Transformation to Concentration Units

The ultimate goal of a basic study of separations is to obtain a description of how component concentration pulses (zones or peaks) move around in relationship to one another. The flux density J tells how solute moves across boundaries into and out of regions, but it does not detail the ebb and flow of concentration. To do the latter we must transform J into a form that directly yields concentration changes. The procedure followed below for this is standard in many fields. It is followed, for example, in treatments of heat conduction and diffusion [14,15]. We shall continue to simplify our treatment to one dimension. 

This gain (or loss if it comes out with a negative sign) occurs between the planes, in a volume equal exactly to dx (inasmuch as the planes have unit area). We recognize the moles gained divided by the volume dx as simply the increment in concentration Ac. All of this acquires a simple form if we multiply the right-hand side of Eq. 3.24 by dxldx = 1 

We now simplify the left-hand side by first making a Taylor s expansion of JM+dM around point x 

We note that (7, v, and D may vary with x in some systems, but these variations are negligibly slow compared to the rapid variation of c with x associated with sharp concentration pulses. More complicated are variations of U, v, and D with concentration. However, since sample amounts are generally small in analytical separations, the concentration dependence can usually be ignored. 

When flow and mean displacement velocities are zero (U = 0 and v = 0), the above reduces to Fick s second law of diffusion 

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