Random Mixing Approximation Research Articles

Isothermal liquid-vapour equilibria of mixtures containing organic compounds. 3. Excess Gibbs free energies of cyclohexane + a linear ether.

Vapour-liquid equilibria of binary mixtures of cyclohexane with a linear mono-, di-, or triether (diethyl, dipropyl, dibutyl, methyl butyl, ethyl butyl ether, formaldehyde dimethyl acetal, formaldehyde diethyl acetal, 1,2-dimethoxyethane, 1,2-diethoxyethane, diethyleneglycol dimethyl ether) were determined at 298.15 K by head-space gas cromatographic analysis of the equilibrated vapour phase. Excess molar Gibbs energies, G E, for the investigated mixtures were obtained by a least-squares treatment of the equilibrium results (x–y). The G E values are positive for all the systems and increase with increasing the ratio of oxygen to carbon atoms in the ether molecule. There is clear evidence for the effect of position of the oxygen atom in the chain structure of monoethers as well as for the effect of distance of two -0- groups in diethers. The behaviour of the activity coefficients at infinite dilution and of the free energies of solvation as well as of molar excess entropies, obtained from G E and the known excess enthalpies, are briefly discussed. The experimental G E values are compared with the predictions of a group contribution model in the random-mixing approximation.

A thermodynamic description of the double layer surrounding hydrous oxides

A thermodynamic description of the double layer surrounding hydrous oxides has been formulated. Parameters of the theory are the surface site density, and the standard free energies of adsorption of the hydrogen ion and the cations and anions of the support electrolyte. The independent variables are pH and concentration of electrolyte. Dependent variables are the ζ-potential and the surface concentrations of adsorbed ions. Adsorption of salts onto charged and neutral sites cause qualitatively different behavior in the computed curves. Adsorption of ions of the supporting electrolyte onto charged sites control the extremum values of the ζ-potential. Unequal adsorption of cations and anions onto charged sites causes shifts in the observed pzc and unequal adsorption onto uncharged sites causes shifts in the observed ieps. Nonlinear regression is used to estimate the parameters. It is shown that information about site density cannot be obtained from electrophoresis data alone. The random mixing approximation for the activity of surface sites becomes inadequate at high concentrations of salt.

Conformal-solution theory of polar fluids: The statistical mechanical basis of shape factors

Abstract Massih, A.R. and Mansoori, G.A., 1983. Conformal-solution theory of polar fluids: the statistical mechanical basis of shape factors. Fluid Phase Equilibria, 10: 57–72. The formalism of conformal-solution theory (CST) has been extended to pure and mixed polar fluids with consideration of dipolar, quadrupolar and induced dipolar forces. The total potential energy is reduced to a form similar to that for neutral molecules by statistical averaging over orientations. The temperature dependences of the conformal parameters faa and haa are derived and it is found that the temperature variations are in opposite directions. These conformal parameters are related to Leland's shape factors θaa and φaa, for which similar trends in temperature variations are observed. Following the theory of Byers-Brown for neutral molecular fluid mixtures, with the random-mixing approximation, an expression is derived for the excess Gibbs free energy of a binary polar mixture, using a generalized mixing-rule approximation.

Phase equilibria calculated for the systems N 2 + CO 2, CH 4 + CO 2 and CH 4 + H 2S

The liquid-vapor equilibria and critical loci of the systems N 2 + CO 2, CH 4 + CO 2 and CH 4 + H 2S have been correlated on the basis of experimental data using an accurate equation of state (EOS) including a semiempirical correction for vanishing ordering effects in the critical regions. The EOS predicts gas-gas equilibria for the N 2 + CO 2 and CH 4 + H 2S systems and confirms the data of Kohn and Kurata (1958) indicating no critical locus for CH 4 + H 2S in the composition region 0.54 ⩽ x CH 4 ⩽ 0.89. The predicted excess enthalpies and compressibilities of the gaseous mixtures agree well with the data. Deviations from the random-mixing approximation as they affect calculated liquid densities are discussed.

Thermodynamics of binary mixtures containing tertiary amines. I. Excess enthalpies of some N,N-dialkylmethylamine+normal alkane mixtures. Measurement and analysis in terms of group contributions. Estimation of excess gibbs energies

A Tian-Calvet type calorimeter was used to determine molar excess enthalpies, HE, at 298.15 K, as a function of concentration, for the twelve mixtures: N,N-dimethylethylamine, N,N-dimethylbutylamine, N,N-diethylmethylamine, and N,N-dipropylmethylamine+n-hexane, +n-heptane, +n-dodecane, or + n-hexadecane. This data and the data available in the literature on the liquid-vapour equilibria of trimethylamine + n-hexane are examined based on the quasi-lattice theory, in the random-mixing approximation, using group surface interactions. With two adjusted group interaction parameters and three alkyl-group increments, the model provides a fairly consistent description of HE of the investigated mixtures and an estimate of the molar excess Gibbs energies, GE. The results are discussed in terms of steric and Patterson effects.

Solid‐liquid equilibria in n‐alkylbenzene + n‐alkane mixtures. A group contribution study of phase diagrams

AbstractThe solid‐liquid equilibrium diagrams of following 11 binary systems were determined: benzene, toluene, ethylbenzene, n‐propylbenzene, and n‐butylbenzene + n‐heptane or n‐tetradecane and benzene + n‐dodecane. All these diagrams could be calculated accurately (maximum deviation ca. 1 K) in terms of the group‐contribution theory (in the random mixing approximation) developed by Kehiaian et al. [7], which is essentially the surface version of the Guggenheim‐Barker lattice theory. Most interchange parameters were identical to those derived previously [7, 31, 33] from experimental data (liquid‐vapor equilibria, excess enthalpies, excess heat capacities). Where experimental data were lacking, the parameters were estimated and finely adjusted to the observed equilibrium diagrams.

Interstitial models of water in the random mixing approximation

The interstitial models of water put forward recently by Bell and Lavis and by Perram are compared. It is shown that the two-dimensional partition function of Bell and Lavis, and the three-dimensional one of Perram, are related by a simple transformation in the important limiting case in which both models are treated according to random mixing statistics.

Statistical thermodynamics of two-fluid models of mixtures

An approximation of a type devised by van der Waals has been shown to be superior to the random mixing approximation, and to theories derived from it, when applied to mixtures of molecules of different sizes. In this paper, van der Waals's approximation is extended to a two-fluid model, and so takes into account the departures from a random distribution induced by differences of intermolecular energy. The treatment is further extended to mixtures of molecules of different shapes. The two-fluid van der Waals approximation is compared with the experimental excess thermodynamic functions for ten simple binary mixtures. These yield unambiguous evidence that the intermolecular energy between unlike molecules is less than the geometric mean of that between the like molecules by up to 2 %. Independent and quantitative confirmation of such differences is provided by recent measurements of the second virial coefficients and of the critical temperatures of the same mixtures.

Ground state energy of an electron gas in a lattice of positive ions

The simple Coulomb potential used to represent the interactions between electrons and lattice charges is replaced by an effective potential. With this potential, the ground state energy of an electron gas in a pure lattice built from identical ions is evaluated. This calculation is generalized to the case of equimolecular mixtures where the two types of sites form a superlattice and of disordered lattices in the random-mixing approximation. (C.E.S.)

Theory of Multicomponent Fluid Mixtures. III. A Pseudo Pair Potential

This article considers some further consequences of the exact statistical mechanical theorem (given in article I of this series) which relates the thermodynamic properties of a mixture to those of a fictitious pure fluid by defining an intermolecular potential function (called the pseudopotential) for this pure fluid. Starting with the perturbation series for the pseudopotential all terms which can be written as a sum of pairwise interactions are collected and summed to give a simple closed expression for a pseudo pair potential function for the mixture. Retaining only these pair terms and neglecting all triplet and higher-order interactions in the pseudopotential gives an approximation which is a rigorous lower bound to the correct pseudo-potential for all configurations. For mixtures of molecules with rigid cores this pseudo pair potential is also a potential with a rigid core, but with a diameter equal to that of the smallest molecule. The extreme nature of this approximation, which severely limits the application of the theory at high densities and pressures, is shown to be a direct consequence of neglecting triplet interaction terms in the pseudopotential. The nature of the limitations inherent in any approximation of the pseudo-potential by a sum of pair-wise interactions is clarified by comparing these results with the random mixing approximation to the pseudopotential which is known to give an upper bound for all configurations. The free-volume theory for the pseudo pair potential, resulting from summing the pair terms, for mixtures of rigid spheres is developed. The results of this calculation are compared with recent Monte Carlo calculations for binary mixtures of rigid spheres to illustrate the extremely ``soft'' equation of state which results when triplet interactions are neglected.

The miscibility of polymers: II. Miscibility and heat of mixing of liquid polyisobutenes and silicones

Heats of mixing of low molecular weight polyisobutene and poly(dimethyl siloxane) fluids have been measured at 50°C and 100°C over the composition range in which they were miscible. Partial molar heats and entropies of transfer across the phase boundary have also been computed. The heats of mixing are comparable in magnitude with the values predicted by the van Laar-Hildebrand equation, suggesting that random mixing occurs. In sharp contrast to this, the entropies of transfer are much lower than the values calculated on the random mixing approximation, and the Flory-Huggins equation fails to predict the form of the coexistence curve.

The second-order theory of conformal solutions

This paper describes a theoretical contribution to the statistical thermodynamics of mixtures of spherical molecules. The second-order perturbation free energy of a conformal solution is obtained by a rigorous Taylor-series expansion of the configuration integral in powers of the differences between intermolecular energy and size parameters, about an ideal unperturbed reference solution. Unlike the first-order terms, those of the second order contain statistical functions of the reference solution which cannot, in general, be related to its thermodynamic properties. All but one of these functions are concerned with departures from a random molecular distribution, and have been called molecular fluctuation integrals ; the remaining function can be related exactly to thermodynamic properties for the Lennard-Jones form of the intermolecular potential. The expressions for the molecular fluctuation integrals implied by the full random mixing approximation and by the semi-random mixing approximation of the cell theory, are derived and compared with the correct expressions given by the cell theory. The role of the Taylor series expansion as a critique of solution theories is discussed.