Theory of tracer diffusion measurements in liquid systems

Abstract

The differential equation of non-steady state tracer diffusion has been solved for liquid systems with general initial and boundary conditions. On the basis of the solution the measurement of the tracer diffusion coefficient can be reduced, regardless of the geometry of the cell, to the determination of the eigenvalues, of which the first one only is necessary after a certain time. An obvious distinction can be made between relative and absolute measuring methods. The planning of new experiments is possible if the roles of the initial condition and the counting efficiency are appreciated.

As a special example of a three dimensional diffusion cell, a slightly conical open-ended capillary has been analysed. To obtain 0.1 % accuracy with the 0.8 mm diameter capillary, a variation of only 0.001 mm in the diameter can be allowed.

The space dependent tracer diffusion coefficient has also been treated. The tracer diffusion current density and the corresponding differential equation of continuity, which take into account the unequal equilibrium distribution of the tracer concentration in the system, have been derived. The solution of the differential equation for this continuous multiphase system has been obtained in an analogous manner to that of the single phase. Here again after a certain time the first eigenvalue can be determined experimentally. However, the eigenvalue may include only a certain part of the tracer diffusion coefficient which is independent on the positional coordinates. In this case the form D=D₀f(r) is used for the tracer diffusion coefficient with the constant D₀.