Hard-sphere Equation Of State Research Articles

On the Effect of the Hard-sphere Term on the Statistical Associating Fluid Theory Equation of State

The hard-sphere system is the reference fluid for most versions of the statistical associating fluid theory (SAFT) equation of state (EoS). In the SAFT-type equations of state, the hard-sphere equation directly affects the reference term and through the radial distribution function, indirectly affects the chain and association terms. Although there are various EoSs for describing the thermodynamic behavior of the hard-sphere fluid, the Carnahan-Starling EoS has traditionally been used to express the hard-sphere contribution in SAFT-type models. In this work, we integrated eight hard-sphere EoS in the simplified SAFT EoS and parameterized the resulting EoS with pressure-density-temperature data. Then, using modified versions of the simplified SAFT (SSAFT), we calculated the thermodynamic properties including pressure, density, temperature, saturated vapor-pressure, isochoric heat capacity, isobaric heat capacity, and speed of sound, for ethane, butane, hexane, and hexatriacontane, and compared the results. In general, the results of calculations show that although the modified Carnahan-Starling, Kolafa, and Carnahan-Starling EoSs have had the best results, considering the simplicity and straightforwardness of the Carnahan-Starling EoS, this EoS is a reasonable choice for developing SAFT-type equations of state.

Soft congestion approximation to the one-dimensional constrained Euler equations

This article is concerned with the analysis of the one-dimensional compressible Euler equations with a singular pressure law, the so-called hard sphere equation of state. We provide a detailed description of the effect of the singular pressure on the breakdown of the smooth solutions. Moreover, we rigorously justify the singular limit for smooth solutions towards the free-congested Euler equations, where the compressible (free) dynamics is coupled with the incompressible one in the constrained (i.e. congested) domain.

Solubility Determination and Correlation of Gatifloxacin, Enrofloxacin, and Ciprofloxacin in Supercritical CO2

The solubilities of gatifloxacin, enrofloxacin, ciprofloxacin in SC–CO2 were determined by the dynamic method under pressures of 12–36 MPa and temperature of 313, 323, and 333 K. The consistency of the resulting solubility data was verified by the Mendez-Santiago and Teja model, Chrastil model, Bartle model, and K-J model and these models give comparable AARDs (between 6.70% and 13.51%). The compressed gas model and modified expanded liquid model were also used to correlate the resulting solubilities of the three fluoroquinolone drugs. By introducing the reference solubility and calculating the fugacity coefficient of the solute using the Carnahan–Starling-VDW hard sphere equation of state (CS-VDW EoS), the compressed gas model gives the AARDs between 0.86% and 23.65%. The intrinsic proximity of the model parameters in this model is also confirmed for the structurally similar fluoroquinolone drugs. This intrinsic proximity can be used in mutual solubility prediction between the fluoroquinolone drugs once the reference solubilities were known. The modified expanded liquid model gives the AARDs between 10.88% and 16.65% in solubility correlation. However, the predictive capability of the modified expanded liquid model based on the group contribution method needs to be improved further.

Equations of state for fluids based on hard sphere repulsion

As is well-known, the structures and thermodynamic properties of fluids are determined by the complex interactions, i.e., the repulsive one and the attractive one, among particles. The simplest equation-of-state (EOS) model maybe the one of hard sphere repulsion plus or multiplying some attraction. Followed by the rapid promotion of the accuracy of hard sphere EOS in the past dozens of years, one question rises as whether more accurate hard sphere repulsion derives better prediction of the structures and properties of fluids with a special attraction. In this work, we used two repulsions with clearly different accuracy and some attractions to construct series equations of state (EOSs) for real fluids, and then we discussed the saturated properties at liquid–gas equilibrium. We found that the answer to the question aforementioned is not definitely standing.

A perturbed hard sphere chain equation of state for refrigerants

The present work shows a successful extension of a recently developed modified perturbed hard sphere equation of state to liquid refrigerants. Our objective is the prediction of a reliable model to describe the volumetric properties of refrigerants. In this regard, we have applied our previously developed equation of state to R12, R13, R13b1, R22, R23, R32, R123, R124, R134a, R141b, R142b, R143a, R170, R218, R227ea, R290, R1150 and R1270. Two temperature dependent parameters which are universal functions of the reduced temperature and just one adjustable parameter that can be calculated by fitting to a few experimental data exist in this equation of state. Boiling point constants are utilized as scaling factors. In order to evaluate the model, we compared the predicted densities to the literature data. This led to very encouraging results. The results of four other equations of state are presented for comparison. Our results seem to be promising in view of application to mixtures of refrigerants and long-chain hydrocarbons. In this respect the paper can be regarded as a necessary step towards the prediction of the mixture behavior of refrigerants and the effect of the presence of the lubricant oil in the mixture.

Evaluation of compressibility factor and mean ionic activity coefficient for aqueous electrolyte solutions with hard sphere equations of state in the MSA model and artificial neural network method

In this study, twenty-five hard sphere equations of state (EOSs) were applied to evaluate the compressibility factor of fifty different single electrolyte solutions (Z1). Then by using two well-known mixing rules (Barrio Solana (BS) and Santos et al. (S)) the compressibility factor (Zm) was calculated for 1086 binary mixtures of electrolytes. The obtained results were compared with the simulation data and it was concluded that the Dehghani–Modarress (DM) EOS when applied along with the Santos et al. mixing rule has the highest accuracy (ARD%Z1DM=0.124 and ARD%ZmDM−S=0.466). Also, the mean ionic activity coefficients of a number of 1:1 electrolytes were calculated by applying the hard sphere EOSs with the mixing rules based on the mean spherical approximation (MSA) model where the concentration of cation was considered a function of electrolyte concentration. By comparing these results with reported experimental data it was found that the Ree–Hoover (RH) EOS with Santos et al. mixing rule predicts the mean ionic activity coefficients with the highest accuracy (ARD%γ(RH−S)=0.206). In continuation of this study the compressibility factor and mean ionic activity coefficient were recalculated by means of the artificial neural network (ANN) method. The optimum architectures of ANN models for predicting the compressibility factor of single and binary mixtures of electrolytes as well as the mean ionic activity coefficient of 1:1 electrolytes were respectively obtained as 2:4:1, 3:3:1 and 3:5:1. The calculated error in the results showed that the mathematical models of the ANN method predicted the abovementioned parameters with better accuracy in comparison with the EOSs and MSA models (ARD%Z1ANN=0.047, ARD%ZmANN=0.022 and ARD%γ(ANN)=0.014).

Applications of statistical mechanical theories for estimating thermodynamic properties of ternary and binary solutions using ultrasonic velocity and density data

Thermodynamic properties of liquids and liquid mixtures play very important role in understanding the nature of molecular interactions occurring in the system. In the present work density (ρ) and ultrasonic velocity (u) of the ternary system benzene+carbon tetrachloride+cyclohexane and its three constituent binaries namely, benzene+cyclohexane, carbon tetrachloride+cyclohexane and benzene+carbon tetrachloride have been measured at 298.15K. Different thermodynamic properties of ternary and binary solutions have also been computed with the help of Flory's statistical theory (FST), hard sphere equation of state (HSE) and hole theory (HT) simultaneously. The calculated values are compared with the experimental findings and quite satisfactory results are obtained.

Helmholtz energies from equations of state and their relative deviations

Extension of the hard-sphere equations of state to non spherical hard-body systems represents a formidable task. For an equation of state that is applicable to liquid and vapor states (and explicit for pressures), the corresponding characteristic function is the residual energy. The equations of state of Carnahan Starling, Boublik and R J Sadus are used to derive an expression for residual Helmholtz energies. Relative deviations of residual Helmholtz energies for these equations of state have been reviewed. The deviations of compressibility factor against packing fraction have been plotted for these equations of state. The agreement of these plots is found to be good for lower n-alkanes but not for longer n-alkanes. Thus it becomes obvious that the compressibility factor depends considerably on the shape of the studied molecules.

Estimation of Thermodynamic Properties of Binary Liquid Mixtures on the Basis of Statistical Mechanical Theories

Thermodynamic properties of liquids and liquid mixtures play very important role in understanding the nature of molecular interactions occurring in the system. In the present work different thermodynamic properties of 15 pure liquids and 34 equimolar binary liquid mixtures of benzene, toluene, p-xylene, chlorobenzene and 1-chloronaphthalene with linear and branched alkanes have been computed with the help of Flory’s statistical theory (FST), Hard sphere equation of state (HSE) and Hole theory (HT) simultaneously. The calculated values are compared with the experimental findings collected from literature and quite satisfactory results are obtained.

Correlating and predicting the solubilities of polycyclic aromatic hydrocarbons in supercritical fluids using the compressed gas model and the reference solubilities

The solubilities of some polycyclic aromatic hydrocarbons (PAHs) in supercritical fluids were correlated and predicted using the compressed gas model and the reference solubilities. By introducing the reference solubilities, the compressed gas models combined with Peng–Robinson equation of state, Carnahan–Starling–van der Waals hard sphere equation of state and van der Waals one-parameter mixing rules can yield the overall averages of the average absolute relative deviations (AARDs) of 7.45% and 8.16% in solubility correlation for PAHs in supercritical fluids, respectively, which are much less than the values obtained using the compressed gas model combined with the corresponding equation of state and the sublimation pressures of the solutes. By introducing the reference solubilities, the calculated solubilities are not sensitive to the binary interaction parameters and the solubilities of PAHs in the corresponding supercritical fluid can be predicted with a specific binary interaction parameter. When using reference solubilities in the compressed gas model, the two equations of state provide comparable average AARDs in solubility prediction for PAHs in supercritical fluids, which are 14.00% and 14.19%, respectively. The introduction of reference solubilities into the compressed gas model provides a feasible and simple solubility prediction method for the compounds in supercritical fluids without using their sublimation pressures. Moreover, combined with the reference solubilities and the compressed gas model, the hard sphere equation of state provides a more simple method to predict the solubilities of solutes in supercritical fluids with only their solid molar volumes.

Osmotic virial coefficients for model protein and colloidal solutions: Importance of ensemble constraints in the analysis of light scattering data

Protein-protein interactions in solution may be quantified by the osmotic second virial coefficient (OSVC), which can be measured by various experimental techniques including light scattering. Analysis of Rayleigh light scattering measurements from such experiments requires identification of a scattering volume and the thermodynamic constraints imposed on that volume, i.e., the statistical mechanical ensemble in which light scattering occurs. Depending on the set of constraints imposed on the scattering volume, one can obtain either an apparent OSVC, A(2,app), or the true thermodynamic OSVC, B(22)(osm), that is rigorously defined in solution theory [M. A. Blanco, E. Sahin, Y. Li, and C. J. Roberts, J. Chem. Phys. 134, 225103 (2011)]. However, it is unclear to what extent A(2,app) and B(22)(osm) differ, which may have implications on the physical interpretation of OSVC measurements from light scattering experiments. In this paper, we use the multicomponent hard-sphere model and a well-known equation of state to directly compare A(2,app) and B(22)(osm). Our results from the hard-sphere equation of state indicate that A(2,app) underestimates B(22)(osm), but in a systematic manner that may be explained using fundamental thermodynamic expressions for the two OSVCs. The difference between A(2,app) and B(22)(osm) may be quantitatively significant, but may also be obscured in experimental application by statistical uncertainty or non-steric interactions. Consequently, the two OSVCs that arise in the analysis of light scattering measurements do formally differ, but in a manner that may not be detectable in actual application.

A perturbed hard-sphere equation of state for liquid metals

A perturbed hard-sphere equation of state, developed previously for liquid alkali metals and liquid refractory metals, has been applied for PVT calculation of some pure liquid metals including alkaline earth metals, tin, lead, antimony, bismuth, and rubidium over a wide range of temperatures and pressures. Two temperature-dependent parameters appear in the equation of state, which are universal functions of the reduced temperature, i.e. two scale parameters are sufficient to calculate the temperature-dependent parameters. The scaling parameters can be easily obtained by employing a corresponding-states principle based on a Lennard-Jones potential energy function. Employing the present equation of state, the liquid densities of aforementioned metals at temperatures ranging from the melting point to 2000 K and at pressures ranging from vapour pressure up to 40,000 bar have been calculated and compared with experimental data. The average absolute deviation in predicted densities compared with experimental data is 1.54%.

A perturbed hard-sphere equation of state extended to imidazolium-based ionic liquids

A perturbed hard-sphere equation of state (EOS) has been previously employed to predict pressure–volume–temperature properties of some ionic liquids (ILs) with phosphonium-, pyridinium-, and pyrrolidinium cations. In this work, we have extended the considered EOS to another class of ILs in compressed states. This class consists of 14 imidazolium-based ILs. The predicted densities were compared with those obtained from the experiment, over a broad pressure range from 0.1 to 200 MPa. From 1,122 data points examined for the aforementioned ILs, the total average absolute deviation was found to be 1.05%.

Thermal expansion coefficient of ternary liquid mixture using hard sphere models and Flory’s statistical theory

The coefficients of thermal expansion (α) of ternary liquid mixtures of heterocyclic compounds viz., pyridine/quinoline with phenol in benzene have been evaluated at 308.15, 318.15 and 328.15 K, respectively. A comparative study of thermal expansivity using different models for hard sphere equation of state and Flory’s statistical theory has been made. The computed values of thermal expansivity are compared with the experimental results. The excess values of thermal expansivity have also been computed and utilized to discuss the existence and strength of intermolecular interactions in the ternary liquid system under investigation.

Application of Restrictive Primitive SAFT Coupled with Different Hard-Sphere Equations to Model Mean Ionic Activity Coefficients of Electrolyte Solutions

In this work, the primitive SAFT equation of state along with three different hard-sphere equations was used to correlate and predict mean ionic activity coefficients of aqueous electrolyte solutions. The mean ionic activity coefficient of aqueous electrolyte solutions was considered as the contribution of hard-sphere and dispersion effects. The Mansoori (M), Wang-Khoshkbarchi-Vera (WKV) and Ghotbi-Vera (GV) hard-sphere equations were applied in correlating the mean ionic activity coefficient of electrolyte solutions. The comparison among above indicated equations was shown. First, vapor pressure and densities of water in the temperature range of 373.15 to 423.15 K was regressed by SAFT equation of state. In the restrictive primitive mean spherical model, ions were hard spheres without any chain structure. Neither association effects were considered in this study. Clearly, in common used five SAFT parameters were decreased to three, which were calculated by using the experimental mean ionic activity coefficients of electrolyte solutions. The comparison among three hard-sphere equations of state approved that Ghotbi-Vera hard-sphere model (GV) correlated the experimental data accurately than the others; two hard-sphere models. The mean ionic activity coefficients of some electrolyte solutions were being predicted by taking the advantage of the regressed values surely, in a wide range of molality.

Colloids with a tunable dipolar interaction: equations of state and order parameters viaconfocal microscopy

Laser scanning confocal microscopy studies were carried out on charged screened colloidal silica microspheres in sedimentation equilibrium in the presence of a tunable external alternating electric field. The external a.c. field controls the averaged dipolar interaction between the silica microspheres. From the equilibrium sedimentation profiles, equations of state were obtained as a function of the dipolar strength and volume fraction. The colloidal fluid at zero field follows the hard-sphere equation of state but with an effective larger hard-sphere diameter. At non-zero fields, the average colloidal fluid isothermal compressibility was obtained as a function of a dipolar strength parameter. We then investigated bond order parameters upon increasing the field, and monitored an orientational order parameter associated with the field dependence of the average orientation of dual-particle bonds. The orientational order parameters were found to be very sensitive measures of anisotropy. The field dependence strongly confirms the use of the point dipolar approximation in this system.

Helium bubbles formation in aluminum: Bulk diffusion and near-surface diffusion using TEM observations

Experimental and analytical investigation of helium bubble formation and growth in aluminum is presented. A pure aluminum with 0.15 wt% of 10B was neutron-irradiated in the Soreq nuclear reactor to get homogeneous helium atoms in the metal according to the reaction 10 B + n → 7 Li + 4 He . Formation and growth of helium bubbles was observed in situ by heating the post-irradiated metal to 470 °C in TEM with a hot stage holder. It was found that above 400 °C the change in the bubble shape takes less than a second. In other experiments the Al– 10B was first heated in its bulk shape and then observed in TEM at room temperature. In this case the helium bubble formation takes hours. Analytical evaluation of the diffusion processes in both cases was done to explain the experimental results. The number of helium atoms in a bubble was calculated from the electron energy loss spectrum (EELS) measurements. These measurements confirmed the hard sphere equation of state (EOS) for inert gases that was used in the analytical diffusion calculations.

A New Hard Sphere Cubic Equation of State for Predicting Fluids’ Properties and Vapor-Liquid Phase Equilibrium Calculations

In this study, a new cubic hard sphere equation of state (EOS) was developed from standard classical thermodynamics. The new equation is applied to calculate properties of fluids and vapor-liquid phase equilibrium calculations. The derived equation is a simplified expression of the hard sphere equation which yields satisfactory agreement with the molecular simulation of hard molecule data. The EOS is written in a cubic form by combining the derived repulsive hard spheres with Redlich-Kwong (RK) empirical attractive term. Satisfactory calculated results for the saturated properties of pure fluids for temperature ranges from 303 to 523 K and pressure ranges from 50 to 5000 psi are obtained. Simplicity and generality of this equation combined with reasonable accuracy makes it a useable EOS for almost all areas of equipment design for separation processes and production operations including refinery and petroleum reservoir industries. The accuracy of the predicted properties from the developed EOS are greater than of other commonly used two parameter cubic equations of state, RK and Pang-Robinson (PR).

Modified equation of state for pure and hard sphere mixtures in mean ionic activity coefficient calculations by mean spherical approximation model

A hard sphere equation of state (EOS) based on tetrakaidecahedron cell geometry (instead of spherical shape) and highly optimized molecular dynamic simulation data is proposed. The EOS is extended to hard sphere mixture and its performance for compressibility factor calculation at different diameter size of hard sphere mixtures by using various mixing rule is compared with Monte Carlo simulation data. The results indicated that for all mixing rules, the proposed EOS has minimum error comparing with computer simulation data. Also the residual prosperities are derived by using the proposed EOS. The residual properties are used in mean spherical approximation model (MSA) to evaluate the mean ionic activity coefficient of aqueous electrolyte solutions. The results are compared with those obtained by similar hard sphere equations of state and it is shown that the proposed EOS has a better performance in predicting the mean ionic activity coefficient.

Speeds of sound and isothermal compressibility of ternary liquid systems: Application of Flory’s statistical theory and hard sphere models

Speeds of sound and densities of three ternary liquid systems namely, toluene + n-heptane + n-hexane (I), cyclohexane + n-heptane + n-hexane (II) and n-hexane + n-heptane + n-decane (III) have been measured as a function of the composition at 298.15 K at atmospheric pressure. The experimental isothermal compressibility has been evaluated from measured values of speeds of sound and density. The isothermal compressibility of these mixtures has also been computed theoretically using different models for hard sphere equations of state and Flory’s statistical theory. Computed values of isothermal compressibility have been compared with experimental findings. A satisfactory agreement has been observed. The superiority of Flory’s statistical theory has been established quite reasonably over hard sphere models.