Biomedical Engineering Reference
In-Depth Information
C
12,
ij
=
4
σ
i
σ
j
6
√
ε
i
ε
j
4
σ
i
σ
j
3
√
ε
i
ε
j
C
6,
ij
=
(2.23)
Finally
α
i
α
j
α
N
i
C
6,
ij
=
C
α
N
j
+
2
C
6,
ij
R
i
+
R
j
6
1
C
12,
ij
=
(2.24)
where α
i
is the dipole polarizability of atom
i
,
N
i
the number of valence electrons and
R
i
the van der Waals radius.
2.10 Comparison with Experiment
You will have got the idea by now that we have to determine the parameters in any pair
potential by appeal to experiment. There are two kinds of experiment to consider. First
there are those that are essentially in the gas phase, where pairs of atoms genuinely interact
with each other unencumbered by other species. This means that the total mutual potential
energy is given by the sum of the interacting pairs. Second there are experiments that
essentially relate to condensed phases, where the interacting particles are sufficiently close
to raise doubts about the credibility of the pair wise additivity assumption.
2.10.1 Gas Imperfections
The deviation of gases from perfect behaviour can be expressed in the form of a virial
equation of state
pV
nRT
=
nB
(
T
)
V
n
2
C
(
T
)
V
2
1
+
+
+···
(2.25)
where the virial coefficients
B
(
T
),
C
(
T
),... depend on the temperature and on the char-
acteristics of the species under study. Here,
n
is the amount of substance,
p
the pressure,
V
the volume,
R
the gas constant and
T
the thermodynamic temperature.
B
(
T
) is called the
second virial coefficient
,
C
(
T
) is called the
third virial coefficient
, and so on. They have to
be determined experimentally by fitting the
pVT
data of the gas under study.
The virial equation of state has a special significance in that the virial coefficients can
be related directly to molecular properties.
B
(
T
) depends on the pair potential
U
(
R
) in the
following way:
1
exp
R
2
d
R
∞
U
(
R
)
k
B
T
B
(
T
)
=
2π
−
−
(2.26)
0
2.10.2 Molecular Beams
In a molecular beam experiment, a beam of mono-energetic molecules is produced and
allowed to collide either with other molecules in a scattering chamber, or with a similar
beam travelling at right angles to the original beam. Measurements of the amount by which
