Diffusion Coefficients of Tracers in Capillary Tube Experiments

Diffusion is an essential topic in solid-state physics, chemistry, and metallurgy. This Demonstration presents a method to compute the diffusion coefficient of a tracer from experimental data obtained in a long capillary or a shear cell experiment.

Consider a capillary tube filled with a fluid separated by an impermeable membrane. Half of the tube's length contains a tracer. The membrane is removed at time zero and the tracer is allowed to diffuse. The space and time distribution of the tracer can be described by the diffusion equation in one dimension: , with initial conditions if and if , and boundary conditions at and .

Here is the tracer concentration, is distance in centimeters, is time in seconds, is the diffusion coefficient of the tracer in , and and are the boundaries of the capillary tube in centimeters. We define a new variable and assuming the diffusion coefficient to be independent of concentration, the diffusion equation can then be written . This equation, which is independent of the diffusion coefficient, is solved with the above initial and boundary conditions. The tracer concentration is shown in two and three dimensions and a plot of the root-mean-square error between the predicted and observed tracer concentration is used to determine the diffusion coefficient.