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2.2. DIFFUSION 41
where r is the distance between centers of the molecules, ε 12 is an intensity constant
and σ 12 is a collision diameter. Consequently, once the form of f is known, the values
of ε 12 and σ 12 determine the potential. When forces between molecules depend on
their relative orientation, as occurs whit polar molecules, more complicated expressions
than (2.13) are needed for ϕ. Such expressions must include the influence of
the relative orientation on the collision. An example of such potentials, which has
been used by Hirschfelder et al. for the computation of transport coefficients of polar
molecules [6], is Stockmayer potential
( (σ12 ) 12 ( σ12
) ) 6
ϕ = 4ε 12 − − µ2 12
r r r 3 f (θ 1, θ 2 , φ 2 − φ 1 ) . (2.14)
The first term in the right hand side is Lennard-Jones’s potential and it represents
the part of interaction independent from the relative orientation of molecules.
The second term gives the attraction between two dipoles whose relative orientation
is defined in Fig. 2.2. So far very few results are available on transport coefficients in
mixtures of polar molecules. 9
φ − φ
1 2
θ 1
θ 2
+
Figure 2.2: Schematic diagram of the angles defining the orientation of two polar
molecules.
The values for the coefficients of diffusion and thermal diffusion can be computed
through a method of successive approximations developed by Chapman and
Enskog.
For this, the form of the interaction potential between molecules as well as its
corresponding parameters must be known. Generally, a first approximation is enough
for practical applications. For D 12 first approximation [D 12 ] 1
, gives
√
[D 12 ] 1
= 3
( )
8 √ kT M1 + M 2 1
2π pσ12
2 RT
(2.15)
M 1 M 2 Ω (1,1)∗
12 (T12 ∗ ).
Here T ∗ 12 is a dimensionless temperature defined by the expression
9 See Ref. [2], p. 597.
+
T ∗ 12 = kT
ε 12
, (2.16)