An effective modification of the Benedict–Webb–Rubin equation of state

A modification of the BWR equation of state is proposed, which is a simplified form of a previously proposed one. It applies to systems formed by hydrocarbons and related compounds, with particular attention to the critical conditions. The range of treatable compounds was extended to a value 0.9 of the acentric factor, corresponding to C₂₀ hydrocarbons. The critical compressibility factor Z_c was made independent of the acentric factor, for a more accurate prediction of pure-component properties (the previous equation did not give the same improvement). Mixing rules require one binary interaction constant for each component pair. Zero binary constants can be used for methane–alkane and alkane–alkane pairs. Examples of applications to pure hydrocarbons and their mixtures are given.     

A modification to the Peng–Robinson-fitted equation of state for pure substances

In this work we present two modifications to the Peng–Robinson-Fitted equation of state where pure component parameters are regressed to vapor pressure and saturated liquid density data. The first modification (PR-f-mod) is a method that enhances the equation of state pure component property predictions through simple temperature dependent pure component parameters. In the second modification (PR-f-prop) we propose a temperature dependency for co-volume b in the repulsive parameter of the EoS, and revise the temperature function in the attractive term. The agreement with experimental data for 72 pure substances, including highly polar compounds, is remarkably good. We obtain average absolute deviations in saturated liquid density of less than 1% for all substances studied.     

Modification of the Peng–Robinson equation of state for helium in low temperature regions

Helium shows the nearest behaviour to ideal gas in the room conditions. In contrast, thermodynamic behaviour of helium in the critical region, in which its liquefaction is possible, is extremely complicated. The equation of state (EOS), which is in common use for helium, is the modified Benedict–Webb–Rubin (MBWR) EOS developed by McCarty and Arp which is a 13th-order equation with 32 substance-dependent parameters. MBWR is a complicated EOS and its use is time consuming. In this work, the modified Peng–Robinson EOS introduced by Feyzi et al. is customised with 10 adjustable parameters for helium in the temperature range of 2.20–15.20 K and pressures up to 16 bar. The proposed EOS is able to predict the properties of helium in the vapour–liquid equilibrium (VLE) conditions and in the single gas-phase region. In addition, a temperature-dependent correlation for constant pressure heat capacity of helium from very low up to normal temperatures is proposed. The liquefaction process of helium, which is being done by cooling it to very low temperatures by passing through a Joule–Thomson valve, is predicted by the proposed EOS. Very accurate results are observed.     

Density and temperature dependence of the relaxation of the 1Δg state of highly compressed O2 fluid

The electronic relaxation of O₂ is investigated by an absorption excitation and fluorescence detection technique. The relaxation rate constant of O₂(¹Δ_g) is measured in the density range from 10²¹ to 3 × 10²² cm⁻³ at temperatures between 90 and 295 K. The experimental results are compared with theoretical models based on the pair distribution functions of the fluid. The effects of intermolecular potentials with hard or soft cores are discussed.     

Simulation of double retrograde vaporization using the Peng–Robinson equation of state

Double retrograde vaporization is a phenomenon characterized by an unexpected retrograde dew point curve at compositions approaching nearly the pure volatile component (component A) and at temperatures very close to the critical temperature of the more volatile component (Tc_A). On the p–x–y diagram, instead of the single-domed dew point curve in the familiar “single” retrograde vaporization, double retrograde vaporization shows two “domes” at temperatures above but close to Tc_A. At temperatures below but close to Tc_A, the dew point curve has an “S”-shape. This results respectively in quadruple- or triple-valued dew points at a specific composition. In this work, the phenomenon of double retrograde vaporization has been simulated using a cubic equation of state. Both the “double-dome” and the “S”-shape curves for the binary systems (ethane+linalool) and (ethane+d-limonene) were successfully modelled, even without the use of binary interaction parameters. Results are also obtained by optimizing interaction parameters using experimental bubble point data. Even though double retrograde vaporization has rarely been observed in literature, we believe that it is the normal behaviour that always occurs in binary mixtures in which the two components differ largely enough in molecular symmetry to produce a very steep dew point curve. To further verify this generality, simulations were performed on a number of binary mixtures of different families. Double retrograde vaporization was estimated in every system with a steep dew point curve.     

A theoretical form of the Martin-Hou equation of state

A new equation of state is derived from the Barker-Henderson hard-sphere perturbation theory. It has the form similar to the Martin-Hou equation of state. The numerical values of the characteristic constants in the equation can be calculated by the method of Martin and Hou. The equation can be used to predict P-V-T properties accurately for fluids when the critical parameters (T_c, P_c and V_c) and one point on the vapor pressure cure are given. By using the functional relationships between the characteristic constants and the microscopic parameters, the molecular microscopic parameters of the substance can be obtained.     

Development of a new form for the alpha function of the Redlich–Kwong cubic equation of state

The dependence on temperature and acentric factor of the attractive term of the Redlich–Kwong equation of state has been modified. A new alpha function is expressed in a generalized form. The new equation allows a good representation of vapor pressure data of a great variety of compounds, as well as thermodynamic properties such as the enthalpy of vaporization and the entropy of vaporization.     

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.     

Twinkling fractal theory of the glass transition: Rate dependence and time–temperature superposition

The twinkling fractal theory (TFT) of the glass transition temperature T_g provides a new method of analyzing rate effects and time–temperature superposition in amorphous materials. The rate dependence of T_g was examined in the light of new experimental and theoretical evidence for the nature of the dynamic heterogeneity near T_g. As T_g is approached from above, dynamic solid fractal clusters begin to form and eventually percolate rigidity at T_g. The percolation cluster is a solid fractal and to the observer, appears to “twinkle” as solid and liquid clusters interchange in dynamic equilibrium with a vibrational density of states g(ω) ∼ ω. The solid-to-liquid twinkling frequencies ω_TF are controlled by the Boltzmann population of intermolecular oscillators in excited energy levels of their anharmonic potential energy functions U(x) such that ω_TF = ω exp −B(T*² − T²)/kT in which T* ≈ 1.2T_g. An oscillator changes from a solid to a liquid when a thermal fluctuation causes it to expand beyond its inflection point in the anharmonic potential. This leads to a continuous solid fraction P_s near T_g given by P_S ≈ 1−[(1 − p_c) T/T_g] where p_c ≈ 1/2 is the rigidity percolation threshold. Since g(ω) is continuous from very low to very high frequencies, the complex twinkling dynamics existing near T_g produces a continuous relaxation spectrum with many different length scales and times associated with the fractal clusters. The twinkling frequencies control the kinetics of T_g such that for a given observation time t when the rate γ > 1/t, only those parts of the twinkling spectrum with ω > γ can contribute to relaxation or percolation upto time t. The most important results in this article are as follows: The TFT describes the rate dependence of T_g, both for DSC thermal heating/cooling rates and DMA frequencies as the classic T_g − lnγ law as T_g(γ) = T_go + (k/2B) ln γ/γ_o in which the constant B = 0.3 cal/mol K². The constant B appears quite universal for the 17 thermoset polymers investigated in this study and 18 linear polymers investigated by others. Many other amorphous metal and ceramic glass materials exhibited the same rate law but required a new B value approximately half that for polymers. The same B = 0.3 value was also used to successfully describe the TTS shift factors using the twinkling fractal frequencies ω_TF = ωexp −B(T*² − T²)/kT, as ln a_T(TFT) = exp B(T_R² − T²)/kT, which gave comparable results with the classical WLF equation, log a_T = [−C₁(T − T_R)]/[C₂ + (T − T_R)]. The advantage of the TFT over the WLF is that C₁ and C₂ are not universal constants and must be determined for every material, whereas the TFT uses one known constant B which appears to be the same for all polymers. The TFT has also been found to describe the strong and fragile nature of the viscosity behavior of liquids and the rate and temperature dependence of the yield stress in polymers. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 2578–2590, 2009     

Prediction of thermodynamic behavior of binary mixtures by applying the crossover approach to Soave–Redlich–Kwong equation of state

Thermodynamic analysis of binary mixtures near the critical region is essential for many chemical process designs. In this research, based on isomorphism principle and incorporating general crossover approach the original Soave–Redlich–Kwong (SRK) equation of state (EOS) was developed for the binary mixtures. We have introduced an additional term in the crossover function in order to take into account the difference between the classical critical parameters and the real critical parameters. The applicability of this crossover EOS was verified against methane–ethane mixture to describe their thermodynamic properties over a wide range of thermodynamic states, because of their wide applications. It is shown that based on this approach we can received too much more accuracy for predicting thermodynamic properties in comparison with classical form of SRK EOS.     

Estimation of the normal boiling points of water and 1,2-dimethoxyethane through the viscosity–temperature dependence in the corresponding binary mixtures

Excess quantities calculated from literature values of experimental density and viscosity in 1,2-dimethoxyethane + water binary systems (from 303.15 to 323.15 K) can lead us to test different correlation equations as well as their corresponding derivative properties. Inspection of the Arrhenius activation energy (Ea) and the enthalpy of activation of viscous flow (ΔH*) shows very close values; here, we can define partial molar activation energies Ea₁ and Ea₂ for 1,2-dimethoxyethane and water, respectively, along with their individual contribution separately. Correlation between the two Arrhenius parameters of viscosity in all compositions shows existence of main distinct interaction behaviours delimited by particular mole fractions in 1,2-dimethoxyethane. Moreover, we add that correlation between Arrhenius parameters reveals interesting Arrhenius temperature which is closely related to the vapourisation temperature in the liquid vapour equilibrium, and the limiting corresponding partial molar properties can permit us to estimate the boiling points of the pure components.     

Liquid–liquid equilibria of copolymer mixtures based on an equation of state

Liquid–liquid equilibria of copolymer mixtures were studied by an equation of state (EoS) for chain-like fluids. The equation consists of a reference term for hetero-nuclear hard-sphere chain fluids developed by Hu et al. where the next-to-nearest-neighbor correlations have been taken into account; and a perturbation term from Alder et al.’s square-well attractive potential. The segment parameters, including number of segments, segment diameter and interaction energy between segments, are obtained by fitting pVT data of pure homopolymer. For the case of different species in the same copolymer, the interaction parameters for unlike segment pairs are obtained by fitting pVT data of pure copolymer. For the interaction between segment of homopolymer and different species in copolymer, the parameters are treated as adjustable by fitting liquid–liquid equilibria data. In the latter case, the difference between different species in a copolymer is simply neglected as an approximation. Therefore, in general, only one pair of adjustable interaction parameter is determined from LLE data. To model miscibility maps of copolymer mixtures having two or three kinds of species, the interaction parameters are obtained from the boundary between miscible and immiscible regions. The EoS used in this work can correlate phase behavior including coexistence curves, miscibility windows and miscibility maps.     

Application of the perturbed Lennard–Jones chain equation of state to solute solubility in supercritical carbon dioxide

The perturbed Lennard–Jones chain (PLJC) equation of state is a thermodynamic model based on the perturbation theory of liquid state. This equation has been shown in the past to be a successful model for phase equilibria calculations of binary and ternary fluid mixtures and polymer solutions. In this work, we employed for the first time the PLJC equation to model the solubility of 39 solids in supercritical carbon dioxide. It was shown that the model achieves good correlation with three temperature independent parameters. A comparison of the PLJC with the commonly used Peng–Robinson equation reveals the PLJC equation gives better correlation to the solubility data than the Peng–Robinson model that utilizes temperature dependent parameters.     

Vapor pressures of pure compounds using the Peng–Robinson equation of state with three different attractive terms

Accurate representation of pure compounds vapor pressures is required to increase the robustness of equations of state when predicting phase equilibria for mixtures. Using cubic equations of state, this representation largely depends on improving the temperature-dependent attractive term of the equation of state (EOS) to cover data from the triple to critical points. With this purpose, the Peng–Robinson equation of state is applied with three different attractive terms: Mathias, Mathias–Copeman, and Carrier–Rogalski–Péneloux. Experimental vapor pressures for 311 pure compounds (9000 experimental values) have been fitted. The studied compounds include nitrogen compounds, oxides, sulfides, chlorides, oxyhalides, inorganic compounds, alkanes, cycloalkanes, alkenes, alkadienes, alkynes, aromatic hydrocarbons, halogenated alkanes, halogenated cycloalkanes, halogenated alkenes, halogenated aromatic hydrocarbons, alcohols, ethers, aldehydes, ketones, alkanoic acids, esters, phenols, heterocyclic oxygen compounds, heterocyclic nitrogen compounds, hydrocarbon nitrogen compounds, and sulfur compounds. Overall average absolute deviations of 0.416, 0.214 and 0.276% have been found for the resulting Peng–Robinson–Mathias (PRM), Peng–Robinson–Mathias–Copeman (PRMC), and Peng–Robinson–Carrier–Rogalski–Péneloux (PRCRP) equations of state, respectively.     

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.     

Phase equilibria study of Lennard–Jones mixtures by an analytical equation of state

The recently developed equation of state (EOS) for Lennard–Jones mixtures [Y. Tang, B.C.-Y. Lu, Fluid Phase Equilibria 146 (1998) 73.] is further investigated in this work for describing phase equilibria of these mixtures. The investigation covers vapor–liquid equilibria (VLE), liquid–liquid equilibria (LLE), vapor–liquid–liquid equilibria (VLLE) and vapor–vapor equilibria (VVE) over a wide range of temperatures, pressures and molecular characteristic parameters. Results from the van der Waals one-fluid (VDW1) theory are included for comparison. The newly proposed theory performs very well for VLE and LLE and the performance is better than the VDW1 theory; but both theories yield only qualitative results for VVE. It is also found that one system should exhibit VLLE, which was not noticed in previous investigations. Results from two other perturbation theories are also compared in some cases.     

Prediction of thermodynamic properties of some polymeric systems using an extended LJ potential-based equation of state up to high temperature–high pressure conditions

In this work, the extended Lennard-Jones potential-based equation of state (ELJ-based EoS) on which the effective near-neighbour pair interactions are LJ (12,6,3) type has been used to predict the specific volume and other thermodynamic properties of some semi-crystalline and liquid polymers and copolymers up to extremely high temperature–high pressure conditions. It seems that, at least in the dense regions, there are no upper- and lower-specific volume limitations in the applicability of the model for different polymeric systems. The parameters can be determined at any temperature for each compound using the temperature dependence of the parameters of ELJ-based EoS. The calculated parameters have been used to calculate the specific volume and other derived thermodynamic properties of different polymeric systems at any temperature and pressure. The ELJ-based EoS has been also compared with some previous studies.