphere determines the polarizability.
Table 6.2 shows the experimental polarizabilities for three inert gas atoms, which
illustrates the dependence on volume.
Table 6.2
Polarizabilities of inert gases
Inert gas
α/
10
−40
C
2
m
2
J
−1
He
0.23
Ne
0.44
Ar
1.83
Table 6.3 shows a comparison between typical <α> QSAR model calculations for a
number of alkanes, together with the result of a serious quantum mechanical calculation.
The latter are recorded in the 'Accurate' column. At the chosen level of quantum mechan-
ical theory (BLYP/6-311
G(3d,2p) with optimized geometries, to be discussed in later
chapters), the results are at least as reliable as those that can be deduced from spectroscopic
experiments. The downside is that the butane calculation took approximately four hours on
my home PC; the corresponding parameterized calculation took essentially no time at all.
++
Table 6.3
Comparison of accurate quantum mechanical calculation with QSAR
Alkane
Accurate
<α>/
10
−40
C
2
m
2
J
−1
QSAR
<α>/
10
−40
C
2
m
2
J
−1
Methane
2.831
2.90
Ethane
4.904
4.94
Propane
7.056
6.99
Butane
9.214
9.02
It looks at first sight that there is an incremental effect. Adding a CH
2
increases the mean
polarizability by about 2.13
×
10
−
40
C
2
m
2
J
−
1
and there has been much speculation over the
