On the Use of the Knox Equation. II. the Efficiency Measurement Problem

Abstract

The Knox equation, which relates the reduced plate height, h, to the reduced linear velocity, v, has a great importance in column and stationary phase testing. Significant A, B, and C terms of the Knox equation cannot be obtain with erroneous plate heights, h. Four methods for efficiency determination, the inflection (0.6H) method, the height/area method, an asymmetry based (0.1H) method and the moment method, were critically compared using 200 real chromatograms. Efficiency obtained using Gaussian assuming methods were highly overestimated, while the asymmetry based method gave quite acceptable results. Using the exponentially modified Gaussian model, it is demonstrated that the b/a ratio (the sum b + a is the peak-width at 10% of the peak height) is not an empirical figure of merit but can be related to a “peak skew” expression.     

The dependence of reduced plate height on reduced velocity in carbon dioxide supercritical fluid chromatography with packed columns

Summary The relation is proposed for the dependence of reduced plate height, h, on reduced velocity in carbon dioxide, supercritical fluid chromatography with packed columns. The classical Knox model does not take into account the density drop in the column which produces a peak-broadening. A supplementary coefficient with a parabolic dependence on velocity must be added and then a good fit between experimental results and theory is observed. Finally, the influence of other parameters such as pressure, capacity factor and solute molecular area is discussed.     

Efficiency for unretained solutes in packed column supercritical fluid chromatography. I. Theory for isothermal conditions and correction factors for carbon dioxide

A general theory for efficiency of nonuniform columns with compressible mobile phase fluids is applied to the elution of an unretained solute in packed-column supercritical fluid chromatography (pSFC). The theoretical apparent plate height under isothermal conditions is given by the Knox equation multiplied by a compressibility correction factor f1, which is equal to the ratio of the temporal-to-spatial average densities of the mobile phase. If isothermal conditions are maintained, large pressure drops in pSFC should not result in excessive efficiency losses for elution of unretained solutes.     

Modeling chromatographic dispersion: a comparison of popular equations

An analyte that is introduced onto a column as a finite band broadens as it moves along the column. This band-broadening is generally attributed to three independent processes, including flow path inequalities, molecular diffusion, and resistance to mass transfer. Many equations have been derived in attempts to mathematically model the process. Some of the more popular of these include the equations of van Deemter, Giddings, Horvath and Lin, and Knox. Although the basis of each equation is theoretically different, the differences among them are minor, and most of the equations can be used to adequately fit plate height data. Chromatographers often collect efficiency data to monitor the performance of their columns, and then use one of the above equations to fit the data. The choice of which equation to use can be daunting, since the theories are conflicting. Using an extensive collection of data, we have compared these equations on the basis of the resulting fit. This study was performed using analytes covering a wide range of retention values and mobile phases of differing strengths. The Foley-Dorsey equation was used to calculate the number of theoretical plates for the efficiency study and special precautions were taken to ensure that the observed broadening was due to only processes occurring in the column and that the peaks were adequately sampled. The variance from extra-column sources was measured and subtracted from the system variance. Although more complex equations gave very slightly better fits, 50 years after its introduction the van Deemter equation remains an incredibly accurate representation of band-broadening processes.     

Non-equilibrium model of liquid column chromatography : I. Exact expressions for elution profile moments and relation to plate height theory

A model for column chromatography that includes non-equilibrated mass transfer, diffusion, and generalized boundary conditions at the top of the column is solved by the method of moments. The theory predicts that as the macroscopic velocity u becomes very small, the plate height approaches a constant (rathe than diverging as u⁻¹. As u increases, the plate height drops to a minimum and then increases. We show that the past assumptions that mass transfer and various types of diffusion contribute additively to the plate height, hold only at or beyond the minimum. We show further that our expression fits observed data with only a single adjustable lumped parameter. The parameter contains the mass transfer rate k⁻¹ for moving out of a bead. The data fit, as well as an analytical approximation that we derive for the position of the minimum, provides a relation between k₋₁ and bead particle size (d_p) and consequently between d_p and profile dispersion. Conditions under which the peak of the elution profile is a good approximation to the mean are described. For a suitably chosen flow-rate, the mass transfer rates can be estimated from the observed dispersion in the elution profile.     

Rapid determination of parabens in seafood sauces by high‐performance liquid chromatography: A practical comparison of core–shell particles and sub‐2 μm fully porous particles

In this work, the chromatographic performance of superficially porous particles (Halo core–shell C₁₈ column, 50 mm × 2.1 mm, 2.7 μm) was compared with that of sub‐2 μm fully porous particles (Acquity BEH C₁₈, 50 mm × 2.1 mm, 1.7 μm). Four parabens, methylparaben, ethylparaben, propylparaben, and butylparaben, were used as representative compounds for calculating the plate heights in a wide flow rate range and analyzed on the basis of the Van Deemter and Knox equations. Theoretical Poppe plots were constructed for each column to compare their kinetic performance. Both phases gave similar minimum plate heights when using nonreduced coordinates. Meanwhile, the flat C‐term of the core–shell column provided the possibilities for applying high flow rates without significant loss in efficiency. The low backpressure of core–shell particles allowed this kind of column, especially compatible with conventional high‐performance liquid chromatography systems. Based on these factors, a simple high‐performance liquid chromatography method was established and validated for the determination of parabens in various seafood sauces using the Halo core–shell C₁₈ column for separation.     

Intraparticle Flow and Plate Height Effects in Liquid Chromatography Stationary Phases

Abstract

Velocity independent plate heights were apparently first recognized for hydrodynamic chromatography columns, packed with nonporous, 115 micron glass beads which were run at reduced mobile phase velocities of 10 to 10,000. Hydrodynamic chromatography separates based on the tendency of small molecules (or particles) to associate with slower moving fluid streamlines near the surfaces of particles, compared to larger molecules which seek faster streamlines. Consequently, the larger molecules elute first. Velocity independent plate heights in liquid chromatography have also been observed for nonadsorbed solutes in paniculate and fibrous stationary phases. These stationary phases have pores which exceed 10^?4 to 10^?5 cm in dimension. The flat plate height is attributed to flow in the channels formed by these large intraparticle spaces. The development of plate height expressions which represent dispersion at interstitial velocities above 10 cm/min are discussed. Explanations of the uncoupling of dispersion from eluent flow rate in continuous stationary phases, membranes, and gigaporous particles is shown to have their origins in the studies of distribution of particles and molecules in hydrodynamic chromatography columns, and to be adequately described by modifications of the van Deemter equation.     

The Effect of Column Diameter on HPLC Separations Using Constant Length Columns

Abstract

The effect of column dimensions on resolution, sample capacity, mobile-phase use and efficiency using both constant flow-rate and constant linear velocity was studied. The four columns selected have the same length (238 mm), but different diameters (column A: 4.6 mm; column B: 4.0 mm; column C: 3.2 mm and column D: 2.1 mm). It was observed that at constant floworate the reduced plate height was the lowest for columns B and C, higher for column A and highest for column D. On the other hand increasing the injected quantity while keeping injected volume constant revealed columns B and C to have a better efficiency only at the low concentrations. Column A with the largest diameter was superior at high concentrations and thus offers the best loadability. Column D offered, as expected, poor loadability. Examination of the columns at the same calculated linear velocity, showed no appreciable changes in efficiency for the four columns. Preferable detection limits and big solvent savings were obtained when small column diameters were used.     

Graphical method for the determination of the complex NMR shift and equilibrium constant for a hetero-association accompanying self-associations

We describe a novel graphical method which, in conjunction with the previously proposed graphical determination of monomer shift, dimer shift, and dimerization constant for self-association, allows us to determine the complex shift and equilibrium constant for a hetero-association, A + B ⇌ AB, accompanying self-associations, A + A ⇌ A₂, and B + B ⇌ B₂. The merit of the new method includes the removal of the restrictions imposed on the conventional Benesi-Hildebrand (B-H) plot: (1) that the concentration of one component must be much less than that of the other; (2) that there be no accompanying self-association. The simultaneous equilibrium of the self-association of 2-pyrrolidone (A) and that of 4-methyl-α-pyrrolidone (B) and the hetero-association between A and B in acetonitrile-d₃ at 25‡C is studied. The inappropriateness of the B-H plot in dealing with this case is also pointed out.     

Radial distribution of the contributions to band broadening of a silica‐based semi‐preparative monolithic column

Using an on‐column local electrochemical microdetector operated in the amperometric mode, band elution profiles were recorded at different radial locations at the exit of a 10 mm id, 100 mm long silica‐based monolithic column. HETP plots were then acquired at each of these locations, and all these results were fitted to the Knox equation. This provided a spatial distribution of the values of the eddy diffusion (A), the molecular diffusion (B), and the resistance to the kinetics of mass transfer (C) terms. Results obtained indicate that the wall region yields higher A values and smaller C values than the central core region. Significant radial fluctuations of these contributions to band broadening occur throughout the exit column cross‐section. This phenomenon is due to the structural radial heterogeneity of the column.     

Effects of Injected Volume and Applied Voltage on Column Efficiency in Capillary Electrochromatography with Open Tubular Columns of 10 μm i. d.

The effect of some instrumental parameters on column efficiency in open tubular capillary electrochromatography (OTCEC) has been evaluated. First, it was investigated whether band broadening due to the sample injection process is within a tolerable range when an open tubular column (OTC) of about 10 μm i. d. is used. As a result of the small injection profile factor (K² = 1.3), injected volumes must be sufficiently small (less than 10 pL) to avoid a significant efficiency loss (>5%) when hydrodynamic injection by siphoning is employed. Secondly, the kinetic performance of OTCs in a CEC system was estimated from the variation of the reduced plate height (h) with the reduced linear velocity (ν) which was controlled by the voltage applied. Reasonable agreement was obtained between the theoretical h versus ν curve and the experimental values for a group of polycyclic aromatic hydrocarbons used as test compounds. Values of 0.25 for minimum h at an optimum ν of 16 are estimated, which permit separations with around 400,000 plates per meter to be obtained in less than 5 min. Finally, the possibility of estimating the diffusion coefficients of the solutes in the mobile phase from the plot of the height of a theoretical plate versus electroosmotic flow velocity is shown.     

A synopsis of empirical gel fraction analytical methods: Application of the Wanxi equation to polyethylenes irradiated in vacuo and in the presence of acetylene

Gel fraction data from linear low-density polyethylene (LLDPE) films and high-modulus polyethylene (HMPE) fibers, which had previously failed to give reasonable results when treated with Charlesby-Pinner and Saito-Kang-Dole theory, have been fitted to a modified form of the Wanxi equation. The fitting yields parameters that compare very well with literature values derived by the Wanxi equation, confirming that fracture and cross-linking densities are proportional to a power function (β) of the radiation dose. Extrapolated G(S) values have been obtained from LLDPE using the fit parameters and UV-Vis spectroscopy data. A brief synopsis of the empirical analytical methods available for gel fraction analysis is given. © 1994 John Wiley & Sons, Inc.     

Treating the polyelectrolytes as polymers with a draining effect. I. The statistical segment length

The statistical segment length of polyelectrolytes, A, in aqueous salt solutions is derived from the unperturbed dimensions parameter, K_θ, that is obtained from the graphical representations based on the Stockmayer–Fixman–Burchard and Dondos–Benoît equations. In order to obtain A from K_θ we use values of the Flory's parameter, Φ, which are given from an empirical equation established for the polymers presenting a draining effect, such as the wormlike polymers. With these values of Φ, the obtained values of A for different polyelectrolytes in aqueous salt solutions of different ionic strength are found very close to the values obtained from other more complicated methods. This result shows that the polyelectrolytes can be considered as polymers presenting a draining effect. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 4225–4229, 2004     

A Modified Method for Estimating Unperturbed Dimensions of Macromolecules from Viscosity Data

Abstract

A new empirical equation is presented which modifies the Krigbaum equation for calculating Kθ values from measurements of intrinsic viscosity in good solvents. This is found to fit the molecular weight data better for the polymer systems investigated.     

Activity coefficients in aqueous mixtures of hydrochloric acid and lanthanum chloride at 25°C

Activity coefficients of hydrochloric acid in aqueous solutions of lanthanum chloride were determined by an emf method at 25°C over an ionic strength range 0.05–3 mol-kg^–1. Harned's rule was obeyed within experimental error by the acid in all the mixtures. However, the fit with Pitzer's equations was not as good as the Harned rule fit, even though the effects of higher-order electrostatic terms were considered. Activity coefficients for the salt in the mixtures were derived using the Pitzer equations and fitted to the Harned equation, whereupon Harned's rule was found to be valid for the salt up to an ionic strength of 0.3 mol-kg^–1 only.     

The influence of pressure on the separating properties of columns in gas chromatography

The Giddings model taking into account the dependence of the coefficients of the van Deemter equation on pressure was used to study changes in the efficiency of a hollow capillary column as the inlet and outlet carrier gas pressures changed. The observed dependence of height equivalent to a theoretical plate (HETP) in the coordinates of inlet and outlet pressures can be approximated by a surface having the shape of a folded sheet of paper, when minimum HETP values are situated along the bend line. Any surface section is actually a van Deemter curve in the corresponding coordinates. The dependence of the minimum HETP on inlet and outlet pressures, which determines the optimum parameters of column service, is of the greatest interest. It was shown that, over the range of pressures studied, the minimum HETP should monotonically decrease as the pressure increases. Experimental model verification showed close correspondence between the inlet and outlet pressures and the values predicted by the model. At the same time, the experimentally found improvement of the efficiency of the column was smaller than that predicted theoretically. Possible reasons for the discrepancy between theory and experiment are considered.     

Viscosity studies of poly(DL‐lactic acid) in supercritical CO2

The effect of supercritical CO₂ on the viscosity and activation energy to viscous flow of P_DLLA is investigated, using a high pressure parallel plate rheometer, over a range of temperatures (50–140 °C) and pressures (5–12 MPa). The Cross model is fitted to the data to enable calculation of the zero shear viscosity and critical shear rate. A significant decrease in the viscosity is observed on increasing both variables; however, at high temperatures, the pressure effect becomes negligible. An increase in the critical shear rate is also observed on raising the pressure, indicative of a reduction in the relaxation time of the polymer. Manipulation of the Arrhenius equation shows a reduction in the activation energy to viscous flow as the pressure is increased. Together, these results show that the melt processing temperature of P_DLLA can be reduced in the presence of supercritical and high pressure CO₂. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012     

Ab initio all-electron periodic Hartree-Fock study of hydrostatic compression of pentaerythritol tetranitrate

We present a computational study of hydrostatic compression effects on the pentaerythritol tetranitrate (PETN) energetic material up to 22.7 GPa by means of the ab initio all-electron periodic Hartree-Fock quantum mechanical method with the STO-3G Gaussian basis set. We fitted the calculated volume-energy relation to the energy SJEOS polynomial function from which we obtained the compression dependence of the pressure (P), the bulk modulus (B), and its pressure derivative (B'). We also fitted the experimental volume-pressure relation to the pressure SJEOS polynomial function, which allowed us to calculate the experimental bulk modulus (B(exp)) and its pressure derivative (). Our calculated values, B = 6.73 GPa and B' = 24.63, are in reasonable agreement with the values B(exp) = 8.48 GPa and = 14.42 from our fit to the experimental X-ray data and with the value B(exp) = 9.8 GPa that was derived from the experimental elastic constants. In addition, we present a discussion on how the lattice vectors and the internal coordinates (i.e., bond lengths, bond angles, and torsion angles) of the C(CH(2)ONO(2))(4) molecules in the PETN lattice change during hydrostatic compression of the crystal. Our calculated results suggest that the C(CH(2)ONO(2))(4) molecules cannot be considered as being rigid but are in fact flexible, accommodating lattice compression through torsions, bendings in their bond angles, and contractions in their bond lengths. At pressures higher than about 8 GPa, however, both the C(CH(2)ONO(2))(4) molecules and the c lattice vector seem to stiffen somewhat. The a lattice vector does not exhibit this stiffening. As a consequence, the pressure dependence of the c/a ratio shows a minimum at about 8 GPa.     

Effects of water and alcohol on the formation of inclusion complexes of d-limonene and cyclodextrins

Abstract

Systematic studies have been carried out on the role of water and alcohol in the formation of inclusion complexes between d-limonene and α-, β- and γ-cyclodextrin (CD) by a micro-aqueous method. The inclusion complex was barely formed at zero water content for all CDs. Above the specific water content for each CD, formation of the inclusion complex correlated well with an equation which was derived on the autocatalytic assumption for the inclusion phenomenon. The inclusion complex correlated well with an equation which was derived on the autocatalytic assumption for the inclusion phrnomenon. The minimum water content, which was defined as 1% of the maximum concentration of the inclusion complex formed, coincided with the number of water molecules inside the cavity of the CD. In the presence of ethanol, a significant amount of the inclusion complex was formed for β- and γ-CD/limonene systems, particularly at lower moisture content. However, for α-CD the inclusion fraction decreased significantly in the presence of ethanol. This means that ethanol inhibits the formation of the inclusion complex between x-CD and d-limonene. For other linear alcohols, the formation of the inclusion complex between d-limonene and β-CD increased with decreasing alkyl chain length. This suggests that the more hydrophilic and the smaller (in molecular size) the alcohol is, the more enhanced is the inclusion of d-limonene to β-CD.