Mass balance equation

Analyte concentration in the eluent is a function of both time and length. Since the volumetric flow rate though the column is F ml/min, therefore during time t the analyte amount of cFdt will move into the layer dx and the amount of (c+dc)Fdt will move out of this layer. The accumulation (positive or negative) in this layer will be

-Fdtdc = Fdt grad(c) dx (1a)

is the gradient of concentration in layer dx which was formed during time t (for a two-component system). Here we have to consider how the component is distributed in this layer. As the column is in equilibrium, then we will have the equilibrium concentration, ce, in the incoming eluent. And we have some excess amount of analyte adsorbed on the surface. We need to emphasise here that we should not determine the space where this excess is located; we will relate it to the surface.

Then, the amount of analyte in the layer is, where vo is the total volume of the liquid phase per unit of column length, ce is the equilibrium concentration, s is the surface area of the adsorbent also per unit of column length, and is the excess adsorption. Changes in the amount of analyte in the cross-sectional layer with thickness dx on the distance x from the column inlet during the time dt will be:

(2a)

On the basis of equilibrium and mass balance conditions, the speed of accumulation of the analyte in the layer dx and the speed of equilibration, (ie. forming the excess amount) should be equal. Thus,

_(3a)

or

_(4a)

Above is the differential dynamic mass balance equation of the component in the chromatographic column.