nbsp; The Gaussian Form of the Elution Equation To change the Poisson form of the elution equation to the Gaussian form, it is necessary to change the position of the origin of the Poisson curve. Consider the elution curve as shown in Figure 9. The origin of the Poisson curve is the point of injection, whereas the origin of the Gaussian curve is at the peak maximum, which will be (n) plate volumes from the injection point.  Now, a point X, (v)  plate  volumes  from the point of injection will be (v-n) = w plate volumes from the peak maximum. Consequently, v = (n+w). This change of origin is depicted in Figure 9. Now, the Poisson form of the elution equation is as follows,                     

;                   and for the nth Plate,                            (12) Equation (12) is the basic elution curve equation and is, in fact, a Poisson function. In due course, it will be shown that if (n) is large, the function tends to the normal Error function or Gaussian function. As in most chromatography systems, (n) >>100 most peaks will be Gaussian or nearly Gaussian in shape. However, due to equipment limitations and other factors, the shape of a peak can be distorted which will also be discussed in due course

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,                                                                    

nbsp; Now,  from the Plate Theory (see Plate Theory and Extensions),                                 , where (n) is the variance of the Gaussian curve. Now, (n) is the volume variance of the Gaussian curve (i.e., ), then, by comparison, (2Dmt) will be the length variance  of the concentration curve where (t) is the elapsed time. Consequently, if a differential equation of the form  is derived that describes some form of dispersion that arises from a random diffusion process, then the solution will be a Gaussian function and, more important from the point of view of the Rate Theory, the Gaussian curve will have a

nbsp;          and,                                                  (41) Equation (41) is the Gaussian form of the elution curve equation and can be used as an alternative to the Poisson form in all applications of the Plate Theory. Retention Measurements on Close Eluting Peaks The retention data, are the most important measurements made in any chromatographic analysis. In addition to providing data for identification (using the capacity ratio or the separation ratio), retention times are also important in column design. It will be shown later that the column efficiency needed to ensure resolution of a pair of

equation (51) in equation (50),                          Now,        Consequently, the term  and those higher can be ignored with respect to . Then,        and                     (52) Equation (52) shows that the points of inflection occur at 0.6065 of the peak height. It follows that the peak width, at 0.6065 of the peak  height, will be equivalent to two standard deviations (2s) of the Poisson or Gaussian curve