Recurrent relations in chromatography

Abstract

One of the main features of first-order recurrent equations y(x + Dx) = ay(x) + b (Dx = const) is that they lead to a linear form of various monotonic functional dependencies, including chemical variables. Such unique capabilities of recurrent relations are due to the fact that they combine the properties of arithmetic and geometric progressions, and this explains the possibility of linearisation of various retention time dependencies, both in gas and high-performance liquid chromatography (HPLC). Examples of the use of recurrent relations to characterize the dependence of analyte retention times on temperature in gas chromatography and on the concentration of organic components of eluents in reversed-phase HPLC are considered. They are effective in approximating variations in the retention indices of homologues and congeners depending on their position in the corresponding taxonomic groups of organic compounds. Moreover, based on the linearity of the recurrent dependence of the retention indices of analytes of various groups, it can be assumed that they belong to homologues or congeners. This identification option is illustrated by the example of previously unknown products of partial hydrolysis of tetraethoxysilane – triethoxysilanol, diethoxysilanediol and ethoxysilanetriol.

Deviations of the recurrent approximation of retention parameters from linearity are also informative. This form of comparison of gas chromatographic retention indices of homologues allows identifying steric interactions in molecules, as illustrated by the example of a series of methyl-substituted benzenes. Recurrent approximation of the retention indices of benzene, toluene, m-xylene (1,3-dimethylbenzene) and mesitylene (1,3,5-trimethylbenzene) are linear, while the retention index of sterically strained isodurol (1,2,3,5-tetramethylbenzene) does not correspond to this dependence. In reversed-phase HPLC, the recurrent representation of the dependence of the retention times of analytes on the content of the organic component of the eluent allows identifying the shift in the positions of acid-base equilibria in solutions. In addition, the deviations of recurrent approximations of retention times from linearity in regions with high water content in the eluent allows detecting the reversible formation of hydrates of the analysed compounds.