Violation of the linearity principle of additivity of sorption energy in chromatography. A universal equation describing retention behavior

Deviations from the additivity of energy contributions to substance sorption energies, determined on the basis of thermodynamic studies of GC behavior of homologous series of organic compounds fall into two categories: one for n-alkanes, the other for homologous series containing a functional group. A previously derived equation is proposed for homologous series, describing the deviation from the linear dependence of retention parameters with a propagating homolog n-alkyl chain. The equation permits calculation of retention parameters of homologs starting from the first member in gas-liquid, gas-solid, and liquid-liquid systems. The results prove its universal applicability.     

An equation for the calculation of retention indices in temperature-programmed gas chromatography with allowance for the nonlinear variation of the retention parameters ofn-alkanes

In view of the nonlinear variation of the temperature increments ofn-alkanes found previously, the accuracy of the calculations of the retention indices (I _pr) of substances in temperature-programmed capillary gas chromatography carried out in terms of six known equations was verified. A new four-parameter equation was proposed, and a general method for the calculation of its coefficients, suitable for all stationary phases, based on the adjusted retention times ofn-alkanes was suggested. The coefficients of the equation for 12 temperature variation programs were determined. Using the homologous series of methyl esters of fatty acids as an example, it was shown that the proposed equation ensures the minimum error of determination ofI _pr under various conditions. The equation also makes it possible to carry out interpolation and extrapolation calculations. The coefficients of the equation are found using the least-squares method based on data for any 4–5 referencen-alkanes. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 2, pp. 323–327, February, 1997.     

A perturbed hard-sphere equation of state for liquid metals

A perturbed hard-sphere equation of state, developed previously for liquid alkali metals and liquid refractory metals, has been applied for PVT calculation of some pure liquid metals including alkaline earth metals, tin, lead, antimony, bismuth, and rubidium over a wide range of temperatures and pressures. Two temperature-dependent parameters appear in the equation of state, which are universal functions of the reduced temperature, i.e. two scale parameters are sufficient to calculate the temperature-dependent parameters. The scaling parameters can be easily obtained by employing a corresponding-states principle based on a Lennard-Jones potential energy function. Employing the present equation of state, the liquid densities of aforementioned metals at temperatures ranging from the melting point to 2000?K and at pressures ranging from vapour pressure up to 40,000?bar have been calculated and compared with experimental data. The average absolute deviation in predicted densities compared with experimental data is 1.54%.     

New insights into pseudo-second-order kinetic equation for adsorption

This study looked into the theoretical basis of the most commonly used pseudo-second-order kinetic equation for adsorption. It was demonstrated that the pseudo-second-order kinetic equation for adsorption indeed is in line with the universal rate law for a chemical reaction, while the same principle can also be applied to the pseudo-first-order kinetic equation for adsorption. This in turn offers a theoretical basis for the pseudo-second-order equation. If the second-order equation is applicable, the rate of adsorption was found to be subject to a second-order rate law with respect to the availability of adsorption sites on the surface of adsorbent rather than adsorbate concentration in bulk solution as proposed in the literature. Based on the rate law for the second-order adsorption process, the original complex second-order kinetic equation can be transformed to a simple hyperbolic form with clearly defined mathematical structure: Q_t = Q_e(t/t + t_r) in which t_r is relaxation time representing time required for adsorption to reach half-saturation state of adsorbent under given experimental conditions, i.e., Q_t = 0.5Q_e.     

Nonlinear variation of sorption parameters of n-alkane homologsl in temperature-programmed gas chromatography (tpgc) and new equation for calculation of retention indixces

It has been demonstated that the considaerable difference in temperature increments of sorption parameters of n-alkanes under isothermal conditions is the main reason for nonlinear dependence of sorption parameters on molecular mass of homolog in temperaturre programmed gas chromatography (TPGC). A new nonlinear 4th parameter equation has been given for calculation of the retention indices. Coefficients of the equation are calculated from n-alkanes. The equation allows extrapolation and interpolation calculations of retention indices under TPGC conditions with experimental precision. The results obtained; for fatty acidkl methyl esters demonstrate the advantage of ovr equation in comparison with van den Dool and Kratz's equation.     

The Ornstein-Zernike equation, critical phenomena, and the equation of state for liquids and gases

It is shown that the Ornstein-Zernike (OZ) equation has two solutions: the standard one, which depends explicitly on the interaction potential, and a second universal one, resulting from the infinity point of the partition function. It is stressed that there are two pressure components: the standard one and a universal one that is valid over the whole of the phase plane. It is concluded that the universal solution parameters depend in general on definite integrals of functions dependent on the interaction potential. In the vicinity of the critical point, however, the dependence on the interaction potential vanishes; i.e., the solution becomes fully universal. It is shown that in this range of the phase diagram, all results of the theory of critical phenomena (scaling theory) follow from the OZ equation.     

The two-parameter taft equation in kinetics of free radical reactions

A new method was developed for the calculation of the resonance substituent constants of the two-parameter Taft equation log k_sub/k₀=ρ*σ*+r*σ^r. It is based on a relationship between the spin density in free radicals and the rate constants of radical substitution reactions of CH₃. Possibilities and limitations of the application of this correlation equation to the investigation of substitution and addition radical reactions are discussed.     

Fluorescence Quenching of Tris(2,2′-bipyridine)Ruthenium(II) Dichloride by Certain Organic Dyes

Fluorescence quenching of [Ru(bpy)₃]²⁺ by a series of organic dyes has been investigated by using the steady state fluorescence technique in aqueous medium. The dyes used are anthraquinone dyes: uniblue, acid blue 129, alizarin, alizarin red S and the azo dyes: congo red, sunset yellow, methyl orange, tartrazine, acid orange 63, methyl red and erichrome black T. The quenching of [Ru(bpy)₃]²⁺ was found to obey the Stern-Volmer equation and the corresponding Stern-Volmer plots were linear indicating dynamic quenching. The quenching rate constants (k _q) were calculated from the fluorescence data. The mechanism of quenching was discussed on the basis of the quenching rate constants as well as the reduction potential of dyes. The electron transfer mechanism has been proved by the calculation of Gibbs energy changes (ΔG _et) by applying the Rehm-Weller equation.     

Argon cluster size‐dependence of sputtering yields of polymers: molecular weights and the universal equation

An analysis is provided of the data of Cristaudo et al. for the sputtering yields of polystyrene and polymethylmethacrylate by argon cluster ion beams as a function of their molecular weight. This analysis is made using the universal sputtering equation of Seah to evaluate the effect of the end‐group density on the parameter A. This parameter is thought to be related to the average energy per atom (excluding hydrogen) to liberate fragments and it is shown that A decreases as the end‐group density rises in the manner expected. Equations are provided relating A to the molecular weight or the end‐group density that are obeyed for both materials. Copyright © 2014 Crown Copyright.     

Energy levels of the weakly interacting radicals

Energy levels of the weakly interacting radicals are estimated on the basis of planar methyl radical interaction as a model. These levels depend on the given separation r between the centers of mass. The appropriate Schrödinger equation is solved by using the Brillouin-Wigner series method. Analytical forms for the integrals used to estimate matrix elements are derived. The principle of total momentum conservation is strictly obeyed. Some energy levels cannot be estimated as a result of divergence.     

The use of a nonlinear equation for calculation of the retention indices of polar substances in gas chromatography with linear temperature programming

New possibilities for using the equation that takes into account the nonlinear variation of parameters of the reference n-alkane scale for the calculation of retention indices of polar substances at different modes of temperature programming were considered. The advantages of this equation over the linear scale used traditionally were demonstrated in relation to C₃—C₁₁ alkan-1-ols. The equation appears to have considerable promise regarding the search for the equivalent isothermal index.     

On the derivation and extension of the uniquac equation

The UNIQUAC equation is derived by phenomenological arguments based on a two-fluid theory. This derivation is free of the inconsistencies which arise when a lattice one-fluid theory is used to derive UNIQUAC or any other local-composition equation. The essential step in the derivation is the adoption of Wilson's assumption that local compositions can be related to overall compositions through Boltzmann factors.In UNIQUAC, the energy terms in the Boltzmann factors are assumed identical to those in the excess energy of mixing. When the assumption of identity is replaced by the much less stringent assumption of proportionality, a three-parameter UNIQUAC equation is derived. This equation often gives a better fit of experimental data whenever all three binary paramters are freely adjusted. Upon assigning a universal value to the third parameter, only marginal improvement is obtained.     

Alternative interpretations of the mole and the ideal gas equation

The fundamental concept of the mole stems from the idea of the gram-atom (or gram-molecule): that an amount of one mole of a given substance has an associated mass, in grams, numerically exactly equal to the relative atomic (or molecular) mass of the substance. This means that the particular number of entities comprising one mole, Avogadro’s number, must be exactly equal to the gram-to-dalton mass ratio. Various interpretations of what a mole actually means and how this affects the ideal gas equation depend on how the amount of a given substance, n, is related to the corresponding number of entities, N. We look at three alternatives: (1) the conventional interpretation in which n is directly proportional to N, where the proportionality factor is the reciprocal of the (evidently poorly understood) Avogadro constant; (2) a more easily comprehended alternative in which n is equal to N entities; and (3) an interpretation in which n and N both represent the number of entities, but expressed in different ways. Regardless of these different interpretations, the form of the stoichiometric equations (relating N, n and the total sample mass) and the form of the relationship between the universal gas constant and the Boltzmann constant remain the same.     

Universal behavior of the enthalpy of vaporization: an empirical equation

Values of the enthalpy of vaporization from the critical to the triple point are correlated by an empirical equation. The equation contains parameters which characterize each substance: the critical and triple point temperatures and the enthalpy of vaporization at the triple point, and for all substances the same universal critical ratio. This work suggests that a wide class of fluids with the exception of quantal liquids shows an universal behavior along the coexistence curve.     

Calculation of salt activities in molten salt hydrates applying the modified BET equation,I: Binary systems

Summary On the basis of the modified BET model according to Stokes and Robinson, an equation for the calculation of salt activities in molten salt hydrates has been derived. The equation is used to describe successfully the liquidus curves of the hydrates of MgCl₂, Mg(NO₃)₂ and CaCl₂ inT-x diagrams. A promissing feature of the model is the small number of adjustable parameters and its extrapolative power.Dedicated to o. Univ._Prof. Dipl.-Ing. Dr. mont. Heinz Gamsjäger on the occasion of his 60th birthday     

Characteristic length of the glass transition

The definition of molecular cooperativity is discussed. The characteristic length of the glass transition describes the size of this cooperativity. Differential scanning calorimetry (DSC) and heat capacity spectroscopy (HCS) results of a series of poly(n-alkyl methacrylates) (alkyl = methyl, ethyl, propyl, butyl, pentyl, hexyl, and octyl) and a series of statistical copolymers poly(n-butylmethacrylate-stat-styrene) are discussed in terms of molecular cooperativity in the αβ splitting region, where a high-frequency dispersion zone a splits off into the main transition zone α and a Goldstein Johari process β at lower frequencies. The characteristic length tends to small values of order one monomer diameter in the splitting region for scenarios with an α relaxation onset. The statements about the size scale of cooperativity are conditional upon certain assumptions leading to the equation used for calculation of this size from HCS and DSC data. The step height of heat capacity (Δc_p) and, with less certainty, the square root of the cooperativity volume or number (V^1/2_α or ^1/2_α) are proportional to the temperature distance from the cooperativity onset, T = T_ons. © 1996 John Wiley & Sons, Inc.     

Effect of the copolymer composition on the K and a constants of the Mark–Houwink equation: Styrene–methyl methacrylate random copolymers

This article reports a study of the effect of composition in styrene–methyl methacrylate random copolymers on K and a constants in the Mark–Houwink equation. Copolymers with a variety of compositions and chain lengths were prepared through a controlled free‐radical copolymerization, with benzyl diethyldithiocarbamate and tetraethyl thiuram disulfide as iniferters. The synthesized products were analyzed with several techniques, including Fourier transform infrared, ¹H NMR, gel permeation chromatography, and viscometry. By relating the determined constants K and a of various copolymers to their composition, we found that the constant varied nonlinearly with the composition. The constant a decreased with increasing poly(methyl methacrylate) (PMMA) content in the copolymer molecule up to 60 wt %. After that, the constant increased with the PMMA content, reaching the value of the PMMA homopolymer. However, the constant K initially increased with the PMMA content up to a critical composition (60 wt %) and subsequently decreased with further increasing PMMA content. These results suggest that the molecular weight of a polystyrene–PMMA random copolymer of known composition cannot be approximated with a simple linear equation comprising the K and a values of each relevant homopolymers. The aforementioned trends are qualitatively discussed in relation to some possible sequential distribution in the copolymer molecules and the resulting conformation. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 562–571, 2002; DOI 10.1002/polb.10119     

Influence of intramolecular interaction on the enthalpy and entropy of methylene unit sorption for the homologous series of carbonyl-containing compounds analyzed by capillary GC

The enthalpy and entropy of sorption of methylene units in the homologous series of n-alkyl acetates; methyl, n-butyl, and phenyl n-alkyl ketones; and the methyl esters and chloroanhydrides of n-aliphatic carboxylic acids have been determined on an SE-54 capillary column. The enthalpy of sorption of the first methylene unit is anomalously high when the growing n-alkyl chain is connected directly to a carbonyl group. This effect is a result of intramolecular interaction between neighboring methyl and carbonyl groups. It has been shown by measurement of the enthalpy and entropy of sorption of the difluoromethylene unit in the series of n-hexyl esters of perfluorinated carboxylic acids that for these compounds the effect is absent. The intramolecular interaction was found to increase in the order methyl butyl ketone < methyl phenyl ketone < methyl acetate < acetyl chloride < acetone.     

Dependence of the glass transition temperature of poly[N-(n-alkyloxy-carbonyl-methyl) Maleimides] and poly[N-(5-n-alkyloxycarbonyl-n-pentyl) Maleimides] on n-alkyl side-chain length

The glass transition temperature has been determined by a refractometric technique for two homologous series of poly[N-(n-alkyloxycarbonyl-methyl) maleimides] (PEMIS 1) and poly[N-(5-n-alkyloxycarbonyl-n-pentyl) maleimides] (PEMIS 5) with the outer part of the n-alkyl side-chain ranging in length from ethyl to decosyl for PEMIS 1 and from ethyl to decyl for PEMIS 5 and including only the even members of the series. The glass transition temperature is directly related to the number of methylene groups in the outer part of the n-alkyl side-chain (including terminal methyl) of the repeating unit. A semiempirical equation is presented that affords estimates of T_g that are in good agreement with the experimental data.