Dispersion in Chromatography Columns - The Random Walk Model > Page 6

To develop an HETP equation it is necessary to first identify the dispersion processes that occur in a column and then determine  the variance that will result from each process per unit length of column. The sum of all these variances will be (H), the Height of the Theoretical Plate, or the total variance per unit column length. There are a number of methods used to arrive at an expression for the variance resulting from each dispersion process and these can be obtained from the various references provided. However, as an example, the Random-Walk Model introduced by Giddings (3) will be employed here to illustrate the procedure. The theory of the Random-Walk processes itself can be found in any appropriate textbook on probability (4) and will not be given here but the consequential equation will be used.

The Random Walk Model

The random-walk model consists of a series of step-like movements for each molecule which may be positive or negative the direction being completely random. After (p) steps, each step having a length (s) the average of the molecules will have moved some distance from the starting position and will form a Gaussian type distribution curve with a variance of s2 .

Now according to the random-walk model,

                                                                                  (1)                      

Equation (1) can be used in a general way to determine the variance resulting from the different dispersion processes that occur in an chromatography column. The application of equation (1) is simple, the problem that often arises is the identification of the average step and sometimes the total number of steps associated with the particular process being considered. As an illustration of its use it will be used to the problem of obtain an expression for the radial dispersion of a sample when it is placed on a packed column in the manner of Horne et al.  (5).