Misinterpreting the effluent profiles obtained during tracer measurements performed for determining packing quality can often lead to excessively large percolation velocities and exaggeration of packing problems. Highly useful and reliable information can be obtained through characterization of tracer effluent curves using the method of moments, information that could be critical for successful scale-up of chromatographic steps. This is the sixth in the "Elements of Biopharmaceutical Production" series.
The plate theory of chromatography was introduced by Martin and Synge in 1941.1 It offered the first description of the development of quasi-Gaussian bands in linear elution chromatography, which is defined as conditions where the solute is partitioned between the mobile phase and stationary phase in the presence of a linear isotherm. Plate theory describes a column as consisting of a number of theoretical plates such that at each plate, the solute is equilibrated between two phases. It is assumed that the band spreading due to diffusion of solute from one plate to another is negligibly small, and that at equilibrium the partition coefficient of a solute is independent of its concentration and of the presence of other solutes. It was observed that after a certain length of time, the zone profile became Gaussian in shape.
Martin and Synge defined a height equivalent to a theoretical plate (HETP) as "the thickness of a layer in the column such that the eluting mobile phase is in equilibrium with the solute concentration in the stationary phase." They argued that the HETP is a constant through a given column, except when the ratio of the mobile phase concentrations at the entrance and exit differ greatly from unity. The plate height depends on the solute diffusivity in the mobile phase, and is directly proportional to the flowrate (v
) and the square of the particle diameter (d
). Though marred by several inconsistencies,2,3 the originality and simplicity of the plate model made it very popular.
In 1955, Giddings and Eyring employed probability (stochastic) concepts to describe the differential migration process of chromatography.2 They developed a rigorous probability theory that gave a precise description of the influence of simple adsorption-desorption processes on the zone profile. Their approach later led to the simplified and approximate theory of zone dispersion, often referred to as the random walk theory of chromatography,3 for the situation when the time spent by the solutes inside the column is large enough to allow each molecule to undergo a large number of individual sorption and desorption steps.
Let us delve into plate theory and the normal probability, or Gaussian, distribution upon which it is based.1 Equation (1) describes the Gaussian distribution and Equation (2) the time elapsed. All symbols are defined in a sidebar.
We plotted two examples in Figure 1. The graphical interpretation of s is the half-width of the distribution at its inflection point. If the curves in this figure represent effluent concentrations from a tracer experiment on a chromatographic column, the number of plates is defined as:
Figure 1. A Gaussian Distribution