The shift operator matrix method applied to the two-dimensional nearest and next nearest neighbor problem

The shift operator matrix (SOM) method is discussed. We show that in the thermodynamic limit, the largest eigenvalue of the SOM determines the grand canonical partition function for situations when simple, nearest-neighbor and next-nearest neighbor interacting particles are distributed on anM ×N lattice space. In addition, we present a method for calculating the appropriate shift operator matrices.     

Mean field kinetic theory for a lattice gas model of fluids confined in porous materials

We consider the mean field kinetic equations describing the relaxation dynamics of a lattice model of a fluid confined in a porous material. The dynamical theory embodied in these equations can be viewed as a mean field approximation to a Kawasaki dynamics Monte Carlo simulation of the system, as a theory of diffusion, or as a dynamical density functional theory. The solutions of the kinetic equations for long times coincide with the solutions of the static mean field equations for the inhomogeneous lattice gas. The approach is applied to a lattice gas model of a fluid confined in a finite length slit pore open at both ends and is in contact with the bulk fluid at a temperature where capillary condensation and hysteresis occur. The states emerging dynamically during irreversible changes in the chemical potential are compared with those obtained from the static mean field equations for states associated with a quasistatic progression up and down the adsorption/desorption isotherm. In the capillary transition region, the dynamics involves the appearance of undulates (adsorption) and liquid bridges (adsorption and desorption) which are unstable in the static mean field theory in the grand ensemble for the open pore but which are stable in the static mean field theory in the canonical ensemble for an infinite pore.     

Stochastic simulation of catalytic surface reactions in the fast diffusion limit

The master equation of a lattice gas reaction tracks the probability of visiting all spatial configurations. The large number of unique spatial configurations on a lattice renders master equation simulations infeasible for even small lattices. In this work, a reduced master equation is derived for the probability distribution of the coverages in the infinite diffusion limit. This derivation justifies the widely used assumption that the adlayer is in equilibrium for the current coverages and temperature when all reactants are highly mobile. Given the reduced master equation, two novel and efficient simulation methods of lattice gas reactions in the infinite diffusion limit are derived. The first method involves solving the reduced master equation directly for small lattices, which is intractable in configuration space. The second method involves reducing the master equation further in the large lattice limit to a set of differential equations that tracks only the species coverages. Solution of the reduced master equation and differential equations requires information that can be obtained through short, diffusion-only kinetic Monte Carlo simulation runs at each coverage. These simulations need to be run only once because the data can be stored and used for simulations with any set of kinetic parameters, gas-phase concentrations, and initial conditions. An idealized CO oxidation reaction mechanism with strong lateral interactions is used as an example system for demonstrating the reduced master equation and deterministic simulation techniques.     

The Hard Rod Fluid in a Uniform Gravitational Field

Abstract

The system of hard rods in an external, uniform gravitational field is studied and exact expressions obtained for the partition functions, and the one- and two-particle distributions. In principle all higher order distributions can be exactly expressed as finite sums and the Laplace transforms of the one-particle density and pressure for a semi-infinite system are reducible to single integrals. Limiting cases of weak field and small rod diameters are examined. In the former case, these results agree with Percus and Lebowitz's local density expansion. In the latter case, corrections to the barometric pressure and density laws are obtained. Finally some mathematical difficulties involved in the calculation of the virial expansion and distribution of roots of the grand partition function are mentioned.     

Classical limit of quantum mechanics in the large

Hilbert space may be regarded as a convenient standard approximation by interpolation and extrapolation to a unitary space, in general decomposable into a sum of semisimple spaces. In the limit we expect a particle theory expanded in powers of h according to the number of interacting particles. In the classical limit h tending to zero, phase space, with equations of motion reducible to Hamiltonian form, replaces Hilbert space. The modification of states by observing them is taken care of by considering probability distributions of petty ensembles. If the equations for any single observer can be made autonomous by replacing empirical time by universal time with an arrow we have a causal system. We then obtain the relation between probability and negentropy required for the second law of thermodynamics. An approximate Newtonian theory provides proximate particles with internal and external variables and admits the Poincaré group. For the internal variables we have approximately the Breit interaction. For the external variables we have equivalence for observers of the homogeneous Lorentz group of relativity. We introduce grand ensembles of ultimate particles, and nebulae as proximate particles in the large. We assume the Einstein principle of equivalence for the ten parameter set of observers suggested by relativity and suppose the second law of thermodynamics holds for each observer. The Einstein law of gravitation follows in classical theory to the order of the reciprocal of the large constant, in general with positive natural curvature as well as that corresponding to mass. Replacing particle interactions by fields we include them in classical theory.     

Thermochemical Investigations of Tautomeric Equilibrium. Variation of the Calculated Equilibrium Constant with Binary Solvent Composition

Abstract

A conventional nonelectrolyte solution model which has led to successful predictive equations for solute solubility and infinite dilution chromatographic partition coefficients is extended to systems containing a tautomeric solute dissolved in a binary solvent mixture. The derived expression predicts that the tautomeric solute concentration in a binary solvent is a geometric average of the pure solvent ratios and permits calculation of solute-solvent association constants from variation of the stoichiometric tautomeric solute concentration ratios as a function of binary solvent composition. Experimental data for phenylazonaphthol dissolved in aqueous-ethanol and aqueous-acetone solvent mixtures is discussed in relation to the theoretical model.     

Configurational probabilities for monomers, dimers and trimers in fluids

A new analytical approach is proposed to model aggregation of molecules with isotropic, nearest-neighbor, attractive interactions. By treating the clustering process as a chain reaction, equations with the exact high temperature limit are derived by evaluating the occupation probabilities of nearest neighbors based on the Ono-Kondo approach for a hexagonal lattice to calculate the configurational probabilities of i-mers (i = 1, 2, 3). Equilibrium constants for dimers and trimers are calculated based on the configurational probability data. The proposed model agrees well with Monte Carlo simulations at medium and high temperatures. At low temperatures, the model can be improved by considering the full set of site densities in the first shell of a central trimer. Approximate analytical solutions derived from exact calculations of the grand partition function for monomer adsorption on a 4 x N hexagonal lattice with cylindrical boundary conditions also are presented.     

Infinite dilution activity coefficients and gas-to-liquid partition coefficients of organic solutes dissolved in 1-sec-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide and in 1-tert-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide

Inverse gas chromatography was used to measure infinite dilution activity coefficients and gas-to-liquid partition coefficients for 48 organic solute probes in either 1-sec-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide or 1-tert-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide in the temperature range from 323.15 to 373.15 K. Partial molar excess enthalpies of solution were calculated from the variation of the infinite dilution activity coefficients with temperature. Abraham model correlations were also derived from the experimental partition coefficient data. The derived Abraham model correlations describe the observed partition coefficients to within 0.11 log units.     

Partition Coefficients at Infinite Dilution from Flory-Huggins Theory

Abstract

An expression relating partition coefficients at infinite dilution to Flory-Huggins interaction parameters has been derived. The equation has been successfully applied to the partitioning of alcohols between water and cellulose acetate. The solvent-solute interaction parameter has a dominant effect on partition coefficients. For the systems studied, solvent-polymer interactions appear to be negligible.     

Determination of infinite dilution activity coefficients using HS-SPME/GC/FID for hydrocarbons in furfural at temperatures of (298.15, 308.15, and 318.15) K

A new methodology using the headspace solid phase microextraction (HS-SPME) technique has been used to evaluate the infinite dilution activity coefficient (${\gamma }_{12}^{\infty }$) of nine hydrocarbons (alkanes, cycloalkanes, and aromatics) in furfural solvent. The main objective of this study was to validate a faster and lower cost methodology expanding the use of HS-SPME to determine infinite dilution activity of solutes in organic solvents. Two approaches were proposed for the determination of ${\gamma }_{12}^{\infty }$ in order to use this technique (HS-SPME). In addition, the fiber–gas partition coefficients (K_fg) for each analyte at each of the studied temperatures were determined. The activity and partition coefficients have been reported at temperatures of (298.15, 308.15, and 318.15) K. The data were compared with the literature infinite dilution data determined by other methods such as liquid–gas chromatography (GLC) and gas stripping. Partial molar excess enthalpies of mixing at infinite dilution for each solute have been determined. The fibers were tested before and after each experiment, using statistical methods to ensure that their properties do not change during the experiments. The fibers were also analyzed by optical microscopy to evaluate possible surface damage by comparing them with new fibers. The activity coefficient values correlated well with the data in the literature and showed average deviations less than 10%.     

Acceleration of Monte Carlo simulations through spatial updating in the grand canonical ensemble

A new grand canonical Monte Carlo algorithm for continuum fluid models is proposed. The method is based on a generalization of sequential Monte Carlo algorithms for lattice gas systems. The elementary moves, particle insertions and removals, are constructed by analogy with those of a lattice gas. The updating is implemented by selecting points in space (spatial updating) either at random or in a definitive order (sequential). The type of move, insertion or removal, is deduced based on the local environment of the selected points. Results on two-dimensional square-well fluids indicate that the sequential version of the proposed algorithm converges faster than standard grand canonical algorithms for continuum fluids. Due to the nature of the updating, additional reduction of simulation time may be achieved by parallel implementation through domain decomposition.     

SOLVENT CHARACTERISTICS OF MOLTEN SULFUR AT 132°C

Abstract

The activity coefficient of 50 solutes in liquid sulfur have been measured at infinite dilution by gas chromatography. The solvent-solute interactions have been examined in terms of chromatographic models, activity coefficients, and solubility parameter theory. The classification of sulfur with polar chromatographic solvents is examined and attributed to enhanced dispersion of sulfur with compounds which have, relative to alkanes, loosely held electron clouds. Solubility parameter theory is found to provide a basis for the correlation of the behavior of n-alkanes in liquid sulfur.     

Infinite Dilution Activity Coefficients and Gas-to-Liquid Partition Coefficients of Organic Solutes Dissolved in 1-Benzylpyridinium Bis(Trifluoromethylsulfonyl)Imide and 1-Cyclohexylmethyl-1-Methylpyrrolidinium Bis(Trifluoromethylsulfonyl)Imide

Infinite dilution activity coefficients and gas-to-ionic liquid partition coefficients were measured for a chemically diverse set of 48 or more organic solute probes dissolved in the ionic liquids 1-benzylpyridinium bis(trifluoromethylsulfonyl)imide ([BzPy][Tf₂N]) and 1-cyclohexylmethyl-1-methylpyrrolidinium bis(trifluoromethylsulfonyl)imide ([ChxmPyrr][Tf₂N]) in the temperature range from 323.15 to 373.15 K using inverse gas chromatography. Selectivities and capacities for different separation problems were calculated from the measured chromatographic data. The measured partition coefficients were correlated using mathematical equations based on the Abraham general solvation parameter model. The derived Abraham model correlations back-calculated the observed partition coefficients to within 0.12 log₁₀ units.     

Distribution of polymer chain lengths inside a spherical volume: Solution in terms of a diffusion problem

The probability W(t) that a given number t of segments of an infinite chain lie within a given sphere can be expressed in terms of the single-pass length probability and the probability of reentrance into the sphere. The problem of calculating these two probabilities is equivalent to that of a diffusing particle exiting or entering the sphere after a given time, when the surface of the sphere is an absorbing wall. It is shown that the boundary condition, c = 0, usually applied to an absorbing surface cannot be used for the present purpose. The boundary condition used instead is the so-called radiation condition ?c/?z = kc; it is shown that when k approaches infinity the final answer for W(t), which is given in the form of an infinite series, approaches the correct limit. In this same limit the ratio 〈t〉²/〈t〉² has the value 2.4     

Microscopic Motion of Atoms in Simple Liquids at Equilibrium and with Shear Flow

Abstract

We investigate the microscopic mechanism of atomic motion and local stress relaxation in Lennard-Jones, LJ liquids using a new class of correlation functions that emphasise the interplay between an abitrary atom in the fluid and its surrounding shells of atoms. We use the linear momenta and stress tensor to characterise the time dependence of this interaction. We consider a series of correlation functions that give complementary information and build a picture of the single particle and small cluster motion. The central particle and first shell undergo a reversal in momentum at different times after the ‘collision’ of the central particle and its first shell of neighbours. This ‘phase difference’ becomes manifest in the subsequent dynamics probed by the new correlation functions. We also consider the effect of a non-newtonian shear flow on this local dynamical relaxation, using profile biased laminar flow equations of motion. In non-newtonian shear flow we find the momentum transfer between particle and cage to be less pronounced and occur over a wider time range.     

Abraham Model Correlations for Triethylene Glycol Solvent Derived from Infinite Dilution Activity Coefficient,Partition Coefficient and Solubility Data Measured at 298.15 K

A gas chromatographic headspace analysis method was used to experimentally determine gas-to-liquid partition coefficients and infinite dilution activity coefficients for 29 liquid organic solutes dissolved in triethylene glycol at 298.15 K. Solubilities were also determined at 298.15 K for 23 crystalline nonelectrolyte organic compounds in triethylene glycol based on spectroscopic absorbance measurements. The experimental results of the headspace chromatographic and spectroscopic solubility measurements were converted to gas-to-triethylene glycol and water-to-triethylene glycol partition coefficients, and molar solubility ratios using standard thermodynamic relationships. Expressions were derived for solute transfer into triethylene glycol by combining our measured experimental values with published literature data. Mathematical correlations based on the Abraham model describe the observed partition coefficient and solubility data to within 0.16 log₁₀ units (or less).     

Reversed-Phase Liquid Chromatography with Mixed Eluents: Partition Model of Retention for Ionogenic Solutes

Abstract

Basing on the pure partition model of solute retention new theoretical equations for ionogenic solutes are discussed. These equations define dependence of the capacity ratio on the mobile phase composition, which is characteristic for the reversed-phase liquid chromatography.     

The combinatorial factor method for investigation of order-disorder in two-dimensional binary alloy

The aim of this study was to investigate order-disorder in the two-dimensional AB alloy to find out whether the number of components, N _A and N _B, was equal or not. To this end, using the nearest neighbor interactions, first, we applied a two-dimensional lattice which consisted of infinite rows, R and columns, L, so that, RL = N _A + N _B. For such a model, using the combinatorial factor method, we derived an exact equation for the partition function. Because, the derived partition function was very complicated, the thermodynamic properties of the lattice could not be calculated; however, but these properties could be estimated for the models with a limited number of rows. Our results show that, for models with finite number of rows, for each mole fraction of A in the specific temperature, a phase transition, like Onsager transition, takes place.