Stochastic theory of size exclusion chromatography by the characteristic function approach

A general stochastic theory of size exclusion chromatography (SEC) able to account for size dependence on both pore ingress and egress processes, moving zone dispersion and pore size distribution, was developed. The relationship between stochastic-chromatographic and batch equilibrium conditions are discussed and the fundamental role of the 'ergodic' hypothesis in establishing a link between them is emphasized. SEC models are solved by means of the characteristic function method and chromatographic parameters like plate height, peak skewness and excess are derived. The peak shapes are obtained by numerical inversion of the characteristic function under the most general conditions of the exploited models. Separate size effects on pore ingress and pore egress processes are investigated and their effects on both retention selectivity and efficiency are clearly shown. The peak splitting phenomenon and peak tailing due to incomplete sample sorption near to the exclusion limit is discussed. An SEC model for columns with two types of pores is discussed and several effects on retention selectivity and efficiency coming from pore size differences and their relative abundance are singled out. The relevance of moving zone dispersion on separation is investigated. The present approach proves to be general and able to account for more complex SEC conditions such as continuous pore size distributions and mixed retention mechanism.     

Vibrational analysis of 3-trifluoromethylphenol

The molecular geometry and molecular vibrations of 3-trifluoromethylphenol have been investigated by means of quantum chemical calculations and vibrational spectroscopy. The computations indicated the preference of the conformer with the OH hydrogen pointing in the direction of the trifluoromethyl group by 0.9 kJ/mol with respect to the anti conformer. FT-IR spectra of the vapour and CCl₄ solution as well as FT-IR and FT-Raman spectra of the pure liquid have been recorded in the range of 4000–150 cm⁻¹. The interpretation of the spectra was based on a scaled quantum mechanical (SQM) analysis for which the initial force field was calculated at the Becke3-Lee-Yang-Parr (B3LYP) DFT level supplemented with a 6-311++G** basis set. Using 11 scale factors refined in the present study an rms deviation of 7.6 cm⁻¹ between the experimental and SQM frequencies has been achieved. On the basis of the computations 40 of the total of 42 fundamentals of the title compound have been assigned.     

Overloaded gradient elution chromatography on heterogeneous adsorbents in reversed-phase liquid chromatography

Overloaded band profiles of phenol were measured on a C18-Kromasil column in gradient elution conditions. The mobile phase used was a mixture of methanol and water. The volume fraction of methanol was allowed to vary between 0 and 0.5. A general adsorption model, which expresses the amount of phenol adsorbed q* as a function of both its concentration C and the composition phi of the organic modifier (methanol) in the mobile phase, was empirically derived from previous independent adsorption experiments based on frontal analysis (FA) and frontal analysis by the characteristic point (FACP). Accordingly, the general model was an extension of the simplest heterogeneous model, the Bilangmuir model, to non-isocratic conditions. The low-energy sites followed the classical linear solvent strength model (LSSM), but not the high-energy sites whose saturation capacity linearly decreased with phi. The general model was validated by comparing the experimental and simulated band profiles in gradient elution conditions, in linear and non-linear conditions, as well. The band profiles were calculated by means of the equilibrium-dispersive model of chromatography with a finite difference algorithm. A very good agreement was observed using steps gradient (delta phi) from 0 to 50% methanol and gradient times t(g) of 20, 25, 30, 40, 60, 80 and 100 min. The agreement was still excellent for steps gradient from 5 to 45% (t(g) = 25 min), 5 to 35% (t(g) = 50 min), 5 to 25% (t(g) = 50 min) and 5 to 15% (t(g) = 50 min). Significative differences appeared between experience and simulation when the slope of the gradient (delta phi/t(g)) became too strong beyond 3.3% methanol per minute. This threshold value probably mirrored the kinetic of arrangement of the G18-bonded chains when the methanol content increased in the mobile phase. It suggested that the chromatographic system was not in a full thermodynamic equilibrium state when very steep mobile phase gradients were applied.     

Permutation functions and (n,m)-commutativity of semigroups

By [4], a semigroupS is called an (n, m)-commutative semigroup (n, m ∈ ?⁺, the set of all positive integers) if $$x_1 x_2 \cdot \cdot \cdot x_n y_1 y_2 \cdot \cdot \cdot y_m = y_1 y_2 \cdot \cdot \cdot y_m x_1 x_2 \cdot \cdot \cdot x_n $$ holds for allx ₁,...,x _n ,y ₁,...,y _m ∈S It is evident that ifS is an (n, m)-commutative semigroup then it is (n′,m′)-commutative for alln′≥n andm′≥m. In this paper, for an arbitrary semigroupS, we determine all pairs (n, m) of positive integersn andm for which the semigroupS is (n, m)-commutative. In our investigation a special type of function mapping ?⁺ into itself plays an important role. These functions which are defined and discussed here will be called permutation functions.     

Complete solutions of a family of quartic Thue and index form equations

Continuing the recent work of the second author, we prove that the diophantine equation

for has exactly 12 solutions except when , when it has 16 solutions. If denotes one of the zeros of , then for we also find all with .

     

Syntheses of Theaspirone and Vitispirane via Palladium(II)-Catalyzed Oxaspirocyclization

Total syntheses of theaspirone (A and B) and vitispirane (A and B) are described. The key step in the syntheses is the palladium(II)-catalyzed intramolecular oxaspirocyclization of diene alcohol 4 to either vitispirane or the allylic alcohol 9. The outcome of the oxaspirocyclization is very much dependent on the solvent employed. In water-acetic acid (4:1) a 1:1 mixture of the diastereomeric alcohols 9A and 9B was exclusively formed. In water with 8 equiv of a strong non-nucleophilic acid, vitispiranes A and B (1:1) were obtained. An alternative procedure to obtain vitispirane with the use of LiCl and K(2)CO(3) is described. In the latter reaction vitispirane B is formed preferentially. This result is explained by an equilibrium between the two possible pi-allyl complexes 5A and 5B, the kinetically favored 5B being transformed into vitispirane 3B before isomerization to 5A occurs.     

Treating singularities present in the Sutcliffe-Tennyson vibrational Hamiltonian in orthogonal internal coordinates

Two methods are developed, when solving the related time-independent Schrodinger equation (TISE), to cope with the singular terms of the vibrational kinetic energy operator of a triatomic molecule given in orthogonal internal coordinates. The first method provides a mathematically correct treatment of all singular terms. The vibrational eigenfunctions are approximated by linear combinations of functions of a three-dimensional nondirect-product basis, where basis functions are formed by coupling Bessel-DVR functions, where DVR stands for discrete variable representation, depending on distance-type coordinates and Legendre polynomials depending on angle bending. In the second method one of the singular terms related to a distance-type coordinate, deemed to be unimportant for spectroscopic applications, is given no special treatment. Here the basis set is obtained by taking the direct product of a one-dimensional DVR basis with a two-dimensional nondirect-product basis, the latter formed by coupling Bessel-DVR functions and Legendre polynomials. With the basis functions defined, matrix representations of the TISE are set up and solved numerically to obtain the vibrational energy levels of H3+. The numerical calculations show that the first method treating all singularities is computationally inefficient, while the second method treating properly only the singularities having physical importance is quite efficient.     

Comparison between adsorption isotherm determination techniques and overloaded band profiles on four batches of monolithic columns

The adsorption isotherms of 4-tert.-butyl phenol were measured on four different monolithic columns, using three different techniques, classical frontal analysis (FA), the perturbation on a plateau method (PP) and the recently introduced numerical procedure known as the inverse numerical method (IN). This last approach requires only the recording of a few overloaded profiles and has the potential advantage of affording a dramatic decrease of the amounts of compounds, solvent, and time needed to determine accurate estimates of the coefficients of the isotherm. The reproducibility of the adsorption data measured on the four columns is discussed with reference to the specific techniques used for obtaining these data and to the most suitable equation used for modeling them. The data obtained for the different columns were highly consistent. The inverse numerical approach was confirmed to provide a powerful, accurate, and economic method for measuring single component adsorption data.     

Modeling of overloaded gradient elution of nociceptin/orphanin FQ in reversed-phase liquid chromatography

The Reversed-phase (RP) gradient elution chromatography of nociceptin/orphanin FQ (N/OFQ), a neuropeptide with many biological effects, has been modeled under linear and non-linear conditions. In order to do this, the chromatographic behavior has been studied under both linear and nonliner conditions under isocratic mode at different mobile phase compositions--ranging from 16 to 19% (v/v) acetonitrile (ACN) in aqueous trifluoracetic acid (TFA) 0.1% (v/v)-on a C-8 column. Although the range of mobile phase compositions investigated was quite narrow, the retention factor of this relatively small polypeptide (N/OFQ is a heptadecapeptide) has been found to change by more than 400%. In these conditions, gradient operation resulted thus to be the optimum approach for non-linear elution. As the available amount of N/OFQ was extremely reduced (only a few milligrams), the adsorption isotherms of the peptide, at the different mobile phase compositions examined, have been measured through the so-called inverse method (IM) on a 5 cm long column. The adsorption data at different mobile phase compositions have been fitted to several models of adsorption. The dependence of the isotherm parameters on the mobile phase composition was modeled by using the linear solvent strength (LSS) model and a generalized Langmuir isotherm that includes the mobile phase composition dependence. The overloaded gradient separation of N/OFQ has been modeled by numerically solving the equilibrium-dispersive (ED) model of chromatography under a selected gradient elution mode, on the basis of the previously determined generalized Langmuir isotherm. The agreement between theoretical calculations and experimental overloaded band profiles appeared reasonably accurate.     

Analysis of cyanobacterial toxins (anatoxin-a, cylindrospermopsin, microcystin-LR) by capillary electrophoresis

Capillary zone electrophoresis (CZE) and micellar electrokinetic chromatography (MEKC) were applied to the simultaneous separation of cyanobacterial toxins (anatoxin-a, microcystin-LR, cylindrospermopsin). The analytical performance data of both methods, optimized for the three toxins, were similar with a precision of migration times smaller than 0.8 RSD% and a detection limit in the range of 1-4 microg/mL, using spectrophotometric detection at 230 nm. Both methods were applied to an analysis of cyanotoxins in water bloom samples and crude cyanobacterial extracts. The results obtained indicate that, for complex matrices, the sequential application of CZE and MEKC is necessary. It is recommended to use both CE techniques for the analysis of the same sample in order to confirm the results by an orthogonal approach.