Corresponding-states correlation for the saturated liquid density of metals and metal mixtures

A corresponding-states correlation for the prediction of the orthobaric liquid density of molten metals has been developed. The correlation is the extension of the recently developed correlation by Iglesias-Silva and Hall, which needs the values of the critical and triple point constants as well as an adjustable parameter. The critical constants are scarce for almost all metals. Our corresponding-states correlation uses the normal boiling and melting point constants plus an adjustable parameter. While the present correlation is simpler in form than the correlation by Iglesias-Silva and Hall, all of its input data are more readily available for almost all metals. In this work, we have applied the present correlation to molten alkali metals, mercury, bismuth, tin, and lead. From about 150 data points for pure liquid metals the average absolute deviation and the maximum deviation are 0.29 and 1.06%, respectively. Also, we have extended the correlation to mixtures of any number of components. The predicted results for the liquid densities of K–Cs and K–Na mixtures over the whole range of concentrations and that of a ternary molten eutectic of K–Na–Cs at temperatures ranging from melting point up to several hundred degrees above the normal boiling point are in excellent agreement with experimental data. From 247 data points examined for molten alloys, the average absolute deviation and the maximum deviation are 0.59 and 1.91%, respectively.     

Corresponding states theory and thermodynamic properties of liquid alkali metals

According to phenomenological scaling and the law of corresponding states, reduced coordinates F ^*-T ^*, where F* represents the reduced thermodynamic properties (enthalpy of vaporization, speed of sound, surface tension, saturated liquid density) and T ^* is the reduced temperature, are introduced for the prediction of the thermodynamic properties of alkali metals. Values of the thermodynamic properties from the melting point up to boiling point are correlated. It has been shown that the correlation between reduced thermodynamic properties, as well as with the reduced temperature, can be expressed as a unique straight-line plot with a linear correlation coefficient of 0.9998. The proposed correlation has a simple form for easy calculation, requires only the melting and boiling point parameters, which are usually easy to acquire, and can predict the thermodynamic properties from the melting temperature up to the boiling temperature accurately.     

Fluid-solid transition in hard hypersphere systems

In this work we present a numerical study, based on molecular dynamics simulations, to estimate the freezing point of hard spheres and hypersphere systems in dimension D = 4, 5, 6, and 7. We have studied the changes of the radial distribution function (RDF) as a function of density in the coexistence region. We started our simulations from crystalline states with densities above the melting point, and moved down to densities in the liquid state below the freezing point. For all the examined dimensions (including D = 3), it was observed that the height of the first minimum of the RDF changes in an almost continuous way around the freezing density and resembles a second order phase transition. With these results we propose a numerical method to estimate the freezing point as a function of the dimension D using numerical fits and semiempirical approaches. We find that the estimated values of the freezing point are very close to the previously reported values from simulations and theoretical approaches up to D = 6, reinforcing the validity of the proposed method. This was also applied to numerical simulations for D = 7 giving new estimations of the freezing point for this dimensionality.     

The vaporization properties of krypton and xenon

The vaporization properties of krypton and xenon are reassessed from the experimental results published in the literature, more sparsely since the 1970s. The measured vapour pressure and the saturated liquid density of both substances are adjusted to empirical, reliable equations. Results are presented for the calculated vapour pressures and the orthobaric densities of the two substances. The coordinates of the triple point, the normal boiling temperature, and the critical point are discussed as well as the corresponding values of the enthalpies of vaporization. Argon, for which recent experimental, accurate work has been published, was kept out of the present review.     

A perturbed Yukawa chain equation of state for refractory liquid metals

The perturbed Yukawa chain equation of state (EoS) has been employed to calculate the liquid density of refractory metals over a wide range of temperatures and pressures. The model uses three independent parameters: m-segment number, σ-segment size, and ε/k-segment energy. For pure components, parameters have been obtained by fitting the models to experimental data on liquid densities. Our calculations on the liquid density of tantalum, rhenium, molybdenum, titanium, zirconium, hafnium and niobium from undercooled temperatures up to several hundred degrees above the boiling point and pressures ranging from 0 to 200?MPa reproduces very accurately the experimental pVT data.     

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%.     

Temperature and density dependence of the structure of liquid lithium from an improved pseudopotential theory

Leribaux, H.R., and Lemarchand, J.L., 1978. Temperature and density dependence of the structure of liquid lithium from an improved pseudopotential theory. Fluid Phase Equilibria, 2: 79–90.The structure of liquid lithium and alloys has recently been measured more accurately near their melting point. We present for liquid lithium a modification of a previous pseudopotential for liquid metals, obtained by including explicitly the Xα exchange energy in the electron pseudopotential. The structure factor S(q), obtained from this model potential via the effective pair potential, and calculated by our solution of the Percus—Yevick equation, is found to have the right oscillations compared with the experimental structure factor, the first peak also having the right magnitude. A Percus—Yevick hard-sphere structure factor is also obtained via thermodynamic perturbation and is found to be surprisingly good even for lithium, except for the high-q oscillations. We present computed structures and radial distributions for several higher temperatures involved typically in the liquid lithium alloys, up to the highest temperatures at which saturated density is measured. Our model potential predicts also the correct cohesive energy for liquid lithium and liquid lithium—magnesium.     

Prediction of vapor pressures and enthalpies of vaporization of organic compounds from the normal boiling point temperature

Recently, our Laboratory proposed a model for the prediction of vapor pressures of organic compounds that requires only the knowledge of the normal boiling point of the compound involved, and a compound specific K_f for which generalized expressions for several classes of organic compounds as functions of the normal boiling point and the molecular weight were developed.In this work our model is compared with the one proposed in Lyman's book, which is similar to our model but uses different K_f values. The results indicate that our model provides very satisfactory results in the temperature range from the melting up to the normal boiling point and up to the critical, where no hydrogen-bonding is involved. Also, it is proven that the accuracy of our model is much better than that proposed by Lyman, especially for the high molecular weight compounds.Finally, our model is used for the prediction of enthalpies of vaporization at the normal boiling point. Excellent results are obtained that are comparable or better than those obtained with two recommended models in “The Properties of Gases and Liquids” book, where the latter, however, require as input information except from the normal boiling point the critical properties of the compound involved as well.     

Solubility of carbon dioxide,methane, and ethane in 1-butanol and saturated liquid densities and viscosities

A designed pressure–volume–temperature (PVT) apparatus has been used to measure the (vapor + liquid) equilibrium properties of three binary mixtures (methane +, ethane +, and carbon dioxide + 1-butanol) at two temperatures (303 and 323) K and at the pressures up to 6 MPa. The solubility of the compressed gases in 1-butanol and the saturated liquid densities and viscosities were measured. In addition, the density and viscosity of pure 1-butanol were measured at two temperatures (303 and 323) K and at the pressures up to 10 MPa. The experimental results show that the solubility of the gases in 1-butanol increases with pressure and decreases with temperature. The dissolution of gases in 1-butanol causes a decline in the viscosity of liquid phase. The saturated liquid density follows a decreasing trend with the solubility of methane and ethane. However, the dissolution of carbon dioxide in 1-butanol leads to an increase in the density of liquid phase. The experimental data are well correlated with Soave–Redlich–Kwong (SRK) and Peng–Robinson (PR) equations of state (EOSs). SRK EOS was slightly superior for correlating the saturated liquid densities.     

Statistical mechanical theory of liquid entropy

The multiparticle correlation expansion for the entropy of a classical monatomic liquid is presented. This entropy expresses the physical picture in which there is no free particle motion, but, rather, each atom moves within a cage formed by its neighbors. The liquid expansion, including only pair correlations, gives an excellent account of the experimental entropy of most liquid metals, of liquid argon, and of the hard-sphere liquid. The pair correlation entropy is well approximated by a universal function of temperature. Higher-order correlation entropy, due to n-particle irreducible correlations for n ≥ 3, is significant in only a few liquid metals, and its occurrence suggests the presence of n-body forces. When the liquid theory is applied to the study of melting, we discover the important classification of normal and anomalous melting, according to whether there is not or is a significant change in the electronic structure upon melting, and we discover the universal disordering entropy for melting of a monatomic crystal. Interesting directions for future research are extension to include orientational correlations of molecules, theoretical calculation of the entropy of water, application to the entropy of the amorphous state, and correlational entropy of compressed argon. We clarify the relation among different entropy expansions in the recent literature. © 1994 John Wiley & Sons, Inc.     

Description of cyclopropane with Backone equation of state

This paper aims to accurately describe the thermodynamic properties of Cyclopropane with a molecular based BACKONE equation of state. The parameters of the BACKONE equation of state found by fitting to experimental vapor pressures and liquid densities are the characteristic temperature T ₀, characteristic density ρ₀, anisotropy factor α, and reduced quadrupolar moment Q*². The values of these parameters are 393.9583 K, 6.076139 mol/L, 1.295445, and 0.699483, respectively. The average absolute deviation between experimental values and those derived from BACKONE EOS is 0.29% for vapor pressures, 0.75% for saturated liquid densities. The prediction power of the BACKONE equation of state are investigated. It is shown that the uncertainties of values derived from the BACKONE equation of state are within 0.90% for isobaric densities in the liquid phase and 2.0% for enthalpy of evaporation.     

Estimation of pure component properties. Part 4: Estimation of the saturated liquid viscosity of non-electrolyte organic compounds via group contributions and group interactions

A new group contribution method for the prediction of pure component saturated liquid viscosity has been developed. The method is an extension of the pure component property estimation techniques that we have developed for normal boiling points, critical property data, and vapour pressures. Predictions can be made from simply having knowledge of the molecular structure of the compound. In addition, the structural group definitions for the method are identical to those proposed for estimation of saturated vapour pressures. Structural groups were defined in a standardized form and fragmentation of the molecular structures was performed by an automatic procedure to eliminate any arbitrary assumptions. The new method is based on liquid viscosity data for more than 1600 components. Results of the new method are compared to several other estimation methods published in literature and are found to be significantly better. A relative mean deviation in viscosity of 15.3% was observed for 813 components (12,139 data points). By comparison, the Van Velzen method, the best literature method in our benchmarking exercise produced a relative mean deviation of 92.8% for 670 components (11,115 data points). Estimation results at the normal boiling temperature were also tested against an empirical rule for more than 4000 components. The range of the method is usually from the triple or melting point to a reduced temperature of 0.75–0.8. Larger than average deviations were observed in the case of molecules with higher rotational symmetry, but no specific correction of this effect was included in this method.     

Upward extrapolation of saturated liquid densities

The design of new energy conversion processes requires equations of state for the working fluids. For their construction saturated liquid densities are needed which are not available for some potential working fluids at higher temperatures. Hence, we investigate how Rackett-type equations behave in extrapolations from saturated liquid densities in the temperature range 0.5 ≤ T/T_c ≤ 0.75 up to the critical temperature T_c. Extrapolation methods with different inputs from the critical point are used: (a) no critical point data, (b) critical temperature, (c) critical temperature and pressure, and (d) critical temperature and compression factor. It is found that upward extrapolations of the saturated liquid densities without using critical point data can be done with some care and that the additional use of the critical temperature improves the quality of the predictions substantially.     

New pressure-density-temperature measurements and new rational equations for the saturated liquid and vapour densities of oxygen

The results of pressure, density, temperature (p, ?, T) measurements in the temperature range from 65 K to 300 K, for pressures up to 7.2 MPa, and for densities from 0.3 mol dm^?3 to 39 mol dm^?3, are presented for pure oxygen. Using the experimental results, new values for the densities of saturated liquid and vapour are evaluated. To check the accuracy of these results, corresponding sets reported in the literature are critically analysed to determine the most reliable p, ?, T set for oxygen. Finally, new equations for the densities of saturated liquid and vapour are developed using a statistical procedure.     

Experimental densities,vapor pressures,and critical point,and a fundamental equation of state for dimethyl ether

Densities, vapor pressures, and the critical point were measured for dimethyl ether, thus, filling several gaps in the thermodynamic data for this compound. Densities were measured with a computer-controlled high temperature, high-pressure vibrating-tube densimeter system in the sub- and supercritical states. The densities were measured at temperatures from 273 to 523 K and pressures up to 40 MPa (417 data points), for which densities between 62 and 745 kg/m³ were covered. The uncertainty (where the uncertainties can be considered as estimates of a combined expanded uncertainty with a coverage factor of 2) in density measurement was estimated to be no greater than 0.1% in the liquid and compressed supercritical states. Near the critical temperature and pressure, the uncertainty increases to 1%. Using a variable volume apparatus with a sapphire tube, vapor pressures and critical data were determined. Vapor pressures were measured between 264 and 194 kPa up to near the critical point with an uncertainty of 0.1 kPa. The critical point was determined visually with an uncertainty of 1% for the critical volume, 0.1 K for the critical temperature, and 5 kPa for the critical pressure. The new vapor pressures and compressed liquid densities were correlated with the simple TRIDEN model. The new data along with the available literature data were used to develop a first fundamental Helmholtz energy equation of state for dimethyl ether, valid from 131.65 to 525 K and for pressures up to 40 MPa. The uncertainty in the equation of state for density ranges from 0.1% in the liquid to 1% near the critical point. The uncertainty in calculated heat capacities is 2%, and the uncertainty in vapor pressure is 0.25% at temperatures above 200 K. Although the equation presented here is an interim equation, it represents the best currently available.     

Evaluation of an improved volume translation for the prediction of hydrocarbon volumetric properties

In the present study the volume translation expression of Ungerer and Batut that is a function of the temperature and molecular weight was evaluated. Instead of being calibrated on the saturated liquid densities, this correlation uses high pressure liquid densities. As a result, it is less sensitive to the inconsistencies shown by other correlations, due to the proximity of the critical region. The mentioned expression is compared to experimental data and with correlations presented by Jhaveri and Youngren, Soreide and Magoulas and Tassios, providing a good prediction of molar volumes at high temperature and pressure and reasonable saturated liquid densities.     

Evaluation of co-volume mixing rules for bitumen liquid density and bubble pressure estimation

The Peng–Robinson cubic equation of state (CEOS) is widely used to predict thermodynamic properties of pure fluids and mixtures. The usual implementation of this CEOS requires critical properties of each pure component and combining rules for mixtures. Determining critical properties for components of heavy asymmetric mixtures such as bitumen is a challenge due to thermolysis at elevated temperatures. Group contribution (GC) methods were applied for the determination of critical properties of molecular representations developed by Sheremata for Athabasca vacuum tower bottoms (VTB). In contrast to other GC methods evaluated, the Marrero–Gani GC method yielded estimated critical properties with realistic, non-negative values, followed more consistent trends with molar mass and yielded normal boiling points consistent with high temperature simulated distillation data. Application of classical mixing rules to a heavy asymmetric mixture such as bitumen yields saturated liquid density and bubble pressure estimates in qualitative agreement with experimental data. However the errors are too large for engineering calculations. In this work, new composite mixing rules for computing co-volumes of asymmetric mixtures are developed and evaluated. For example, composite mixing rules give improved bubble point predictions for the binary mixture ethane + n-tetratetracontane. For VTB and VTB + decane mixtures the new composite mixing rules showed encouraging results in predicting bubble point pressures and liquid phase densities.     

The embedded atom model of molten lead

The method for calculating the embedded atom potential for liquid metals from the diffraction structural data close to the melting point was applied to lead at temperatures from 613 to 20000 K. The embedded atom potential parameters were adjusted using the data on the lead structure at 613–1173 K, the thermodynamic properties of lead over the temperature range 613–2000 K, and the results of shock wave experiments. The embedded atom potential and molecular dynamics method allowed the structural characteristics of the liquid metal to be successfully predicted up to 1173 K. The calculated bulk compression modulus at 613 K was close to its actual value. The self-diffusion coefficients along the liquid-vapor equilibrium line increased as the temperature rose following the power law with the exponent close to 2.03. The properties of lead under extremal conditions were calculated up to the temperature 20000 K and density 20.721 g/cm³. At 1000 K and a density of 18.156 g/cm³, close agreement with the experimental pressure (101.5 GPa) was obtained. The potential found fairly well described the properties of crystalline lead. At the same time, the embedded atom potential adjusted to describe the properties of the crystalline phase only poorly described the properties of liquid lead at increased densities.     

Thermodynamic modeling of saturated liquid compositions and densities for asymmetric binary systems composed of carbon dioxide,alkanes and alkanols

The present study mainly focuses on the phase behavior modeling of asymmetric binary mixtures. Capability of different mixing rules and volume shift in the prediction of solubility and saturated liquid density has been investigated. Different binary systems of (alkane + alkanol), (alkane + alkane), (carbon dioxide + alkanol), and (carbon dioxide + alkane) are considered. The composition and the density of saturated liquid phase at equilibrium condition are the properties of interest. Considering composition and saturated liquid density of different binary systems, three main objectives are investigated. First, three different mixing rules (one-parameter, two parameters and Wong–Sandler) coupled with Peng–Robinson equation of state were used to predict the equilibrium properties. The Wong–Sandler mixing rule was utilized with the non-random two-liquid (NRTL) model. Binary interaction coefficients and NRTL model parameters were optimized using the Levenberg–Marquardt algorithm. Second, to improve the density prediction, the volume translation technique was applied. Finally, Two different approaches were considered to tune the equation of state; regression of experimental equilibrium compositions and densities separately and spontaneously. The modeling results show that there is no superior mixing rule which can predict the equilibrium properties for different systems. Two-parameter and Wong–Sandler mixing rule show promoting results compared to one-parameter mixing rule. Wong–Sandler mixing rule in spite of its improvement in the prediction of saturated liquid compositions is unable to predict the liquid densities with sufficient accuracy.     

A perturbed hard-sphere equation of state for alkali metals

A perturbed hard-sphere equation of state, employing a basic frame proposed by Eslami [H. Eslami, J. Nucl. Mater. 336 (2005) 135–139] has been developed for alkali metals. Following the approach introduced by Ihm et al. [G. Ihm, Y. Song, E.A. Mason, J. Chem. Phys. 94 (1991) 3839–3848], the temperature dependence of the parameters a and b has been fitted to liquid density data for potassium. The scaling parameters that are used to reduce the temperature are the temperature and density at normal boiling point. The important improvement is to omit the adjustable parameters, the well depth and the location of the minimum of pair potential, which are required to apply the earlier equation of state of Eslami. The present EoS, which can be used without fitting parameters, reproduces the volumetric behavior of liquid alkali metals with a very good accuracy. Six hundred and ninety four data points at different pressures and temperatures are examined and the average absolute deviation of predicted liquid density data compared to experiment is 1.41%.