Page 112 - Separation process principles 2

3.2 Diffusion Coefficients 77

The Stokes-Einstein and Wilke-Chang equations predict
an inverse dependence of liquid diffusivity with viscosity.
The Hayduk-Minhas equations predict a somewhat smaller
dependence on viscosity. From data covering several orders
of magnitude variation of viscosity, the liquid diffusivity is
found to vary inversely with the viscosity raised to an expo-
nent closer to 0.5 than to 1.0. The Stokes-Einstein and
Wilke-Chang equations also predict that DABpB/T is a
Use the Vignes equations to estimate diffusion coefficients over
constant over a narrow temperature range. Because p~ de-
the entire composition range.
creases exponentially with temperature, DAB is predicted to
increase exponentially with temperature. For example, for a
dilute solution of water in ethanol, the diffusivity of water SOLUTION
increases by a factor of almost 20 when the absolute temper-
Using a spreadsheet to compute the derivatives in (3-45) and
ature is increased 50%. Over a wide temperature range, it is (3-46), which are found to be essentially equal at any composition,
preferable to express the effect of temperature on DAB by an and the diffusivities from the same equations, the following results
Arrhenius-type expression, are obtained with DAB = DBA at each composition. The calcula-
tions show a minimum diffusivity at a methanol mole fraction
of 0.30.

where, typically the activation energy for liquid diffusion, E,
is no greater than 6,000 callmol.
Equations (3-39), (3-40), and (3-42) for estimating diffu-
sivity in binary liquid mixtures only apply to the solute, A, in
a dilute solution of the solvent, B. Unlike binary gas mix-
tures in which the diffusivity is almost independent of com-
position, the effect of composition on liquid diffusivity is
complex, sometimes showing strong positive or negative If the diffusivity is assumed linear with mole fraction, the value at
deviations from linearity with mole fraction. XA = 0.50 is 1.625 x lop5, which is almost 40% higher than the
Based on a nonideal form of Fick's law, Vignes [9] has predicted value of 1.18 x lop5.
shown that, except for strongly associated binary mixtures
such as chloroform/acetone, which exhibit a rare negative de-
viation from Raoult's law, infinite-dilution binary diffusivi- Diffusivities of Electrolytes
ties, (D),, can be combined with mixture activity-coefficient In an electrolyte solute, the diffusion coefficient of the dis-
data or correlations thereof to predict liquid binary diffusion solved salt, acid, or base depends on the ions, since they are
coefficients DAB and DBA over the entire composition range. the diffusing entities. However, in the absence of an electric
The Vignes equations are: potential, only the molecular diffusion of the electrolyte
molecule is of interest. The infinite-dilution diffusivity of a
single salt in an aqueous solution in cm2/s can be estimated
from the Nernst-Haskell equation:

where
EXAMPLE 3.7 n+ and n- = valences of the cation and anion,
respectively
At 298 K and 1 atm, infinite-dilution diffusion coefficients for
the methanol (A)/water (B) system are 1.5 x lop5 cm2/s and A+ and A- = limiting ionic conductances in (A,/cm2)
1.75 x cm2/s for AB and BA, respectively. (~/cm)(g-equiv/cm3), where A = amps and
Activity-coefficient data for the same conditions as estimated V = volts
from the UNIFAC method are as follows:
F = Faraday's constant
= 96,500 coulombs/g-equiv
T = temperature, K
R = gas constant = 8.3 14 Jlmol-K

Values of A+ and A- at 25'C are listed in Table 3.7. At other
temperatures, these values are multiplied by T/334pB,