Phase equilibrium, 4.28

Simulating Phase Equilibria by the Gibbs Ensemble Monte Carlo Method 

Null, H. R., "Phase Equilibria in Process Design," Wiley-Interscience (1970). 

Scott R L 1965 Phase equilibria in solutions of liquid sulfur. I. Theory J. Phys. Chem. 69 261-70 

In the case of three-phase equilibria, it is also necessary to account for the solubility of hydrocarbon gases in water. This solubility is proportional to the partial pressure of the hydrocarbon or, more precisely, to its partial fugacity in the vapor phase. The relation which ties the solubility expressed in mole fraction to the fugacity is the following  

A. Muan and E. F. Osborn, Phase Equilibria Among Oxides in Steelmaking, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. 

Figure 4 shows experimental and predicted phase equilibria for the acetonitrile/benzene system at 45°C. This system exhibits moderate positive deviations from Raoult s law. The high-quality data of Brown and Smith (1955) are very well represented by the UNIQUAC equation. 

PRAUSNITZ J.M. MOLECULAR THERMODYNAMICS OF FLUID PHASE EQUILIBRIA, PRENTICE-HALL. ENGLEWOOD CLIFFS. N.J.I19691. 

Panagiotopoulos A Z 1994 Moleoular simulation of phase equilibria Supercritical Fluids—Fundamentals for Application NATO ASI Series ed E Kiran and J M H Levelt Sengers (Dordreoht Kluwer) 

Prausnitz, J. M., "Molecular Thermodynamics of Fluid-Phase Equilibria," Prentice-Hall, Englewood Cliffs, N.J. (1969) 

Panagiotopoulos A Z 1989 Exaot oaloulations of fluid-phase equilibria by Monte Carlo simulation in a new statistioal ensemble Int. J. Thermophys. 10 447-57 

Laso M, dePablo J J and Suter U W 1992 Simulation of phase equilibria for chain molecules J Chem. Phys. 97 2817 

Panagiotopoulos A Z 1992 Direot determination of fluid phase equilibria by simulation in the Gibbs ensemble a review Mol. SImul. 9 1 -23 

Selected References for Calculation of Multicomponent Fluid-Phase Equilibria 

Panagiotopoulos A Z, Quirke N, Stapleton M and Tildesley D J 1988 Phase equilibria by simulation in the Gibbs ensemble. Alternative derivation, generalization and applioation to mixture and membrane equilibria Mol. Phys. 63 527-45 

Smit B, Karaborni S and Siepmann J I 1995 Computer simulations of vapor-liquid phase equilibria of n-alkanesJ. Chem. Phys. 102 2126-40 

Levelt Sengers J M H 1999 Mean-field theories, their weaknesses and strength Fluid Phase Equilibria 158-160 3-17 

The use of group contribution methods for the estimation of properties of pure gases and Uquids [20, 21] and of phase equilibria [22] also has a long history in chemical engineering. 

In Secs. 4.3 and 4.4 we discussed the thermodynamics of the crystal -> Uquid transition. This and other famiUar phase equilibria are examples of what are called first-order transitions. There are other less familiar but also well-known 

Kostrowicka Wyczalkowska A, Anisimov M A and Sengers J V 1999 Global crossover equation of state of a van der Waals fluid Fluid Phase Equilibria 158-160 523-35 

Martin M G and J I Siepmann 1999. Novel Configurational-bias Monte Carlo Method for Blanche Molecules. Transferable Potentials for Phase Equilibria. 2. United-atom Description of Branchi Alkanes. Journal of Physical Chemistry 103 4508-4517. 

For a multicomponent system, it is possible to simulate at constant pressure rather than constant volume, as separation into phases of different compositions is still allowed. The method allows one to study straightforwardly phase equilibria in confined systems such as pores [166]. Configuration-biased MC methods can be used in combination with the Gibbs ensemble. An impressive demonstration of this has been the detennination by Siepmaim et al [167] and Smit et al [168] of liquid-vapour coexistence curves for n-alkane chain molecules as long as 48 atoms. 

In the petroleum refining and natural gas treatment industries, mixtures of hydrocarbons are more often separated into their components or into narrower mixtures by chemical engineering operations that make use of phase equilibria between liquid and gas phases such as those mentioned below  

This chapter concentrates on describing molecular simulation methods which have a counectiou with the statistical mechanical description of condensed matter, and hence relate to theoretical approaches to understanding phenomena such as phase equilibria, rare events, and quantum mechanical effects. 

With the rapid development of computer power, and the continual hmovation of simulation methods, it is impossible to predict what may be achieved over the next few years, except to say that the outlook is very promising. The areas of rare events, phase equilibria, and quantum simulation continue to be active. 

The maximum-likelihood method, like any statistical tool, is useful for correlating and critically examining experimental information. However, it can never be a substitute for that information. While a statistical tool is useful for minimizing the required experimental effort, reliable calculated phase equilibria can only be obtained if at least some pertinent and reliable experimental data are at hand. 

Figure A2.5.26. Molar heat capacity C y of a van der Waals fluid as a fimction of temperature from mean-field theory (dotted line) from crossover theory (frill curve). Reproduced from [29] Kostrowicka Wyczalkowska A, Anisimov M A and Sengers J V 1999 Global crossover equation of state of a van der Waals fluid Fluid Phase Equilibria 158-160 532, figure 4, by pennission of Elsevier Science.

Lattice models have been studied in mean field approximation, by transfer matrix methods and Monte Carlo simulations. Much interest has focused on the occurrence of a microemulsion. Its location in the phase diagram between the oil-rich and the water-rich phases, its structure and its wetting properties have been explored [76]. Lattice models reproduce the reduction of the surface tension upon adsorption of the amphiphiles and the progression of phase equilibria upon increasmg the amphiphile concentration. Spatially periodic (lamellar) phases are also describable by lattice models. Flowever, the structure of the lattice can interfere with the properties of the periodic structures. 

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