Moment Analysis Method for Measurement of Reaction Equilibrium and Rate Constants by Using High-Performance Liquid Chromatography

The moment analysis method was developed for the determination of association equilibrium constant (KA) and association (ka) and dissociation (kd) rate constants of intermolecular interactions between solute and ligand molecules. They are accurately determined by using moment equations from elution peak profiles because they are measured by using high-performance liquid chromatography (HPLC) under preferable conditions that neither immobilization nor chemical modification (i.e., fluorescence labeling) of solute and ligand molecules is required. To demonstrate the effectiveness of the method, it was applied to the inclusion complex formation system between dibenzo-18-crown-6 (DB18C6) and alkaline earth metal cations, i.e., Mg2+, Ca2+, and Sr2+, as a concrete example. Because the diameter of the three metal cations is smaller than that of the inner cavity of DB18C6, the values of KA, ka, and kd were analytically determined by assuming the stoichiometry of 1:1 between DB18C6 and the metal cation. They reflected the influence of the difference in the size between the inner cavity of DB18C6 and the metal cations on the inclusion complex formation. It seems that the moment analysis method based on HPLC separation is effective for the multifaceted analysis of chemical reactions because some characteristics of the method are different from those of other conventional methods. It must contribute to the dissemination of an opportunity for the study of chemical reactions to many researchers because of the versatility of HPLC.

Simplification of Moment Analysis Procedure for Kinetic Study of Chromatographic Behavior of Core-shell Particles

The moment analysis method for chromatographic behavior in core-shell columns was simplified. Mass-transfer phenomena other than intra-stationary phase diffusion are analyzed while considering that the packing materials are spherical particles. The manner of intra-stationary phase diffusion is analyzed while assuming a hypothetical flat plate. For most core-shell particles commercially available, the geometry of a spherical thin layer can be supposed as a hypothetical flat plate with a relative error of less than ca. 2% because the thickness of the shell layer is sufficiently smaller than the diameter of whole particle. This supposition makes moment analysis easier because the moment equations for flat plates are simpler than those strictly developed for core-shell particles. Some chromatographic data measured using a core-shell column were analyzed by the simple moment analysis method to confirm its usefulness. It was demonstrated that the method is effective for a preliminary study of mass-transfer kinetics in core-shell columns.

Three-dimensional graphing representing six variables for speed and separation performance in liquid chromatography

The two variables, flow rate and column length, enable naive determination of the number of theoretical plates (N) in isocratic elution; this, in turn, enables the formation of a three-dimensional graph with N as the z-axis. An alternate three-dimensional graph with N as the z-axis can be drawn, then, with the alternate basal plane illustrating the pressure drop and hold-up time. In this article, the pressure drop and hold-up time are formulated so as to be represented unitarily in the former graph, because the flow rate and column length interact simultaneously as operational variables. This formulation manipulates both the pressure drop and the hold-up time as logarithmic axes, to evaluate the landscape. Also of use is the representation, in the same graph, of the height equivalent to a theoretical plate, as the fundamental property of the packing supports. For this purpose, the number of theoretical plates per unit length are here introduced as the sixth variable, instead of the height equivalent to a theoretical plate. Representing the six variables in three-dimensional graphs enables a clear understanding both of the separation condition optimization methods and the relation among variables for the speed and separation performance. The linear velocity, column length, N, velocity-length product, hold-up time, and number of theoretical plates per unit length, are here selected as the six elementary variables for the three-dimensional graphs; and, based on the packing supports of 2, 3, and 5-μm particle and monolithic columns. Finally, the usage of logarithmic three-dimensional graph is illustrated for understanding the speed and separation performance.

Simple Moment Analysis for a Kinetic Study of the Chromatographic Behavior of Spherical Particles and Silica Monoliths

A simple procedure of moment analysis was proposed for a kinetic study of the rate processes in the columns packed with full-porous spherical particles and silica monoliths. Previous chromatographic data measured in reversed-phase HPLC systems using Mightysil and Chromolith columns were analyzed by a simple moment analysis. The surface of the packing materials is chemically modified with octadecyl alkyl ligands. A mixture of methanol and water (80/20, v/v) and alkylbenzene homologous series (C₆H₅C_nH_2n+1, n = 0 - 7) were used as the mobile-phase solvent and sample probes, respectively. More detailed information about the experimental conditions is provided in Supporting Information. The values of the intra-stationary phase diffusivity (D_e) and the surface diffusion coefficient (D_s), derived by the simple moment analysis, were almost the same as those by the conventional moment analysis. The simple moment analysis is effective for quantitative studies of mass transfer in chromatographic systems. The previous chromatographic data were also analyzed by assuming external porosity (ε_e) as typical values, i.e., 0.40 for spherical particles and 0.70 for silica monoliths. The resulting values of D_e and D_s were of the same order of magnitude as those derived by using ε_e experimentally measured. Even if ε_e is assumed to be typical values, the simple moment analysis is effective for preliminary studies of the mass-transfer kinetics in the columns.

Effect of the polydispersity on the dispersion of polymers through silicas having different morphologies (fully porous and core-shell particles and monoliths)

The effect of the polydispersity of polystyrenes on the dispersion through silicas having different morphologies (fully porous, core-shell particles and monoliths) was investigated. The heights equivalent to a theoretical plate (HETP) of those columns were measured for a small molecule (toluene) and a series of polystyrenes of different sizes in non-adsorbing conditions. The different contributions to the total HETP including polydispersity were determined experimentally. The longitudinal diffusion and the mass transfer resistance term were obtained from peak parking experiments. The eddy dispersion was obtained from models and experiments. The effect of polydispersity on the HETP values (H_poly) can thus be calculated from the total HETP by substraction of the other contributions. The results were compared to the Knox model which surestimates the H_poly values for porous and core-shell particles which is usually explained by an overestimation of the polydispersity index (PDI) given by the manufacturer. The PDI of two polymers (P02, M_w= 690 g.mol^-1 and P03, M_w=1380 g.mol^-1) was verified by liquid chromatography by separating each fraction of the polymer on the silica columns by using adsorbing conditions which are obtained with a mixture of heptane and THF. The PDI obtained are comparable to the PDI given by the manufacturer meaning that the assumptions made by Knox are not entirely valid. A direct method is proposed in this paper in order to determine H_poly. In this method the excess of spreading as compared with a polymer with only one size corresponding to the average size is studied assuming the polymer size distribution is gaussian. The H_poly values obtained by the direct method are comparable to the experimental values.

Three approaches to improving performance of liquid chromatography using contour maps with pressure, time, and number of theoretical plates

Attempts to improve HPLC performance often focus on increasing the speed or separation performance. In this article, both the flow rate and column length are optimized as separation conditions, while observing the number of theoretical plates and hold-up time with isocratic elutions. In addition, the upper pressure limit must be simultaneously considered as the boundary condition. Approaches based on the optimal velocity (Opt.) are often adopted; but the kinetic performance limit (KPL) in Desmet's method can also be utilized for three-dimensional graphing with axes of pressure, time, and number of theoretical plates. Here, two approaches involving pressure increase are introduced, beginning with the condition of optimal linear velocity: one aimed at greater speed and the other at higher resolution. Coefficients of pressure-application are derived to measure the effectiveness of the intermediate conditions between the Opt. and KPL methods. In the third approach, the hold-up time is extended while maintaining a fixed pressure. Coefficients of time-extension are also derived, to determine the effectiveness to improve the separation performance.

Moment theory for the analytical determination of rate constants for solute permeation at the interface of spherical molecular aggregates

Moment equations were developed on the basis of the Einstein equation for diffusion and the random walk model to analytically determine the rate constant for the interfacial solute permeation from a bulk solvent into molecular aggregates (k_in ) and the inverse rate constant from the molecular aggregates to the bulk solvent (k_out ). The moment equations were in good agreement with those derived in a different manner. To demonstrate their effectiveness in one concrete example, the moment equations were used to analytically determine the values of k_in and k_out of three electrically neutral solutes, i.e. resorcinol, phenol, and nitrobenzene, from the first absolute (μ_1A ) and second central (μ_2C ) moments of their elution peaks, as measured by electrokinetic chromatography (EKC), in which the sodium dodecyl sulfate (SDS) micelles were used as a pseudostationary phase. The values of k_in and k_out should be determined with no chemical modifications and no physical action with the molecular aggregates because they are dynamic systems formed through weak interactions between the components. The moment analysis of the elution peak profiles measured by EKC is effective to unambiguously determine k_in , k_out , and the partition equilibrium constant (k_in /k_out ) under appropriate experimental conditions.

Moment analysis for mass transfer kinetics at the interface of spherical molecular aggregates

New moment equations were developed on the basis of the principle of relativity for explaining some characteristics of elution peaks measured by electrokinetic chromatography (EKC) using spherical molecular aggregates. Basic equations representing mass balance and mass transfer kinetics in EKC system in a Galilean coordinate system S were transformed to those in another coordinate system S', which imaginarily moved with respect to S. Moment equations for EKC peaks in S' in the time domain were derived from the analytical solution of the modified basic equations in the Laplace domain. Moment equations for EKC peaks in S were derived from those in S' by the inverse Galilean transformation. The moment equations were used to the re-analysis of EKC data previously measured. The values of permeation rate constants of thymol at the interface of sodium dodecylsulfate micelles were fairly in agreement with those determined in a previous study. The moment equations were also used to the numerical simulation of elution peaks in EKC systems. The influence of some experimental parameters on elution peak profiles was quantitatively analyzed. The moment equations are useful for determining the rate constants of interfacial solute permeation from elution peak profiles measured by EKC.

Moment analysis for reaction kinetics of intermolecular interactions

Moment equations were developed on the basis of the principle of relativity for analyzing elution peak profiles measured by ACE to analytically determine the association (k_a ) and dissociation (k_d ) rate constants of intermolecular interactions. Basic equations representing the mass balance, mass transfer rate, and reaction kinetics in ACE system in a Galilean coordinate system S were transformed to those in another coordinate system S', which imaginarily moved with respect to S. Moment equations for ACE peaks in S' in the time domain were derived from the analytical solution of the modified basic equations in the Laplace domain. Moment equations for ACE peaks in S were derived from those in S' by the inverse Galilean transformation. The moment equations were used for analyzing some ACE data previously published to determine k_a and k_d values. It was demonstrated that the moment equations were effective for extracting the information about affinity kinetics of intermolecular interactions from the elution peak profiles measured by ACE. The moment equations were also used to discuss the influence of mass transfer and reaction kinetics on ACE peak profiles. Some results of the numerical calculations are also indicated in Supporting Information.

Moment Analysis of Mass Transfer Kinetics in Micellar Electrokinetic Chromatography Systems

Moment equations were developed for quantitatively studying the separation characteristics of micellar electrokinetic chromatography (MEKC). They explain how the first absolute and second central moments of elution peaks are correlated with some fundamental parameters of the partition equilibrium and mass transfer kinetics in MEKC systems. In order to discuss the influence of the mass transfer kinetics on peak broadening, the moment equations were used to analyze the separation behavior in MEKC systems. Separation conditions were chosen on the basis of practical MEKC experiments previously conducted. It was quantitatively clarified that both the solute permeation at the interfacial boundary of surfactant micelles and axial diffusion of solute molecules in a capillary had a predominant contribution to the spreading of the elution peaks in MEKC systems. This is a preliminary study for the analytical determination of rate constants concerning solute permeation at the interface of surfactant micelles from elution peak profiles measured by MEKC. In addition, it was also indicated that the experimental conditions of MEKC systems could be controlled so that the interfacial solute permeation would have a predominant role for the band broadening. For example, the contribution of the interfacial permeation was about 33 times larger than that of the axial diffusion of solute molecules under the MEKC conditions in a previous study. This means that the rate constants could appropriately be determined for the interfacial solute permeation.

Moment Analysis Theory for Size Exclusion Capillary Electrochromatography with Chemical Reaction of Intermolecular Interaction

New moment equations were developed for size exclusion capillary electrochromatography (SECEC), in which intermolecular chemical reactions simultaneously took place. They explain how the first absolute and second central moments of elution peaks are correlated with some fundamental equilibrium and kinetic parameters of mass transfer and chemical reaction in SECEC column. In order to demonstrate the effectiveness of the moment equations, they were used to predict chromatographic behavior under hypothetical SECEC conditions. It was quantitatively studied how the association and dissociation rate constants of intermolecular interaction affected the position and spreading of elution peaks. It was indicated that both the intermolecular reaction kinetics and axial dispersion of solute molecules in a capillary column had a predominant contribution to the band broadening.

Recent advancements and future directions of superficially porous chiral stationary phases for ultrafast high-performance enantioseparations

This review focuses on the use of superficially porous particles (SPPs) as chiral stationary phases for ultra-high performance liquid enantioseparations.

Kinetic Study of Interaction between Solute Molecule and Surfactant Micelle

We developed moment analysis of affinity kinetics by chromatographic capillary electrophoresis (MKCCE) method for the kinetic study of intermolecular interactions. Association and dissociation rate constants of the interaction in a micellar electrokinetic chromatography (MEKC) system between thymol and sodium dodecylsulfate micelle were determined by the MKCCE method. It is a method based on the moment theory for the kinetic study of intermolecular interactions under the conditions that neither immobilization nor chemical modification of molecules is required. In CCE mode, experimental conditions are controlled so that the migration of solute-micelle complex is stopped and only solute molecules migrate in a capillary. Mass transfer behavior of solute molecules in the CCE system is analogous to that in a chromatographic system. However, because it was difficult in practice to really perform CE experiments under the CCE conditions, CE data were measured with changing experimental conditions, i.e., applied pressure, under the conditions that the migration velocity of solute-micelle complex was around zero. The rate constants could be analytically determined from the CE data. In the MKCCE method, it is not necessary to fit elution curves numerically calculated to those experimentally measured for the determination of the rate constants. Regarding the interaction between thymol and SDS micelles, association equilibrium constant and association and dissociation rate constants were determined as 6.35 × 10(3) dm(3) mol(-1), 5.6 × 10(4) dm(3) mol(-1) s(-1), and 8.7 s(-1), respectively. It was demonstrated that the MKCCE method was effective for the kinetic study of intermolecular interactions.

Temperature Effects on Retention and Separation of PAHs in Reversed-Phase Liquid Chromatography Using Columns Packed with Fully Porous and Core-Shell Particles

Effects of temperature on the reversed-phase chromatographic behavior of PAHs were investigated on three columns. The first was the recent C18column (250 mm × 4.6 mm) packed with 5 µm core-shell particles while the others were more conventional C18columns (250 mm × 4.6 mm) packed with fully porous particles. Among the 16 PAHs studied, special attention has been paid to two pairs of PAHs, fluorene/acenaphthene and chrysene/benzo[a]anthracene, which often present coeluting problems. Due to the low surface area of the core-shell particles, lowest retention time of each PAH was highlighted and effects of the temperature on the separation of PAHs were negligible in regard to those using columns packed with fully porous particles. For each PAH studied, it was demonstrated that peaks were symmetrical and may be considered as Gaussian peaks when the column packed with core-shell particle was employed. In the best condition, the separation of PAHs was conducted at 16°C under very low pressure values (670–950 psi = 46–65 bars). Depending on PAHs, the limit of detection ranged from 0.88 to 9.16 μg L−1. Analysis of spiked acetonitrile samples with PAHs at 10 and 50 µg L−1and tap water at 10 µg L−1gave very good recoveries (94%–109.3%) and high precision (1.1%–3.5%).