The kinetic theory for the diffusion in binary liquids is not as well developed or as simple as the kinetic theory for gases at low density. Therefore, there have arisen a variety of analytical correlations to predict liquid diffusivities at infinite dilution. Many of these empirical expression take the Nernst-Einstein equation as a starting point. This maple session demostrates the development of one such expression (the Wilke-Chang equation), and its subsequent implementation.

> restart;

The Nernst-Einstein equation describing the diffusivity of solute molecule 1 through solvent medium 2. mu1/F1 is the steady-state velocity given a unit force.

> D12:=kappa*T*(mu1/F1);

>

The mobility is solved for by using the "creeping flow assumption" and solving the Navier-Stokes equation and discarding the acceleration terms (see Bird, et al section 3.5). B12 is teh coefficient of sliding friction and R1 is the radius of the solvent particle.

> mu1:=((3*mu2+R1*Beta12)/(2*mu2+R1*Beta12))*F1/(6*Pi*mu2*R1);

Substitute the expression for the mobility to get a new expression for the diffusivity.

> D12;

The no-slip condition. In this limit, the coefficient of sliding friction goes to infinity. This is known as the Stokes-Einstein equation and applies well to large, spherical, low molecular weight molecules.

> noslip:=limit(D12,Beta12=infinity);

The complete slip condition. In this limit, the coefficient of sliding friction goes to zero, and another simplified expression is obtained.

> completeslip:=limit(D12,Beta12=0);

For self-diffusion, with molecules in a cubic lattice and barely touching, the following approximation gives results to with 12% error.

> R1:=1/2*(V1/N1)^(1/3): D11:=simplify(completeslip);

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> restart;

The Wilke-Chang expression. It is one of the most widely used empirical expressions as a derivitive of the Stokes-Einstein equation and is used only for very dilute solution. Phi2 is an association factor that accounts for hydrogen bonding in the solvent.

The units added to this expression are necessary for the final result to retain the appropriate units.

> D12:=7.4e-8*g^(.5)*cm^(14/5)/gmol^(1/10)/K/s^2*sqrt(Psi2*M2)*T/mu2/V1^(0.6);

>

Here are parameters for a dilute solution fo 2,4,6-trinitrotoluene (TNT) in benzene at 15 degC. The accepted values for Psi2 are as follows:

Water: 2.26

Methanol: 1.9

Ethanol: 1.5

Propanol: 1.2

Others: 1.0

Note: Bird et al. gives a value of Psi2 = 2.6 for water, however, a reevaluated empirical best fit (Perry's Handbook, 2001) gives a value of Psi2 = 2.26. However, in either case the Wilke-Chang expression does not provide consistent predictions with either value.

> mu2:=0.705*g/cm/s; V1:=140*cm^3/gmol; Psi2:=1.0; M2:=78.11*g/gmol; T:=(273+15)*K;

The simplified expression checks with the solution to Example 17.4-1. The units of the diffusivity are in cm^2/sec.

> simplify(D12,assume=positive);