Van Der Waals Partition Function Research Articles

Extension of a square-well equation of state for chain-like molecules using perturbed hard chain theory

The Square-Well equation of state developed by Haghtalab and Mazloumi [A. Haghtalab, S.H. Mazloumi, Fluid Phase Equilib., 280 (2009) 1–8.] is extended for chain like molecules using a generalized van der Waals partition function for polyatomic fluids suggested by Donohue and Prausnitz [M.D. Donohue, J.M. Prausnitz AICHE J., 24 (1978) 849-860.]. The proposed EOS has three physical parameters per each pure fluid that are close-packed volume, characteristic temperature and one-third of external degree of freedom of the molecule. Here in, the parameters of the new EOS for a selection of pure fluids containing normal alkanes from C1 up to C20 are reported that are obtained by correlating saturated vapor pressure and liquid density of the fluids. Then, using proper mixing rules, vapor-liquid equilibrium calculation of a number of binaries are successfully carried out. The comparisons of the results with SPHCT EOS demonstrate the good accuracy of the new EOS.

Phase Transition and Criticality of Methane Confined in Nanopores.

For the first time, the phase transition and criticality of methane confined in nanoporous media are measured. The measurement is performed by establishing an experimental setup utilizing a differential scanning calorimeter capable of operating under very low temperatures as well as high pressures to detect the capillary phase transition of methane inside nanopores. By performing experiments along isochoric cooling paths, both the capillary condensation and the bulk condensation of methane are detected. The pore critical point of nanoconfined methane is also determined and then used to derive the parameters of a previously developed self-consistent equation of state based on the generalized van der Waals partition function. Using these parameters, the equation of state can predict the capillary-condensation curves that agree well with the experimental data.

Prototype equation of state for phase transition of confined fluids based on the generalized van der Waals partition function.

A simple self-consistent prototype equation of state (EOS) based on the generalized van der Waals (vdW) partition function has been demonstrated to describe the phase transition of simple fluids in nanopores with uniform size. Different from those commonly presented in the literature, the new EOS does not need an auxiliary equation that is conventionally applied to provide the capillary pressure derived from surface tension. The encouraging performance of the EOS calls for further extension to applications with more complex fluids and porous media.

A new adsorption model based on generalized van der Waals partition function for the description of all types of adsorption isotherms

A new adsorption model is proposed based on the Generalized van der Waals partition function by introducing the concept of attractive region, which integrates the effects of the adsorption sites of adsorbent and represents pore space where adsorbates concentrate. A theoretical expression for the chemical potential of adsorbates in the attractive region is derived and used to calculate the adsorbate density in each attractive region. The model can generate all the six types of isotherms classified by the International Union of Pure and Applied Chemistry (IUPAC) report including those with hysteresis and is easy to use. It is a generalized model that can be applied to various types of porous materials. With the fitted parameters, the model can predict the adsorption isotherms even for adsorbents with wide pore size distributions. It also well captures the isotherm-type transition as temperature changes and can estimate the hysteresis when capillary condensation exists.

A generalized van der Waals model for light gas adsorption prediction in IRMOFs.

By making reasonable simplifications to the structures of isoreticular metal-organic frameworks (IRMOFs) and defining important attractive regions of square-well potential, an adsorption model derived from the generalized van der Waals partition function is proposed to describe the isotherms of light gas adsorption in IRMOFs. The simplification of the structures is based on the geometries of the accessible surfaces and the dimensions of the frameworks, and the locations of the attractive regions are defined by examining the distribution of the adsorbate molecules. Grand Canonical Monte Carlo (GCMC) simulations using the simplified structures with square-well potentials and the complete atomic structures with Lennard-Jones and coulombic potentials are performed and compared to verify the reliability of the simplification. The adsorption model proposed in this work can predict adsorption isotherms of IRMOFs accurately by calculating the adsorbed amounts in different attractive regions of the simplified frameworks. It is also demonstrated that the model with the five parameters fitted to the adsorption isotherm at one temperature can accurately predict the isotherms at other temperatures.

Adsorption of simple square-well fluids in slit nanopores: Modeling based on Generalized van der Waals partition function and Monte Carlo simulation

A model based on the Generalized van der Waals partition function is derived to predict the adsorption of square-well fluid in slit pores of any size, the walls of which also have square-well potential. The space inside the pore is divided into several regions based on the extent of the attractive regions generated by the walls. Closed-form expressions of the chemical potentials of the confined fluid in different regions in the pore are obtained. The densities of fluid in different regions are calculated by equalizing the chemical potentials of fluid in those regions to that of the bulk phase. To examine the accuracy of the model, the Grand Canonical Monte Carlo (GCMC) simulation is also conducted. We find that the model well captures the effects of the bulk conditions and the properties of adsorbate and pore on the density of adsorbate in the pore. The model is also shown to be able to predict the adsorption of real gases in various activated carbons.

Exact solution of the van der Waals model in the critical region

The celebrated van der Waals model describes simple fluids in the thermodynamic limit and predicts the existence of a critical point associated to the gas–liquid phase transition. However the behavior of critical isotherms according to the equation of state, where a gas–liquid phase transition occurs, significantly departs from experimental observations. The correct critical isotherms are heuristically re-established via the Maxwell equal areas rule. A long standing open problem in mean field theory is concerned with the analytic description of van der Waals isotherms for a finite size system that is consistent, in the thermodynamic limit, with the Maxwell prescription. Inspired by the theory of nonlinear conservation laws, we propose a novel mean field approach, based on statistical mechanics, that allows to calculate the van der Waals partition function for a system of large but finite number of particles N. Our partition function naturally extends to the whole space of thermodynamic variables, reproduces, in the thermodynamic limit N→∞, the classical results outside the critical region and automatically encodes Maxwell’s prescription. We show that isothermal curves evolve in the space of thermodynamic variables like nonlinear breaking waves and the criticality is explained as the mechanism of formation of a classical hydrodynamic shock.

The generalized van der Waals partition function as a basis for excess free energy models

The important insight of J. D. van der Waals in developing the equation of state that bears his name was the analysis of the separate contributions of the attractive and hard-core repulsive interactions to the equation of state. This insight led him to important advances in understanding fluids and their phase transitions. This same separation of attractive and hard core interactions have been used in statistical mechanics in the form of perturbation theory, and also as here in a form referred to as the Generalized van der Waals partition function. This partition function has been used in the literature to understand the assumptions that underlie equations of state, and to develop equations with a better theoretical basis. Here, we demonstrate how the generalized van der Waals partition function can be used to elucidate the assumptions inherent in all of the commonly used correlative activity coefficient models.

A modified UNIFAC model for the prediction of phase equilibrium for polymer solutions

A new simple activity coefficient model for the prediction of solvent activities in polymer solutions is presented. The model consists of three terms, the combinatorial and the residual terms, which are taken from UNIFAC model, and a free-volume term derived from the van der Waals partition function with the mixed volume, the mixed free-volume and the mixed c-value being supposed to be given by linear mixing rules. Twenty-one polymer solution systems were used to test the applicability of the present model. Compared with the UNIFAC-FV, Entropic-FV, and Entropic-FV-1.2 models, it is found that the model proposed in this work can produce better results than those from the other three models. Furthermore, the model is valid for a broad range of polymer concentration and a wide range of temperature. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 2541–2547, 2005

A free-volume term based on the van der Waals partition function for the UNIFAC model

The objective of this work was to develop a model that takes into account the unique free-volume effect of water. The free-volume percentage of water is less than that of most polymers. This is opposite to the free-volume behavior of most solvents with polymers. As a result, the free-volume term in the UNIFAC-FV model introduced by Oishi and Prausnitz fails for aqueous systems. In the new model a free-volume term derived from the generalized van der Waals partition function is added to the combinatorial and residual terms of UNIFAC. In UNIFAC the geometric parameters were generally obtained directly from the van der Waals values. For water, however, these parameters were regressed. With the new free-volume term better predictions were obtained when the standard van der Waals geometric parameters for water were used. The new model was compared with the UNIFAC-FV model of Oishi and Prausnitz and the Entropic-FV model of Elbro et al. The new model was found to give better predictions for the aqueous polymer systems as well as for the non-aqueous polymer systems.

Salting-out in the aqueous single-protein solution: the effect of shape factor

A molecular–thermodynamic model is developed to describe salt-induced protein precipitation. The protein–protein interaction goes through the potential of mean force. An equation of state is derived based on the generalized van der Waals partition function. The attractive term including the potential of mean force is perturbed by the statistical mechanical perturbation theory. The precipitation behaviors are studied by calculating the partition coefficient with various conditions such as the ionic strength and the shape of protein. Our results show that the protein shape plays a significant role in the protein precipitation behavior.

Improved models for the prediction of activity coefficients in nearly athermal mixtures Part II. A theoretically-based GE-model based on the van der Waals partition function

This work, which is a continuation of a recent publication on the same topic (G.M. Kontogeorgis, P. Coutsikos, P. Tassios and Aa. Fredenslund, Fluid Phase Equilibria, 92 (1994) 35), presents a new theoretically-based G E-model (Chain-FV) for alkane systems. The model is based on a modified form of the generalized van der Waals partition function and attempts to account for all non-energetic effects of solutions of both short- and long-chain alkanes, including alkane polymers. Both the free-volume effects and the density-dependent rotational degrees of freedom are considered. The resulting G E-model which, despite its derivation from a partition function resembles the Flory-Huggins formula, is suitable for vapor-liquid and solid-liquid equilibrium calculations for nearly athermal polymer solutions as well as for alkane systems. We show that using plausible assumptions for the free-volume and the external-degree-of-freedom parameter, very good predictions are obtained for activity coefficients of asymmetric alkane systems at both concentration ends, for solid-liquid equilibrium calculations, as well as in extreme cases (polymer solutions, activity coefficients of heavy model alkane polymers in short-chain compounds recently available from molecular simulation studies). The predictions of the Chain-FV model are better than the Flory-Huggins and Elbro terms and equivalent to the recently proposed empirical R-UNIFAC and p-FV models. However, unlike the latter two models, the one proposed in this work offers a successful combination of simplicity and accuracy, with a solid theoretical basis.

An equation of state for fluids by applying the tower—Well potential model

A simple theoretical equation of state is derived by applying the Tower-Well potential model about the molecular distribution based on the generalized van der Waals partition function. It needs only three molecular parameters which have distinct physical meanings. The resulting equation of state predicts rather well the vapor pressures, saturated liquid volumes, saturated vapor volumes and PVT thermodynamic properties of polar and structurally complex molecules over a wide temperature and pressure range.

Application of generalized van der Waals theory and approximations of radial distribution functions to the development of mixing rules

Li, M.-H. and Huang, F.-N., 1994. Application of generalized van der Waals theory and approximations of radial distribution functions to the development of mixing rules. Fluid Phase Equilibria, 96: 155-172. Based on generalized van der Waals partition function theory and approximations of radial distribution functions, mixing rules for the attractive part of an equation of state have been presented for extension of an augmented hard-core equation of state to mixtures. Two approximations, the van der Waals one-fluid approximation and the mean density approximation, are applied in this study. Within the framework of the conformal solution theory, the properties of a hypothetical fluid were calculated from the attractive part of an augmented hard-core equation of state. As a preliminary test of the method, an augmented hard-sphere equation of state has been applied. For polar systems in which components do not differ appreciably in size, both the MDA and the VDW models yield satisfactory results for the vapor-liquid equilibrium calculations. However, for strongly non-ideal fluid systems, the MDA is superior to the VDW. Adopting the mixing rules for the repulsive hard-core mixtures to describe structure effects, the proposed method can, in principle, be used to extend any accurate augmented hard-core equation of state to mixtures.

Equations of state for pure and mixture square-well fluids. II. Equations of state

New coordination number models for square-well (SW) fluids are incorporated with the generalized van der Waals partition function to develop equations of state for both pure and mixture SW fluids. The equations of state have been extensively tested with the Monte Carlo simulation data, of which three sets (18 data points) for the pure SW fluids are produced in this work. The results show that, without any parameters, our model reasonably describes not only the PVT behaviors but also the second and third virial coefficients of the model fluids. In addition, a comprehensive comparison has been made between our models and the other equations of state derived from the Lee-Chao and Lee-Sandler coordination number correlations.

The generalized van der waals partition function. V. Mixture of square-well fluids of different sizes and energies

In this communication we report the results of Monte Carlo simulations for mixtures of square-well molecules of different sizes and interaction energies. We use these results to test various local composition models which have been suggested, and find that only the model proposed earlier is in reasonable agreement with the simulations. This local composition model is then used in the generalized van der Waals partition function to derive a new density-dependent equation of state mixing rule. This mixing rule increases the order of the volume dependence of any equation to which it is applied, though a simple modification to the solution algorithm proposed here easily handles this problem. We find that this new density-dependent mixing rule leads to more accurate predictions than the van der Waals one-fluid mixing rule when no binary interaction parameters are used, and retains only the advantage of smaller values of these parameters and slightly better critical region predictions when binary interaction parameters are used.

Monte Carlo Study of the Buckingham Exponential-six Fluid

Abstract In this paper we report the results of extensive Monte Carlo simulations of a pure fluid of Buckingham modified exponential-six molecules. Data are presented for the configurational energy and pressure covering a wide range of temperatures and densities. These data are interpreted using the generalized van der Waals partition function with a novel separation into free volume and mean potential terms. We find, surprisingly, that the Buckingham fluid is described by a simple van der Waals-like equation of state provided that the b parameter is temperature dependent and chosen in a theoretically correct manner.

The generalized van der waals partition function—IV. Local composition models for mixtures of unequal-size molecules

In this study, we present the results of Monte Carlo computer simulations for binary mixtures of square-well molecules of equal energy but differing size ratios at a variety of temperatures and densities. These data are used to determine the extent of the local composition effects in these mixtures, and to test various local composition models which had been proposed. We then show how, by use of the most successful of the local composition models and the generalized van der Waals partition function, a new two-parameter activity coefficient model and a density-dependent mixing rule can be derived. The new activity coefficient model is found to be quite successful in correlating data for the aqueous nonelectrolyte solutions on which it was tested, with the important advantages of having two parameters which are not intercorrelated and only slightly temperature-dependent. The density-dependent equation of state mixing rule we derived was found, when compared to simulation data, to be of some improvement over the van der Waals one-fluid mixing rule for mixtures of molecules of different square-well depths, but of no advantage for molecules of similar energies but different size.

The generalized van der Waals partition function. III. Local composition models for a mixture of equal size square-well molecules

New local composition models for mixtures of equal size molecules with differing attractive forces are presented, and compared with our results of Monte Carlo computer simulations for mixtures of square-well molecules which are also reported here. Unlike most previous local composition models, these new models predict random mixing in the high density limit and Boltzmann factor nonrandomness at low density; both limiting behaviors are in agreement with statistical mechanical theory. The predictions of these new models as a function of composition, density and temperature are in good agreement with the Monte Carlo computer simulation results.

The generalized van der waals partition function. I. basic theory

Sandler, S.I., 1985. The generalized van der Waals partition function. I. Basic theory. Fluid Phase Equilibria, 19:233-257 In this paper we provide a new derivation of the generalized van der Waals partition function for pure fluids and mixtures, and show how this partition function can be used as a basis for understanding equations of state, their mixing rules, and excess free energy (activity coefficient) models. The results presented here clarify some of the confusion which presently exists in the literature concerning the ramifications of local composition thermodynamic models, and provide the theory for papers to follow which combine the generalized van der Waals partition function and our computer simulation results to obtain new, statistical mechanical-based thermodynamic models.