Conformal Solution Theory Research Articles

Mean density approximation and hard sphere expansion theory: A review

This review surveys research dealing with the Mean Density Approximation (MDA) and the Hard Sphere Expansion (HSE) theory developed by Leland and coworkers. MDA and its modifications provide a simple way to predict radial distribution functions of mixtures from the pure fluid information. Comparisons with computer simulation data show the MDA to be superior to the van der Waals approximation for mixture radial distribution functions. Derivations of the HSE theory and the HSE Conformal Solution Theory (HSE-CST) are also described. For Lennard-Jones mixtures, the HSE theory is proven to be superior to the van der Waals theory by using a proper method to determine the hard-sphere diameter. The major problem associated with the development of a consistent method to determine the hard-sphere diameter of the HSE-CST and the requirements regarding the extension of the HSE-CST to polar mixtures are discussed. It is concluded that the HSE-CST is capable of predicting, satisfactorily, the vapor-liquid equilibrium properties of mixtures containing hydrocarbons and/or slightly quadrupolar or polar compounds such as CO 2 and H 2S.

Density expansion (DEX) mixing rules: Thermodynamic modeling of supercritical extraction

Conformal solution theory and the density expansion expression of the radial distribution function of fluids are used to derive a set of mixing rules. The new mixing rules are composition, density, and temperature dependent. To test the new mixing rules they are used for thermodynamic modeling of supercritical extraction. Comparison of the result of calculation by the mixing rules with the van der Waals mixing rules indicates a profound improvement over the latter in prediction of properties of mixtures consisting of species with large molecular size and shape differences.

Conformal solution theory for simple liquid mixtures

A new conformal solution theory using a single pure fluid as a reference substance for the calculation of thermodynamic properties of fluid mixtures is developed. The perturbation theory developed by Weeks, Chandler and Andersen (WCA) and by Verlet and Weis (VW) is used to calculate the reference properties. The mean density approximation and corresponding state principle are used to eliminate the higher order terms in the mixture system and to derive the pseudo-parameters for the reference system. The mixture properties are obtained from the reference properties and their corresponding hard sphere excess functions defined as the properties of the mixture less the value of the properties for the hard sphere mixture. The excess functions of mixing for several liquid mixtures of Lennard-Jones fluids, obeying the Lorentz-Berthelet rule, are calculated by the new method (VW-HSE). Comparison with the results of other theories and Monte Carlo data shows definite improvement. Since only the properties of a pure reference fluid are directly calculated, the method can be applied to more complicated multicomponent systems without additional computational effort as required by other theories.

Thermodynamic properties of liquid mixtures of krypton + methane

The equations of state of liquid methane and of a mixture of composition 0.475 Kr + 0.525 CH4 have been measured at 129.32, 134.32, 142.68 and 147.08 K, from just above the saturation vapour pressure to the freezing pressure of krypton. Similar studies have been made at 142.68 K for two other mixtures with mole fraction of krypton x= 0.645 and 0.339.The results show that the excess volume is small and negative at low pressures and increases slightly with pressure, becoming almost zero at high pressures. A plot of VE against x has a rather symmetric curve at low pressures, becoming less so at high pressures. The changes in the excess functions GE, HE and TSE with increasing pressure have been calculated; GE is less affected by pressure than HE or TSE, a feature common to other simple liquid mixtures.Neither the conformal solution theory (in the van-der-Waals–one-fluid form) nor a perturbed hard-sphere equation could account for the quantitative behaviour of the system.

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.

Thermodynamics of liquid mixtures of argon+ carbon monoxide

The heat of mixing of liquid argon (component 1) and liquid carbon monoxide has been measured by direct calorimetry for several mixtures of composition ranging from x 1 ⋍ 0.2 to x 1 ⋍ 0.8 at 85.79 K. At this temperature H E is positive and an almost parabolic function of mole fraction, with a maximum value of H E = 95.9 J mol −1 at x 1 = 0.569. The experimental results are compared with those derived from the Gibbons-Rigby equation of state, and from conformal solution theory.

Conformal‐Solution‐Theorie für Mischungen konvexer Moleküle

AbstractThe Hard Spheres Expansion Conformal Solution Theory [1], originally developed to predict the thermodynamic properties of mixtures of spherical molecules, is generalized for the treatment of fluids that are composed of particles interacting via Kihara type potentials [2].—A perturbation expansion is generated by dividing the intermolecular pair potential into a repulsive and an attractive part. The hard‐core reference system is represented by a mixture of hard convex bodies. It's thermodynamic functions are obtained from a new accurate equation of state. The attractive contributions are evaluated from the known attractive interactions of a pure reference fluid using the principles of conformal solution theory and a new type of pseudoparameters.—A method for the determination of temperature and density dependent effective hard‐core parameters is presented.

CALCULATION OF PHASE DIAGRAMS OF IONIC SYSTEMS FROM FUNDAMENTAL SOLUTION THEORIES∗

Fundamental concepts and theories of molten salt solutions appear to be useful in a number of technologies (e.g., high-temperature batteries, fuels cells, aluminum production). A simple but exact cycle permits one to calculate the solubility product of, for example, oxides and sulfides in halide melts. The use of this cycle is well illustrated by calculations related to nuclear fuel processing and to advanced battery technology. For concentrated solutions, the conformal ionic solution theory which is a statistical mechanical perturbation theory is used to calculate the phase diagrams of terna ry systems solely based on information from the lower order (unary and binary) subsystems. Thus, one can numerically generate phase diagrams a priori, reliably extend phase diagrams into unmeasured regions and check the thermodynamic self consistency of published measurements.

Behavior of a nonconformal mixture via computer simulation

A binary 50% mixture of soft spheres is studied via nonequilibrium molecular dynamics, and the equilibrium and nonequilibrium radial distribution functions for a nonconformal mixture with a mass ratio of 10 and a size ratio of about 2 are examined. This model system is related to the real methane/decane mixture, and it is shown that apparently anomalous properties of this mixture, especially the viscosity, could perhaps be understood in terms of the local or ambient mole fraction. In addition, the postulates of the Van der Waals one fluid conformal solution theory are discussed, and a mixing rule for the mass is derived.

Vapour—liquid equilibrium at elevated pressures from the perturbation theory. II. Binary systems

Vapour—liquid equilibria in binary systems of non-polar non-spherical molecule compounds were studied theoretically by combining the perturbation theory of convex molecule fluids with a new variant of the conformal solution theory. The recently proposed equation of state of hard convex body mixtures and the corresponding expressions for the contact values of distribution functions were employed to determine the reference thermodynamic functions and the perturbation terms. Ten binary systems, i.e. ArCH 4, N 2CH 4, N 2C 2H 4, N 2C 2H 6, CH 4C 2H 4, CH 4C 2H 6, C 2H 4C 2H 6, CO 2C 2H 4, CO 2C 2H 6, and ArCO 2 were studied at constant temperatures. Comparison of theoretical predictions with experimental data is given.

Calculation of thermodynamic properties of multicomponent ionic reciprocal systems

Thermodynamic properties of multicomponent ionic reciprocal systems are derived using the conformal ionic solution theory. The equations obtained are more general than previous equations and depend solely on the properties of the components and on those of the binary subsystems. The behavior of dilute solutions is carefully studied leading to a priori predictions of solubility products in multicomponent systems. The solubility products and the specific bond free energy for making an ion pair, e.g., the pair (A–X) in the binary solvent BY–CY, are shown to depend upon specific ionic interactions in the binary subsystems. The equations presented are compared with equations derived from prior theories.

Statistical thermodynamics of fluid phase equilibrium in a conformal polydisperse system

Fluid phase equilibrium in a polydisperse system can be conveniently described by an extended theory of conformal solutions. The distribution of particles between coexisting phases appears more sensitive to the interparticle forces than to the equation of state. Under certain conditions a polydisperse system may separate into two liquid phases below an upper critical solution point. The relationship to phase transitions in electrostatically-stabilized colloidal dispersions is discussed.

Theory of Surface Tension in Liquid Mixtures

Abstract Equations determining the density profiles of a multicomponent liquid are first given in terms of the partial direct correlation functions in the presence of the surface. The fluctuation theory of surface tension is also generalized to liquid mixtures. The density gradient procedure simplifies the above problem to knowledge of direct correlation functions in the bulk mixture. For the binary case, the density profile equations are then usefully written in terms of number-concentration correlation functions. The total density and surface segregation profiles are finally considered in a simplified model by utilizing the theory of conformal solutions. Such a model, though limited in practice, should apply, for example, to the Na-K alloy system.

Vapor-liquid equilibrium data for binary systems of chlorobenzene with acetone, acetonitrile, ethyl acetate, ethylbenzene, methanol, and 1-pentene

RbBr Figure 6. LL isotherms ais calculated by means of the CIS theory. along with the temperatures of the corresponding eutectics and of the LL equilibrium limits. The stratification lens (dashed) impinges on the LiF crystallization field and occupies 4 1.5 % of the composition square!. The main axis of the lens coincides with the stable diagonal. Along this axis PCT is constant at 834 f 1 oc. Figure 5 reports the projections of the interpolated LL isotherms at 850, 900, 950, 1000, and 1030 OC. The upper critical solution point of the system is at xLF = 0.76 and 1050 OC. The SL isotherms at 500, 600, 700, and 800 OC are also reported. Moreover, the figure brings into evidence the four invariant points presented by the system with the following coordinates: eutectic E, at t= 240 OC and xLLIF = 0.04, xLiBr = 0.58, xRbBr = 0.38; eutectic E2 at t= 436 C and xL~F = 0.14, xRbF = 0.55, xRbBr = 0.31; peritectic P, at t = 268 OC and xLiF = 0.04, xLiBr = 0.55, xRB = 0.41; peritcsctic P4 at t = 467 OC and xW = 0.39,, xRbF = 0.57, xRm = 0.C135. The general topology of the system agrees with the triangulation rules. Discussion In order to extend the test of the conformal ionic solution (CIS) theory (7) carried out in ref 2, we made attempts to predict the LL equilibria in the present system. The required data (see ref 2) are the standard Gibbs free energy change for the meLiterature Cited

Miscibility Gaps in Fused Salts. Note X. The Reciprocal Ternary System K, Li/Br, F

Solid-liquid and liquid-liquid equilibria in the reciprocal ternary system K, Li/Br, F were fully measured. The projection of the miscibility gap occupies 20.8% of the composition square; the upper critical point of the gap is along the stable diagonal at 953°C and xLIF = 0.70. Attempts to predict the liquid-liquid equilibria by means of the conformal ionic solution theory using temperature independent interaction parameters lead to an incorrect symmetry of the gap. The symmetry can be improved if the temperature dependence of the interaction parameters of the binary mixtures is taken into account.

Thermodynamics of polyatomic fluid mixtures—I theory

Solution theories currently in use by chemical engineers (regular solutions, conformal solution theory, etc.) are based on an oversimplified view of the intermolecular forces, so that they are strictly valid only for such simple fluids as argon. We present here a solution theory (based on thermodynamic perturbation theory) which allows strong directional inter-molecular forces to be taken into account when calculating thermodynamic properties. This theory is in excellent agreement with computer simulation results for strongly polar or quadrupolar fluids. It is applied to the prediction of phase equilibrium surfaces, critical loci, three-phase lines, and azeotropic loci for mixtures containing constituents with (a) polar, (b) quadrupolar, and (c) anisotropic shape (charge overlap) intermolecular forces. The relation between these intermolecular forces and the type of phase behaviour is explored for binary mixtures. The anisotropic shape forces have little effect on the phase diagram; however, a wide variety of phase behaviour is found for the other cases. Mixtures with polar or quadrupolar constituents are shown to give rise to five of the six classes of binary phase diagrams found in nature.

Multiparameter Conformal Solution Theory — Application to Hydrocarbon Pure Fluid and Mixture Thermodynamic Properties

AbstractThe basis for a multiparameter conformal solution theory which offers potential for applications to wide classes of fluid mixtures is discussed. The concept of conformality is invoked for functionals which are dimensionless integrals over the reference fluid radial distribution function in the Pople type of perturbation expansion rather than the usual requirement of conformality of reduced properties. For mixtures, a van der Waals one fluid type model is used to express mixture properties as hypothetical pure fluid properties with intermolecular potential parameters expressed as functions of the component potential parameters and composition. In addition to the usual separation and energy parameters, the intermolecular potential can include dipole, quadrupole, octupole and overlap potential parameters. A simplified version of the method yields accurate predictions for light hydrocarbon thermodynamic properties.

Statistical thermodynamic approach to the prediction of vapor-liquid equilibria of multi-component mixtures

A Hard Sphere Expansion ( HSE) conformal solution theory, which is a result of the application of statistical mechanics, is developed in order to predict the vapor-liquid equilibria properties of multicomponent mixtures. The HSE mixing rules are derived with consideration of the Kihara spherical-core function as the pair potential together with the 3-body potential function. Based on this HSE conformal solution theory, an equation of state for multi-component fluid is developed. In this conformal solution theory perturbation equation of state of pure fluids is used as the reference equation of state. The vapor-liquid equilibria properties of several properly chosen binary and ternary non-polar mixtures are predicted. These mixtures include: argon-krypton, argon-methane, methane-krypton, nitrogen-methane, nitrogen-argon, argon-neopentane, methane-neopentane, methane-perfluoromethane, carbon dioxideethylene, carbon dioxide-ethane, nitrogen-argon-oxygen, and nitrogen-argon-methane. The predicted results obtained in the present investigation compare quite favorably with the experimental vapor-liquid equilibria values. Furthermore, the present approach, being a purely molecular theory of fluid mixtures, provides detailed insight into the peculiar behavior of the azeotropic fluid mixture. It is shown that the vapor-liquid equilibria properties of multicomponent fluid systems may be predicted correctly through the present statistical mechanic approach provided that like- and unlike-interaction parameters of the species of the mixtures are known a priori. Generally, the present investigation shows that the use of the molecular theory of fluid mixtures in predicting the mixture properties is very promising.

Application of the perturbation theory to vapor-liquid equilibria of systems containing ammonia.

The perturbation theory in conjunction with the conformal solution theory was applied to the correlation of vapor-liquid equilibria of systems containing ammonia; hydrogen-ammonia, nitrogen-ammonia, argon-ammonia, methane-ammonia, and nitrogen-methane-ammonia. The Lennard-Jones potential, with temperature-dependent parameters which were determined from the P-V-T and related properties of pure substance, was used to represent the intermolecular potentials for both nonpolar and polar molecules. By comparing the calculated results with the experimental data, it was found that the perturbation theory is useful in correlating the vapor-liquid equilibria of binary and ternary mixtures containing ammonia.

A theoretical attempt to interpret patterns of trace element partitioning in silicate minerals.

Using the conformal solution theory of molten salts (REISS et al., 1962; DAVIS and RICE, 1964; BLANDER, 1967), an attempt was made to interpret the patterns of the partition coefficients of trace elements in silicate minerals as a function of ionic radius. It is shown that the partition coefficient is represented to the zeroth order approximation by the quadratic equation of ionic radius in a “simple” homovalent substitution where host and guest cations are of primarily ionic nature. Pressure effects on the partition coefficient of a trace component are qualitatively discussed with emphasis on the interrelationship of melting points between major and minor components.