Analytical Equation Of State Research Articles

Note: Equation of state and the freezing point in the hard-sphere model

The merits of different analytical equations of state for the hard-sphere system with respect to the recently computed high-accuracy value of the freezing-point packing fraction are assessed. It is found that the Carnahan–Starling–Kolafa and the branch-point approximant equations of state yield the best performance.

Some electrically charged relativistic stellar models in general relativity

Some new families of electrically charged stellar models of ultra-compact star have been studied. With the help of particular form of one of the metric potentials the Einstein–Maxwell field equations in general relativity have been transformed to a system of ordinary differential equations. The interior matter pressure, energy–density, and the adiabatic sound speed are expressed in terms of simple algebraic functions. The constant parameters involved in the solution have been set so that certain physical criteria satisfied. Based on the analytic model developed in the present work, the values of the relevant physical quantities have been calculated by assuming the estimated masses and radii of some well known potential strange star candidates like X-ray pulsar Her X-1, millisecond X-ray pulsar SAX J 1808.4-3658, and 4U 1820-30. The analytical equations of state of the charged matter distribution may play a significant role in the study of the internal structure of highly compact charged stellar objects in general relativity.

The isotropic-nematic and nematic-nematic phase transition of binary mixtures of tangent hard-sphere chain fluids: an analytical equation of state.

An analytical equation of state (EoS) is derived to describe the isotropic (I) and nematic (N) phase of linear- and partially flexible tangent hard-sphere chain fluids and their mixtures. The EoS is based on an extension of Onsager's second virial theory that was developed in our previous work [T. van Westen, B. Oyarzún, T. J. H. Vlugt, and J. Gross, J. Chem. Phys. 139, 034505 (2013)]. Higher virial coefficients are calculated using a Vega-Lago rescaling procedure, which is hereby generalized to mixtures. The EoS is used to study (1) the effect of length bidispersity on the I-N and N-N phase behavior of binary linear tangent hard-sphere chain fluid mixtures, (2) the effect of partial molecular flexibility on the binary phase diagram, and (3) the solubility of hard-sphere solutes in I- and N tangent hard-sphere chain fluids. By changing the length bidispersity, two types of phase diagrams were found. The first type is characterized by an I-N region at low pressure and a N-N demixed region at higher pressure that starts from an I-N-N triphase equilibrium. The second type does not show the I-N-N equilibrium. Instead, the N-N region starts from a lower critical point at a pressure above the I-N region. The results for the I-N region are in excellent agreement with the results from molecular simulations. It is shown that the N-N demixing is driven both by orientational and configurational/excluded volume entropy. By making the chains partially flexible, it is shown that the driving force resulting from the configurational entropy is reduced (due to a less anisotropic pair-excluded volume), resulting in a shift of the N-N demixed region to higher pressure. Compared to linear chains, no topological differences in the phase diagram were found. We show that the solubility of hard-sphere solutes decreases across the I-N phase transition. Furthermore, it is shown that by using a liquid crystal mixture as the solvent, the solubility difference can by maximized by tuning the composition. Theoretical results for the Henry's law constant of the hard-sphere solute are in good agreement with the results from molecular simulation.

Thermodynamic properties of solid face centered cubic Rb3C60 at high temperature and pressure

Analytic equation of state and thermodynamic quantities of solid fcc Rb3C60 are derived by using an analytic mean field potential method. For intermolecular forces, the double-exponential potential is utilized. Four potential parameters are determined by fitting experimental compression data of Rb3C60 up to 14 GPa at 296 K. Various physical quantities including isothermals, thermal expansion, isochoric heat capacity, Helmholtz free energy and internal energy are calculated and analyzed. Calculated results are consistent with available experimental data in literature. Furthermore, spinodal temperature for Rb3C60 is found to be 2,860 K. Results verify that analytic mean field potential method is a useful approach to consider the anharmonic effect at high temperatures. Numerous reasonable predictions and the change trend of the properties for Rb3C60 at high temperature and pressure have been given.

Analytical equation of state with three-body forces: Application to noble gases

We developed an explicit equation of state (EOS) for small non polar molecules by means of an effective two-body potential. The average effect of three-body forces was incorporated as a perturbation, which results in rescaled values for the parameters of the two-body potential. These values replace the original ones in the EOS corresponding to the two-body interaction. We applied this procedure to the heavier noble gases and used a modified Kihara function with an effective Axilrod-Teller-Muto (ATM) term to represent the two- and three-body forces. We also performed molecular dynamics simulations with two- and three-body forces. There was good agreement between predicted, simulated, and experimental thermodynamic properties of neon, argon, krypton, and xenon, up to twice the critical density and up to five times the critical temperature. In order to achieve 1% accuracy of the pressure at liquid densities, the EOS must incorporate the effect of ATM forces. The ATM factor in the rescaled two-body energy is most important at temperatures around and lower than the critical one. Nonetheless, the rescaling of two-body diameter cannot be neglected at liquid-like densities even at high temperature. This methodology can be extended straightforwardly to deal with other two- and three-body potentials. It could also be used for other nonpolar substances where a spherical two-body potential is still a reasonable coarse-grain approximation.

An equation of state for solid–liquid–vapor equilibrium applied to gas processing and natural gas liquefaction

The analytical equation of state for the representation of solid, liquid, and vapor (SLV-EoS), proposed by Yokozeki, has been applied to the representation of phase equilibrium of substances involved in gas processing and in the natural gas liquefaction process. A new procedure for the parameters regression has been set up in order to achieve a better representation of phase equilibrium. This new procedure has been applied for obtaining new parameters for the original SLV-EoS equation, with the van der Waals attractive term. Different attractive terms have also been studied, in order to achieve an optimized representation for solid–liquid, solid–vapor, and liquid–vapor equilibrium of pure substances. The obtained equations have been used for representing phase equilibrium of carbon dioxide, methane, ethane, and propane in a wide range of temperature and pressure. The new equations have also been applied to the description of phase equilibrium, involving also solid phases, of the mixtures methane–carbon dioxide, ethane–carbon dioxide, and propane–carbon dioxide.

Extension of Tao-Mason Equation of State to Heavy n-Alkanes

In our previous paper we extended the Tao and Mason equation of state (TM EOS) to refrigerant fluids, using the speed of sound data. This is a continuation for evaluating TM EOS in predicting PVT properties of heavy n-alkanes. Liquid density of long-chain n-alkane systems from C9 to C20 have been calculated using an analytical equation of state based on the statistical-mechanical perturbation theory. The second virial coefficients of these n-alkanes are scarce and there is no accurate potential energy function for their theoretical calculation. In this work the second virial coefficients are calculated using a corresponding state correlation based on surface tension and liquid density at the freezing point. The deviation of calculated densities of these alkanes is within 0.5% from experimental data. The densities of n-alkanes obtained from the TM EOS are compared with those calculated from Ihm-Song-Mason equation of state and the corresponding-states liquid densities (COSTALD). Our results are in favor of the preference of the TM EOS over other two equations of state.

Equation of state for mercury: revisited

An analytical equation of state by Song and Mason is developed to calculate the PVT properties of mercury. The equation of state is based on the statistical-mechanical perturbation theory of hard convex bodies and can be written as a fifth-order polynomial in the density. There exists three temperature-dependent parameters in the equation of state; the second virial coefficient, an effective molecular volume, and a scaling factor for the average contact pair distribution function of hard convex bodies. The temperature-dependant parameters have been calculated using corresponding-states correlations based on the surface tension and the liquid density at the normal boiling point. Employing the present equation of state, we have calculated the PVT properties of mercury over a wide range of temperatures and pressures. The average absolute deviation for the calculated density of mercury in the saturation and compressed states is 1.09%.

Correlation for the viscosity of liquid alkali metals

A modified Enskog equation is combined with an analytical equation of state to calculate the viscosity coefficient of liquid alkali metals. The temperature-dependent van der Waals covolume, appearing in the correlation and in the equation of state, is correlated with temperature using two scaling constants. The resulting correlation equation needs two adjustable constants, for the calculation of the viscosity coefficients of liquid alkali metals from the freezing point (about 350K) to several hundred degrees above the boiling point (1800K). It is shown that the present correlation well predicts the experimental liquid viscosities with an average absolute deviation of 1.7%. A comparison of the predictive ability of the present correlation with two recent theoretical methods in the literature shows that the present correlation is more accurate and covers a much wider range of temperatures and densities than those.

Grand canonical Monte Carlo simulations of vapor-liquid equilibria using a bias potential from an analytic equation of state

This article introduces an efficient technique for the calculation of vapor-liquid equilibria of fluids. Umbrella Sampling Monte Carlo simulations in the grand canonical ensemble were conducted for various types of molecules. In Umbrella Sampling, a weight function is used for allowing the simulation to reach unlikely states in the phase space. In the present case this weight function, that allows the system to overcome the energetic barrier between a vapor and liquid phase, was determined by a trivialized Density Functional Theory (DFT) using the PC-SAFT equation of state. The implementation presented here makes use of a multicanonical ensemble approach to divide the space of fluctuating particle number N into various subsystems. The a priori estimate of the weight function from the analytic DFT allows the parallelization of the calculation, which significantly reduces the computation time. In addition, it is shown that the analytic equation of state can be used to substitute sampling the dense liquid phase, where the sampling of insertion and deletion moves become demanding.

Developing a new equation of state for carbon capture and storage pipeline transport

As the development of more advanced capture and storage technologies continues along with gradual scale-up of demonstration projects, the imminent deployment of commercial-scale carbon capture and storage is becoming increasingly likely. The transport stage needs to undergo similar development as there is much that needs to be better understood. In particular, there is a need for an equation of state which can accurately describe the physical behaviour of mixtures of carbon dioxide with impurities during the transport stage of carbon capture and storage in a simple, effective way. Presented here is the early development of a new, analytic equation of state, specifically designed to be highly accurate within the window 0 to 30.9782 ℃ and up to 150 bar (the anticipated usual region of operation for carbon capture and storage pipelines). In this paper, the potential of the proposed equation of state is demonstrated with descriptions for the pressure (from which density can be directly calculated) and phase behaviour of pure CO2. Comparisons with the Peng–Robinson equation of state are made in order to demonstrate the potential of the model, which has both simplicity similar to that of the Peng–Robinson equation and accuracy similar to that of the Span–Wagner equation. Presently, it is found that inside this window, the maximum percentage error of the new equation in the description of pressure for the case of pure carbon dioxide is not more than approximately 2.53%, compared with 40.85% for the Peng–Robinson equation, whilst for the description of phase behaviour, it is not more than 2.41%, compared with 18.23% for the Peng–Robinson equation.

Experimental Measurement of Speeds of Sound in Dense Supercritical Carbon Monoxide and Development of a High-Pressure, High-Temperature Equation of State

We report the adiabatic sound speeds for supercritical fluid carbon monoxide along two isotherms, from 0.17 to 2.13 GPa at 297 K and from 0.31 to 3.2 GPa at 600 K. The carbon monoxide was confined in a resistively heated diamond-anvil cell, and the sound speed measurements were conducted in situ using a recently reported variant of the photoacoustic light scattering effect. The measured sound speeds were then used to parametrize a single site dipolar exponential-6 intermolecular potential for carbon monoxide. PρT thermodynamic states, sound speeds, and shock Hugoniots were calculated using the newly parametrized intermolecular potential and compared to previously reported experimental results. Additionally, we generated an analytical equation of state for carbon monoxide by fitting to a grid of calculated PρT states over a range of 0.1-10 GPa and 150-2000 K. A 2% mean variation was found between computed high-pressure solid-phase densities and measured data-a surprising result for a spherical interaction potential. We further computed a rotationally dependent fluid to β-solid phase boundary; results signal the relative magnitude of short-range rotational disorder under conditions that span existing phase boundary measurements.

Note on the Calculation of the Second Osmotic Virial Coefficient in Stable and Metastable Liquid States

The second osmotic virial coefficient is calculated from analytical equations of state as illustrated with the van der Waals two-component equation. It is shown that when the fixed solvent chemical potential or pressure at which the virial coefficient is calculated is taken to be that of the pure solvent in coexistence with its vapor, as in a recent report, the liquid solution is in a metastable state. When, by contrast, that fixed chemical potential or pressure is that of the pure solvent in its one-phase liquid state, the solution, with increasing solute concentration, is initially in a stable state; then, on crossing the liquid-vapor equilibrium line, it becomes metastable and ultimately approaches a spinodal and incipient instability. Nevertheless, in practice, as seen in a numerical illustration for a hydrocarbon dissolved in water, there is scarcely any difference in the virial coefficient calculated with the fixed solvent chemical potential or pressure of the pure solvent at its vapor pressure (metastable states of the solution) or at 1 bar (initially stable states). It is also seen in that example that the virial coefficient may be reliably calculated only for solute concentrations that are neither too small nor too large; typically only for mole fractions roughly from 10(-7) to 10(-3.5).

Equation of state for magnetized Coulomb plasmas

We have developed an analytical equation of state (EOS) for magnetized fully-ionized plasmas that cover a wide range of temperatures and densities, from low-density classical plasmas to relativistic, quantum plasma conditions. This EOS directly applies to calculations of structure and evolution of strongly magnetized white dwarfs and neutron stars. We review available analytical and numerical results for thermodynamic functions of the nonmagnetized and magnetized Coulomb gases, liquids, and solids. We propose a new analytical expression for the free energy of solid Coulomb mixtures. Based on recent numerical results, we have constructed analytical approximations for the thermodynamic functions of harmonic Coulomb crystals in quantizing magnetic fields. The analytical description ensures a consistent evaluation of all astrophysically important thermodynamic functions based on the first, second, and mixed derivatives of the free energy. Our numerical code for calculation of thermodynamic functions based on these approximations has been made publicly available. Using this code, we calculate and discuss the effects of electron screening and magnetic quantization on the position of the melting point in a range of densities and magnetic fields relevant to white dwarfs and outer envelopes of neutron stars. We consider also the thermal and mechanical structure of a magnetar envelope and argue that it can have a frozen surface which covers the liquid ocean above the solid crust.

Theoretical Equation of State for Highly Anharmonic Solids

Thermodynamic properties of Lennard–Jones (LJ) face-centered cubic crystals are calculated in a wide range of temperatures along sublimation and melting lines on the basis of statistical mechanical theory which accounts for anharmonicity, pair and triple correlations in the displacements of atoms. Results are compared with the most accurate Monte Carlo data available. Excellent agreement was found. Anharmonic and correlation contributions to entropy, internal energy, and the compressibility factor of LJ solids are computed and their relative importance analyzed. An analytical equation of state approximating the theoretical results is developed.

Analytic equation of state and thermodynamic properties for solid FCC C60 over a wide range of pressures and temperatures

As an analytic approximation to the free volume theory (FVT), the analytic mean field potential (AMFP) approach is applied to the multi-exponential model solid. The analytic expressions for the equation of state (EOS) and thermodynamic quantities are derived. The formalism for the case of the double-exponential (DE) model is applied to the solid fcc C60. The four potential parameters are determined by fitting the experimental data of cohesive energy, lattice constant and compression curve of solid C60. The numerical results calculated show that the Girifalco potential is too hard and gives compression curve prominently deviated from experiments at high pressure, whereas the DE potential can well describe the thermo-physical properties of C60 system within wide pressure range. The calculated variation of lattice constant versus temperature relationship and compression curve for solid C60 by using the DE potential agree well with experimental data available. Especially the AMFP with DE potential predicted that the fcc C60 solid is stable up to 2440 K, the spinodal temperature, in good agreement with the result from molecular dynamic (MD) simulation. The AMFP is a useful approach to consider the anharmonic effects at high temperature for solid C60.

Percolation line, response functions, and Voronoi polyhedra analysis in supercritical water

The problem of a physical relevance (meaning) of percolation in supercritical fluids is addressed considering a primitive model of water. Two different criteria, physical and configurational, are used for the cluster definition in Monte Carlo simulations over a range of pressures to determine the percolation line and skewness, and a theoretical analytic equation of state is used to evaluate response functions. It is found that both criteria yield practically the same percolation line. However, unlike the findings for simple fluids, the loci of the response function extrema exhibit density/pressure dependence quite different from that of the percolation line. The only potential coincidence between the loci of the extrema of a thermodynamic property and a detectable structural change is found for the coefficient of isothermal compressibility and Voronoi neighbors distribution skewness maximum.

P–V–T properties of polymer melts based on equation of state and neural network

An accurate and efficient analytical equation of state (EOS) and artificial neural network (ANN) methods are developed for the prediction of volumetric properties of polymer melts. To apply EOS, the second virial coefficients B2(T), effective van der Waals co-volume, b(T) and correction factor, α(T) were determined. The second virial coefficient was calculated from a two-parameter corresponding states correlation, which is constructed with two constants as scaling parameters, i.e., temperature (Tf) and density at melting (ρf) point. The new correlations were used to predict the specific volumes of polypropylene glycol (PPG), polyethylene glycol (PEG), polypropylene (PP), polyvinylchloride (PVC), poly(1-butene)(PB1), poly (ϵ-caprolactone) (PCL), polyethylene (PE) and polyvinylmethylether (PVME) at compressed state in the temperature range of 298.15–634.6K. The obtained results show that the two models have good agreement with the experimental data with absolute average deviation of 0.28% and 0.39% for ANN and EOS, respectively. The Comparison of the results with ISM model shows that the proposed models represent an efficient method and are more accurate.

Equation of state and jamming density for equivalent bi- and polydisperse, smooth, hard sphere systems

We study bi- and polydisperse mixtures of hard sphere fluids with extreme size ratios up to 100. Simulation results are compared with previously found analytical equations of state by looking at the compressibility factor, Z, and agreement is found with much better than 1% deviation in the fluid regime. A slightly improved empirical correction to Z is proposed. When the density is further increased, excluded volume becomes important, but there is still a close relationship between many-component mixtures and their binary, two-component equivalents (which are defined on basis of the first three moments of the size distribution). Furthermore, we determine the size ratios for which the liquid-solid transition exhibits crystalline, amorphous or mixed system structure. Near the jamming density, Z is independent of the size distribution and follows a -1 power law as function of the difference from the jamming density (Z → ∞). In this limit, Z depends only on one free parameter, the jamming density itself, as reported for several different size distributions with a wide range of widths.

Improved analytical thermodynamic properties of the triangular-well fluid from perturbation theory

In this short communication, it is shown that the triangular-well fluid analytic equation of state (EOS) recently proposed by Betancourt-Cárdenas et al. [Mol. Phys. 105 (2007) 2987] within the framework of the Barker–Henderson perturbation theory of fluids with the macroscopic compressibility approximation (BH-MCA) can be improved in two ways. First it is cast into a more compact form, which avoids the calculations of seventy-five numerical coefficients. Secondly, by using a more accurate expression of the hard-sphere radial distribution function (HS-RDF), gHS(r), the accuracy of the analytical expression of the free energy second-order term is improved, being in agreement with its BH-MCA numerical evaluation. For short-range triangular-well potentials, the present EOS also leads to second-order residual internal energies and reduced pressures in better agreement with the molecular simulations than the Betancourt-Cárdenas et al. results.