Untitled - Aerobib - Universidad Politécnica de Madrid

42 CHAPTER 2. TRANSPORT PHENOMENA IN GAS MIXTURES
and Ω (1,1)∗
12 is a function of T ∗ 12 whose form depends on the potential of molecular
interaction. For instance, if the molecules behave as rigid spheres Ω (1,1)∗
12 = 1 for all
values of T ∗ 12.
For numerical computations it is more convenient to express [D 12 ] 1
in the form
[D 12 ] 1
= 0.002628
√
M1 + M 2
2M 1 M 2
T 3
(2.17)
pσ12 2 Ω(1,1)∗ 12 (T12 ∗ ),
where p is measured in atm, T in K, σ 12 in o A and [D 12 ] 1
in cm 2 /s.
The above mentioned theory does not determine the interaction potential that
should be applied in each case. To the contrary a comparison between the theoretical
results corresponding to several potentials and experimental measurements, can
furnish information on such potentials. In Ref. [2], pp. 589 and following, some examples
of such comparisons can be found showing the agreement that can be expected
between theoretical and experimental results. It is verified that for many cases the best
agreement is attained with a Sutherland interaction potential or with that of Lennard-
Jones. Sutherland’s potential is the one that produces between rigid spheres attracting
one another with a force inversely proportional to a given power of its distance r.
Therefore, it has the form
This potential gives for Ω (1,1)∗
12
r < σ 12 : ϕ(r) = ∞,
r > σ 12 : ϕ(r) = −ε 12
( σ12
r
) δ
.
(2.18)
Ω (1,1)∗
12 = 1 + S 12
T , (2.19)
where S 12 = g(δ) ε 12
is a constant whose value depends on the parameters of Eq. (2.18)
k
and generally is determined experimentally. Substituting Eq. (2.19) into (2.15) one
obtains
[D 12 ] 1
= 3
8 √ 2π
k
pσ 2 12
√ ( ) 5
M1 + M 2 T 2
R
M 1 M 2 S 12 + T . (2.20)
If the interaction potential is of Lennard-Jones type, no simple expression can
be given for Ω (1,1)∗
12 , whose values must be calculated by numerical integration. Table
2.1 gives the values calculated by Hirschfelder et al. [7].