Moment Analysis and Plate Height Equations

As mentioned earlier, a closed-form, analytical solution of the inverse Laplace transform has not been derived yet in the case of the general rate model for a pulse injection. However, Kucera [30] and Kubin [31] have shown that the first five moments of the solution can easily be calculated from their solution of the general rate model giving the band profile in the Laplace domain (Eqs. 6.58 to 6.64a). 

By definition of the moments of a distribution, the nth moment of the band profile at the exit of a chromatographic bed of length z = L is 

The theorem of Van der Laan [42] permits calculation of the moments of the response of the chromatographic column to a pulse injection, from the analytical solution in the Laplace domain 

Moments are often used in connection with the different formulations and applications of the general rate model because this model can often be solved algebraically in the Laplace domain and, although this solution cannot be inverted into the spatial domain, the moments of this solution can most often be derived as analytical expressions. However, the use of band moments encounters serious problems both on the calculation and the application fronts. 

The derivation of analytical expressions for the moments of a band in chromatography is tedious. It involves successive differentiations of the Laplace transform solution of the chromatography model used. Several more expedient methods have been proposed to simplify these derivations for axial chromatography [43,44]. A simple and generalized method was described by Lee et al. [45] for the moments in chromatographic elution peaks with any geometric configuration (axial or radial) and any kinetic models. 

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