Calculation of pVT and vapor–liquid equilibria of copolymer systems based on an equation of state

A molecular thermodynamic model for copolymers and their mixtures has been established by adopting the hard-sphere-chain fluid as a reference and a square-well (SW) term as well as an association term as a perturbation. The latter is introduced to consider various associating functions in a chain-like molecule based on the shield-sticky model of chemical association. The model adopts five molecular parameters, i.e. r_i, σ_ii ε_ii/k, δε_ii/k and ω_ii, for a polymer species i, where the last two are responsible for association. These parameters can be obtained from the pVT data of the corresponding molten homopolymer i. The model can be used to correlate pVT data for molten copolymers with an adjustable parameter describing the interaction between different polymer species. The model can also be used to calculate vapor–liquid equilibria (VLE) for copolymer solutions with three adjustable interaction parameters.     

Soft-SAFT modeling of vapor–liquid equilibria of nitriles and their mixtures

Nitriles are strong polar compounds showing a highly non-ideal behavior, which makes them challenging systems from a modeling point of view; in spite of this, accurate predictions for the vapor–liquid equilibria of these systems are needed, as some of them, like acetonitrile (CH₃CN) and propionitrile (C₂H₅CN), play an important role as organic solvents in several industrial processes. This work deals with the calculation of the vapor–liquid equilibria (VLE) of nitriles and their mixtures by using the crossover soft-SAFT Equation of State (EoS). Both polar and associating interactions are taken into account in a single association term in the crossover soft-SAFT equation, while the crossover term allows for accurate calculations both far from and close to the critical point. Molecular parameters for acetonitrile, propionitrile and n-butyronitrile (C₃H₇CN) are regressed from experimental data. Their transferability is tested by the calculation of the VLE of heavier linear nitriles, namely, valeronitrile (C₄H₉CN) and hexanonitrile (C₅H₁₁CN), not included in the fitting procedure. Crossover soft-SAFT results are in excellent agreement with experimental data for the whole range of thermodynamic conditions investigated, proving the robustness of the approach. Parameters transferability has also been used to describe the isomers n-butyronitrile and i-butyronitrile. Finally, the nitriles soft-SAFT model is further tested in VLE calculation of mixtures with benzene, carbon tetrachloride and carbon dioxide, which proved to be satisfactory as well.     

A new three-parameter cubic equation of state for calculation physical properties and vapor–liquid equilibria

A new three-parameter cubic equation of state is presented by combination of a modified attractive term and van der Waals repulsive expression. Also a new alpha function for the attractive parameter of the new EOS is proposed. The new coefficients of alpha function and the other parameters of the attractive term are adjusted using the data of the saturated vapor pressure and liquid density of almost 60 pure compounds including heavy hydrocarbons. The new EOS is adopted for prediction of the various thermophysical properties of pure compounds such as saturated and supercritical volume, enthalpy of vaporization, compressibility factor, heat capacity and sound velocity. Following successful application of the new EOS for the pure components, using vdW one-fluid mixing rules, the new EOSs are applied to prediction of the bubble pressure and vapor mole fraction of the several binary and ternary mixtures. The accuracy of the new EOS for phase equilibrium calculation is demonstrated by comparison of the results of the present EOSs with the PT, PR, GPR and SRK cubic EOSs.     

Modification of the Furter equation and correlation of the vapor–liquid equilibrium for mixed-solvent electrolyte systems

Isobaric vapor–liquid equilibrium values at 1 atm pressure were measured for the systems 1-propanol–water–potassium acetate and 2-propanol–water–potassium acetate under fixed salt mole fractions using a modified Othmer recirculation still. A modified Furter equation, ln(α_s/α)=k₁z+k₂z², was proposed for correlating the effect of dissolved salts on vapor–liquid equilibrium (VLE). The modified equation contains two parameters that are applicable to the entire salt/solvent composition range. Correlation of VLE for 15 mixed-solvent electrolyte systems was made by means of the proposed modified equation with better results than those obtained from the original equation.     

The liquid–liquid coexistence curves of {benzonitrile + n-pentadecane} and {benzonitrile + n-heptadecane} in the critical region

Liquid + liquid coexistence curves for the binary solutions of {benzonitrile + n-pentadecane} and {benzonitrile + n-heptadecane} have been measured in the critical region. The critical exponent β and the critical amplitudes have been deduced and the former is consistent with the theoretic prediction. It was found that the coexistence curves may be well described by the crossover model proposed by Gutkowski et al. The asymmetries of the diameters of the coexistence curves were also discussed in the frame of the complete scaling theory.     

On the applicability of Young–Laplace equation for nanoscale liquid drops

Debates continue on the applicability of the Young–Laplace equation for droplets, vapor bubbles and gas bubbles in nanoscale. It is more meaningful to find the error range of the Young–Laplace equation in nanoscale instead of making the judgement of its applicability. To do this, for seven liquid argon drops (containing 800, 1000, 1200, 1400, 1600, 1800, or 2000 particles, respectively) at T = 78 K we determined the radius of surface of tension R_s and the corresponding surface tension γ_s by molecular dynamics simulation based on the expressions of R_s and γ_s in terms of the pressure distribution for droplets. Compared with the two-phase pressure difference directly obtained by MD simulation, the results show that the absolute values of relative error of two-phase pressure difference given by the Young–Laplace equation are between 0.0008 and 0.027, and the surface tension of the argon droplet increases with increasing radius of surface of tension, which supports that the Tolman length of Lennard-Jones droplets is positive and that Lennard-Jones vapor bubbles is negative. Besides, the logic error in the deduction of the expressions of the radius and the surface tension of surface of tension, and in terms of the pressure distribution for liquid drops in a certain literature is corrected.     

Thermodynamics for fluid mixtures near to and far from the vapor–liquid critical point

To describe thermodynamic properties of fluid mixtures near to and far from the vapor–liquid critical point, we need a method where a classical equation-of-state is augmented with a correction based on renormalization group (RG) theory. The advantage of the method described here is that, subject to well-defined assumptions, it can be applied not only to binary mixtures but to mixtures containing any number of components. While our method is based on White’s recursion procedure, our extension to mixtures is based on the isomorphism assumption and on an approximation suggested by Kiselev. To illustrate, calculations are presented for the critical loci of some alkane mixtures containing one discrete component and one pseudo-component whose composition is characterized by a continuous distribution of molecular weight. While critical loci calculated with the RG correction are similar to those calculated by the classical equation-of-state alone, inclusion of the RG correction provides better agreement with experiment.     

Modelling of high-pressure phase equilibria using the Sako–Wu–Prausnitz equation of state: II. Vapour–liquid equilibria and liquid–liquid equilibria in polyolefin systems

Phase equilibria in binary and ternary polyolefin systems are calculated using the cubic equation of state proposed by Sako–Wu–Prausnitz (SWP). Calculations were done for high-pressure phase equilibria in ethylene/polyethylene (LDPE) systems and for liquid–liquid equilibria (LLE) in systems containing either high-density polyethylene or poly(ethylene-co-propylene). The calculations for the copolymer/solvent systems are compared with those using the SAFT EOS. The two equations of state can describe UCST, LCST as well as U-LCST behaviour with similar accuracy. Whereas, the binary interaction parameter is temperature-independent for SAFT, it is found to be a function of temperature for the SWP model. Moreover, the influence of an inert gas on the LCST of the polyethylene/hexane system is investigated using the SWP EOS. The polydispersity of the different polyethylenes is considered in the phase equilibrium calculations using pseudocomponents chosen by the moments of the experimental molecular weight distributions.     

Molecular simulation of the phase behaviour of ternary fluid mixtures: the effect of a third component on vapour–liquid and liquid–liquid coexistence

The Gibbs ensemble algorithm is implemented to determine the vapour–liquid and liquid–liquid phase coexistence of dilute ternary fluid mixtures interacting via a Lennard–Jones potential. Calculations are reported for mixtures with a third component characterised by different intermolecular potential energy parameters. Comparison with binary mixture data indicates that the choice of energy parameter for the third component affects the composition range of vapour–liquid substantially. The addition of a third component lowers the energy of liquid phase while slightly increasing the energy of the vapour phase.     

Vapor–liquid equilibria of systems containing acetic acid and gaseous components. Measurements and calculations by a cubic equation of state

Isothermal pressure-composition VLE data have been measured for four systems containing acetic acid and a gaseous component. The gaseous components are carbon monoxide, carbon dioxide, hydrogen and methane. The measurements were made in a static cell and the compositions of the gas and the liquid phases were measured by a gas chromatograph. A new correlation model (ACE, Association+Cubic Equation of state) was developed. It is based on a cubic equation of state and a model for the dimerization of acetic acid. It was applied to correlate the experimetnal VLE data with good results.     

Use of normal boiling point correlations for predicting critical parameters of paraffins for vapour–liquid equilibrium calculations with the SRK equation of state

Correlations between normal boiling points and critical parameters (critical temperatures and critical pressures) and between normal boiling points and acentric factors of normal and branched paraffins in the range from C₇ to C₁₀₀ have been developed. These correlations can be used to quickly and easily compute critical properties that allow to reproduce closely, by means of the Soave–Redlich–Kwong (SRK) equation of state (EoS), the vapour pressure of pure compounds. In the range of 0.5–5 mmHg the values of vapour pressure calculated by means of the SRK EoS become less accurate and they can be improved using a different equation for the temperature-depending attractive parameter (i.e., the Mathias–Copeman alpha function) instead of the classical Soave function.     

Solid–liquid equilibria for binary and ternary systems with the Cubic-Plus-Association (CPA) equation of state

A systematic investigation of the CPA model’s performance within solid–liquid equilibria (SLE) in binary mixtures (methane + ethane, methane + heptane, methane + benzene, methane + CO₂, ethane + heptane, ethane + CO₂, 1-propanol + 1,4-dioxane, ethanol + water, 2-propanol + water) is presented. The results from the binary mixtures are used to predict SLE behaviour in ternary mixtures (methane + ethane + heptane, methane + ethane + CO₂). Our results are compared with experimental data found in the literature.     

Low density observations of Rb and Cs chains along the liquid–vapour coexistence curves to the critical point in relation to quantum-chemical predictions on the metal-insulator transitions in Li and Na rings

Recent ab initio predictions concerning the metal-insulator (MI) transition in rings of the light alkali atoms, Li and Na, are compared and contrasted with experimental facts concerning diluted Rb and Cs alkalis. The main focus here is on the local coordination number as a function of density as these two heavy alkali metallic fluids are taken along the liquid–vapour coexistence curve towards the critical point, which in these cases coincides with the MI transition. Also recorded are the results of experiments in which Cs chains are observed at large interatomic spacing outside semiconducting substrates of InSb and GaAs.     

A saturated liquid density equation in conjunction with the Predictive-Soave–Redlich–Kwong equation of state for pure refrigerants and LNG multicomponent systems

An equation and a set of mixing rules for the prediction of liquid density of pure refrigerants and liquified natural gas (LNG) multicomponent systems have been developed. This equation uses the parameters of Mathias and Copeman [P.M. Mathias, T.W. Copeman, Fluid Phase Equilib. 13 (1983) 91–108] temperature dependent-term for the Predictive-Soave–Redlich–Kwong [T. Holderbaum, J. Gmehling, Fluid Phase Equilib. 70 (1991) 251–265] equation of state and hence it could be used together with this equation. The equation uses a characteristic parameter for each refrigerant; however, if it is not available, a value of zero is recommended. This model gives an average of absolute errors less than 0.42% for the prediction of liquid density of 28 pure refrigerants consisting of 2489 data points and 0.33% for 18 multicomponent LNG systems involving 132 data points. The model parameters were determined from pure component properties and reported. These parameters were then used without any adjustment to predict liquid density of multicomponent LNG mixtures and excellent results were obtained. The model was also compared with other available methods.     

Isothermal vapor–liquid equilibria of the ternary system formed by ethanol,tert-amyl methyl ether and toluene

Vapor–liquid equilibrium (VLE) data are presented for the ternary system ethanol–tert-amyl methyl ether (TAME)–toluene at 333.15 K. The experimental results were measured by using a Boublik vapor–liquid recirculation still. The results are compared with values predicted from the PRSV equation of state with the modified Huron–Vidal first order (MHV1) and Wong–Sandler (WS) mixing rules. Good agreement is obtained.     

Determination of non-steroidal anti-inflammatory drugs in urine by the combination of stir membrane liquid–liquid–liquid microextraction and liquid chromatography

A stir membrane liquid phase microextraction procedure working under the three-phase mode is proposed for the first time for the determination of six anti-inflammatory drugs in human urine. The target compounds are isolated and preconcentrated using a special device that integrates the extractant and the stirring element. An alkaline aqueous solution is used as extractant phase while 1-octanol is selected as supported liquid membrane solvent. After the extraction, all the analytes are determined by liquid chromatography (LC) with ultraviolet detection (UV). The analytical method is optimized considering the main involved variables (e.g., pH of donor and acceptor phases, extraction time, stirring rate) and the results indicate that the determination of anti-inflammatory drugs at therapeutic and toxic levels is completely feasible. The limits of detection are in the range from 12.6 (indomethacin) to 30.7 μg/L (naproxen). The repeatability of the method, expressed as relative standard deviation (RSD, n = 5) varies between 3.4% (flurbiprofen) and 5.7% (ketoprofen), while the enrichment factors are in the range from 35.0 (naproxen) to 72.5 (indomethacin).     

Application of the Wilson—Wegner expansion to represent vapor—liquid equilibrium surfaces of binary and ternary systems from the critical locus to the vapor pressure of the heavier component

The Torquato—Stell modification of the Wilson—Wegner expansion has been used to present and to represent the vapor—liquid equilibrium surfaces of several binary methane—alkane mixtures and one ternary system from the critical locus to the heavier component's vapor pressure within the experimental accuracy of the data. Inconsistent phase compositions are revealed without recourse to equations of state computations. Since the critical point of the mixtures are not precisely known, an isothermal regression analysis of the data is first used to obtain an improved estimate of the critical point of mixtures using the scaled expressions. Then, along isotherms, the phase compositions are analyzed in terms of the reduced proximity to the critical pressure. Subsequently, the entire compositional surfaces are presented in terms of both the reduced proximity to the critical pressure of the mixture and the reduced proximity to the critical temperature of the heavier component.     

A comparison of mixing rules for the combination of COSMO-RS and the Peng–Robinson equation of state

In this work, the COSMO-RS model is combined with a volume-translated Peng–Robinson equation of state (EOS) via a G^E$G^{\text{E}}$-based mixing rule. The performance of several mixing rules previously published for this purpose is compared and semi-empirical modifications to one of them are introduced to improve its performance in our application. The new mixing rule contains three internal parameters that are adjusted to achieve consistency between the mixing rule and COSMO-RS. No experimental binary data is needed for our EOS. The new COSMO-RS-based, predictive EOS introduces a density dependence into COSMO-RS and extends its applicability to higher pressures and to mixtures containing supercritical components.