Learning Resources

Example CD11-5: Measuring Liquid Phase Diffusivities

The molecular diffusivity of potassium ferricyanide in a solution was determined by filling a capillary with the solution and immersing it in a bath (at 25 ) of the solution from which the ferricyanide was omitted. The solution compositions were:

Solute
Bath conc.
Capillary conc. (gmoles/lt.)
Sodium hydroxide
2.0270
2.0270
Potassium ferrocyanide
0.1919
0.1919
Potassium ferrocyanide
0
0.1963






The capillary tube is 1.0 mm in diameter and 2.1 cm long. After 15 hours the capillary was removed from the bath and its contained fluid titrated for ferricyanide. It was found to contain 2.308 micro-moles of potassium ferricyanide.

  1. Treating the process as diffusion from an infinite capillary, what is the molecular diffusivity of potassium ferricyanide in this solution at 25?
  2. Does the small (2.1 cm) length of the capillary tube cause any appreciable error?
  3. What is the molecular diffusivity of potassium ferricyanide in this solution at 10 ?

(Derive the differential equation and boundary conditions for this process and put the problem in dimensionless form. The error function solution may then be written down, if known.)

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Solution

Let: A = potassium ferricyanide
       B = all other components

Initial condition

t = 0, CA = 0, Z 0

Boundary Conditions

  1. t > 0, CA = 0, Z = 0
  2. t > 0, CA = CAo, Z

A mass balance on component "A" in the capillary element SZ gives:

Dividing by SZ and taking the limit

Fick's first law for dilute solutions is

from which

Thus we arrive at Fick's second law in the form

Utilizing:

  1. The Boltzman transformation   
  2. We will use a modified form of the Boltzman Transform   
  3. Changing Variables



  4. Let    
  5. Transformed boundary conditions
    1. CA = 0,
    2. CA = CAo,
  6. Probability function definition (and value )

The following solution results

differentiating for (dCA/dZ) and substituting

evaluating @ Z = 0

for the period of the experiment, the weight loss (M)



Rearranging and evaluating

To evaluate the error introduced by assuming Z utilize

0.9999

2.0

2.17

0.99

1.81

1.97

Thus (0.99 < < 0.9999) @ t = 15 hr.,

the assumption (Z , = 1.0) was pretty good.

For liquid diffusivity

Assuming for this dilute solution