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In Secs 162, 163, and 164 we describe how macroscopic thermodynamic properties, such as internal energy, entropy, and the Helmholtz energy are connected to properties of individual molecules and their assemblies These connections are exploited quantitatively through computer simulation of collections of molecule^^^ This approach has also been applied directly to VLE, primarily to pure species and to binary and simple ternary mixtures Descriptions of os ~ assumptions, methods, and calculational procedures are given by ~ a n a g i o t o ~ o u l and~in an extended review by ~ u b b i n sTheir extensive bibliographies are a guide to relevant literature ~~ The method requires suitable intermolecular potential energy functions U(r) and solution of the equations of statistical mechanics for the assemblies of molecules As mentioned in Sec 161, potential energy functions are as yet primarily empirical Except for the simplest molecules, U(r) cannot be predicted by ab i n i t i ~ ~ ~ calculations, because of still-inadequate computer speed Therefore, semi-empirical functions based on quantum-mechanical theory and experimental data are employed Two procedures are used for the very large number of computer calculations required to treat molecular assemblies The first, proposed by ~ a n a ~ i o t o ~ o u l ocalled the direct Gibbsis s , ~ ~ ensemble Monte Carlo method One considers two separate phases, each represented by a finite number of molecules contained in a simulation box Both are at the same specified temperature but are of different initial densities and compositions The idea is to implement a sequence of perturbations which gradually bring these two phases into equilibrium This ultimately insures internal equilibrium for each phase separately, equality of pressures between the two phases, and equality of the chemical potentials for each species in the two phases The perturbations ("moves") designed to bring about these conditions of equilibrium are therefore: 1 Random displacement of molecules within each box These are the usual moves of Monte Carlo simulation, insuring internal equilibrium and generating the ensemble upon which the partition function is based, thus leading to a set of thermodynamic properties for the molecules of each box 2 Random equal and opposite volume changes in the two boxes These moves alter the pressures in the boxes and ultimately bring about their equality As the pressures change, the ongoing Monte Carlo simulations in the boxes generate evolving thermodynamic properties.

2 1 ~ Allen and D J Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1989; D Frenkel and P B Smit, Understanding Molecular Simulations: From Algorithms to Applications, Academic Press, San Diego, 1996 2 2 ~ Panagiotopoulos, Molecular Simularion, vol 9, pp 1-23, 1992 Z 2 3 ~ E Gubbins, "Applications of Molecular Theory to Phase Equilibrium Predictions" in Models for Thermody namic and Phase Equilibrium Calculations, S I Sandler, ed, pp 507-600, Marcel Dekker, Inc, New York, 1994 2 4 ~ e a n i n gfrom the beginning, ie, from first principles : 2 5 ~ Panagiotopoulos, Molecular Physics, vol 61, pp 813-826, 1987 Z

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3 Random transfer of molecules between the two boxes These moves alter compositions and chemical potentials pi of the species in the boxes, ultimately bringing about equality of the chemical potentials for each species in the two boxes These moves also contribute to the evolution of the thermodynamic properties of the molecules in the boxes Moves may be accepted or rejected on the basis of certain probabilities that insure progression on average to states of lower Gibbs energy for the two boxes considered together Phase equilibrium obtains at the state of minimum total Gibbs energy A limitation of the Gibbs-ensemble Monte Carlo method is that the successful transfer of molecules between phases (item 3) becomes difficult (improbable) for dense fluids, leading to excessive calculation time In this event, a second, more indirect, procedure can be employed The idea here is to calculate the chemical potentials of the species in each box for a range of states Equilibrium compositions are then identified as those states having the same temperature, pressure, and chemical potential for each species in the two boxes Much more computation is needed by this procedure than for the direct procedure, except where the molecular-transferstep becomes difficult For high-density fluids and for solids, the indirect procedure is preferred Molecular simulation is not a routine method for VLE calculations nor does it substitute for experimental data At present, it is most useful for extrapolating to conditions not accessible by other means The Gibbs-ensemble procedure has also been employed to estimate adsorption isotherms for simple systems The approach is i l l u ~ t r a t e d ~ ~ by calculations for a straight cylindrical pore where both fluidtfluid and fluidladsorbent molecular interactions can be represented by the Lennard-Jones potential-energy function [Eq (16 I)] Simulation calculations have also been made for isotherms of methane and ethane adsorbed on a model carbonaceous slit pore27 Isosteric heats of adsorption have also been c a l c ~ l a t e d ~ ~.

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