An Analytical Equation of State for Some Saturated Liquid Metals

An analytical equation of state (EoS) is developed for some saturated molten metals. The equation is that of Ihm, Song and Mason in which the three temperature–dependent parameters, second virial coefficient, van der Waals co–volume, and a scaling parameter, are calculated by means of corresponding states correlations. The required characteristic constants are the heat of vaporization and the density at the melting point, H _vap and _m, respectively. The EoS is applied to these liquid metals to calculate the density at temperatures higher than their melting points. The results are fairly consistent with experiment, maximum difference less than ±4%.     

Equation of State for Molten Alkali Metals from Surface Tension. Part II

This work presents a new method for predicting the equation of state for molten alkali metals, based on statistical–mechanical perturbation theory from two scaling constants that are available from measurements at ordinary pressures and temperatures. The scaling constants are the surface tension and the liquid density at the boiling temperature (_b, _b). Also, a reference temperature, T _Ref, is presented at which the product (T _Ref T _b ^1/2 ) is an advantageous corresponding temperature for the second virial coefficient, B ₂(T). The virial coefficient of alkali metals cannot be expected to obey a law of corresponding states for normal fluids, because two singlet and triplet potentials are involved. The free parameter of the Ihm–Song–Mason equation of state compensates for the uncertainties in B ₂(T). The vapor pressure of molten alkali metals at low temperatures is very low and the experimental data for B ₂(T) of these metals are scarce. Therefore, an equation of state for alkali metals from the surface tension and liquid density at boiling temperature (_b, _b) is a suitable choice. The results, the density of Li through Cs from the melting point up to several hundred degrees above the boiling temperature, are within 5%.     

Equation of State for Complex Liquid Mixtures from Surface Tension

The present work shows a successful extension of previous studies to molecular liquids for which the second virial coefficients are not known. Recent advances in the statistical mechanical theory of equilibrium fluids can be used to obtain an equation of state (EOS) for compressed normal liquids and molten alkali metals. Three temperature-dependent quantities are needed to use the EOS: the second virial coefficient, B(T), an effective van der Waals covolume, b(T), and a scaling factor, (T). The second virial coefficients are calculated from a correlation that uses the surface tension, _tr, and the liquid density at the triple point. Calculation of (T) and b(T) follows by scaling. Thus, thermodynamic consistency is achieved by use of two scaling parameters (_tr, _tr). The correlations embrace the temperature range T _tr<T<T _c and can be used in a predictive mode. The remaining constant parameter is best found empirically from _tr data for pure dense liquids. The equation of state is tested on 42 liquid mixtures The results indicate that the liquid density at any pressure and temperature can be predicted within about 5%, over the range from T _tr to T _c.     

Analytical Equation of State for Solid-Liquid-Vapor Phases

A simple analytical equation of state has been proposed for describing the phase behavior of three thermodynamic states (solid, liquid, and vapor) of matter. In terms of reduced parameters, it can be written as:

Saturated Liquid Densities for 33 Binary Refrigerant Mixtures Based on the ISM Equation of State

In this work, the ISM equation of state based on statistical-mechanical perturbation theory has been extended to liquid refrigerant mixtures by using correlations of Boushehri and Mason. Three temperature-dependent parameters are needed to use the equation of state: the second virial coefficient, B₂(T), an effective van der Waals covolume, b(T), and a scaling factor, α (T). The second virial coefficients are calculated from a correlation based on the heat of vaporization, ΔH_vap, and the liquid density at the normal boiling point, ρ_nb. α(T) and b(T) can also be calculated from second virial coefficients by a scaling rule. The theory has considerable predictive power, since it permits the construction of the PVT surface from the heat of vaporization and the liquid density at the normal boiling point. The equation of state was tested on 33 liquid mixtures from 12 refrigerants. The results indicate that the liquid densities can be predicted to at most 2.8% over a wide range of temperatures, 170–369 K.     

A Perturbed Hard-Sphere Equation of State for Alkali Metal Alloys

A perturbed hard-sphere equation of state for liquid alkali metals has been employed to calculate the liquid density of alkali metal alloys over a wide range of temperature. 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, and hence, to predict the equation of state. In this article, calculated results of the liquid density of binary molten alloys of Na–K and K–Cs over the whole range of concentration at temperatures from the freezing point up to several hundred kelvin above the boiling point reproduce accurately the experimental PVT data points.     

Equation of State for Molten Alkali Metal Alloys

Calculated results of the liquid density of binary molten alloys of Na–K and K–Cs over the whole range of concentrations and that of a ternary molten eutectic of Na–K–Cs from the freezing point up to several hundred degrees above the boiling point are presented. The calculations were performed with the analytical equation of state proposed by Ihm, Song, and Mason, which is based on statistical-mechanical perturbation theory. The second virial coefficients were calculated from the corresponding-states correlation of Mehdipour and Boushehri. Calculation of the other two temperature-dependent parameters was carried out by scaling. The calculated results cover a much wider range of temperatures and are more accurate than those presented in our previous work.     

Tait's Equation of State for Liquid Mixtures

The methods of statistical theory of liquid state are used to validate the well-known Tait's equation of state for liquid mixtures. An expression is derived which relates the coefficients A and B of Tait's equation of state to the parameters of steepness of repulsion forces m and to the thermodynamic properties of the system. P–V–T–x measurements for a water-acetone system are performed to check the theoretical results. The method of molecular dynamics is used to calculate the parameters of steepness of repulsion forces of a water-acetone mixture at different temperatures and concentrations. It is demonstrated that m in the treated ranges of temperature and pressure assumes a constant value of 15. The theoretically obtained coefficient A coincides with the experimentally obtained value within the experimental error, and the coefficient B describes qualitatively correctly the temperature and concentration dependences obtained as a result of P–V–T–x measurements.     

Linear Isotherm for Compressed Molten Alkali Metals

Molten alkali metals are shown to be in the domain of the newly developed linear regularity that is valid for pure compressed liquids and liquid mixtures. It holds in the range of melting to boiling temperature and shows deviations as the critical temperature is approached. The agreement with experimental data is better than 1.4% when it is used to predict the density of molten Li, Na, K, Rb, and Cs metals. A reasonable conformity with the ISM statistical mechanical equation of state is manifested.     

3种典型面心立方金属的超声物态方程比较研究

旨在通过对各种解析物态方程的对比研究,试图寻找一种可以利用较低压力下的超声测量数据建立在较宽广的压力范围内适用于简单面心立方金属的等温物态方程模型。对于Al、Cu、Ag3种面心立方金属,结合实测DAC实验数据对比研究了基于较低压力下的超声测量数据计算的各种解析状态方程适用的压力范围,发现目前尚不存在一种完全适用于材料从低压段到极高压力区的物态方程模型,而从较低压力下得到的超声测量数据出发,选取Vinet等温物态方程模型,可以在相当高的压力范围内(对铝约200GPa,对于铜和银约100GPa)很好地描述几种面心立方金属的压缩性。     

An Extended Equation of State Modeling Method II. Mixtures

This work is the extension of previous work dedicated to pure fluids. The same method is extended to the representation of thermodynamic properties of a mixture through a fundamental equation of state in terms of the Helmholtz energy. The proposed technique exploits the extended corresponding-states concept of distorting the independent variables of a dedicated equation of state for a reference fluid using suitable scale factor functions to adapt the equation to experimental data of a target system. An existing equation of state for the target mixture is used instead of an equation for the reference fluid, completely avoiding the need for a reference fluid. In particular, a Soave–Redlich–Kwong cubic equation with van der Waals mixing rules is chosen. The scale factors, which are functions of temperature, density, and mole fraction of the target mixture, are expressed in the form of a multilayer feedforward neural network, whose coefficients are regressed by minimizing a suitable objective function involving different kinds of mixture thermodynamic data. As a preliminary test, the model is applied to five binary and two ternary haloalkane mixtures, using data generated from existing dedicated equations of state for the selected mixtures. The results show that the method is robust and straightforward for the effective development of a mixture- specific equation of state directly from experimental data.     

Equation of state for compressed liquids from surface tension

A method for predicting an analytical equation of state for liquids from the surface tension and the liquid density at the freezing temperature ( ₁, ₁) as scaling constants is presented. The reference temperature. T_ref. is introduced and the product (T _ref T ₁ ^{1 2} ) is shown to be an advantageous corresponding temperature for the second virial coeflicienls. B₂(T). of spherical and molecular fluids. Thus, B₂(T) follows a promising corresponding states principle and then calculations for(T) andb(T), the two other temperature-dependent constants of the equation of state, are made possible by scaling. As a result, ( ₁, ₁) are sufficient for the determination of thermophysical properties of fluids from the freezing line up to the critical temperature. The present procedure has the advantage that it can also be used in cases whereT _c andP _c are not known or the vapor pressure is too small to allow accurate measurements. We applied the procedure to predict the density of Lennard-Jones liquids over an extensive range of temperatures and pressures. The results for liquids with a wide range of acentric factor values are within 5%.     

An Extended Equation of State Modeling Method I. Pure Fluids

A new technique is proposed here to represent the thermodynamic surface of a pure fluid in the fundamental Helmholtz energy form. The peculiarity of the present method is the extension of a generic equation of state for the target fluid, which is assumed as the basic equation, through the distortion of its independent variables by individual shape functions, which are represented by a neural network used as function approximator. The basic equation of state for the target fluid can have the simple functional form of a cubic equation, as, for instance, the Soave–Redlich–Kwong equation assumed in the present study. A set of nine fluids including hydrocarbons, haloalkane refrigerants, and strongly polar substances has been considered. For each of them the model has been regressed and then validated against volumetric and caloric properties generated in the vapor, liquid, and supercritical regions from highly accurate dedicated equations of state. In comparison with the underlying cubic equation of state, the prediction accuracy is improved by a factor between 10 and 100, depending on the property and on the region. It has been verified that about 100 density experimental points, together with from 10 to 20 coexistence data, are sufficient to guarantee high prediction accuracy for different thermodynamic properties. The method is a promising modeling technique for the heuristic development of multiparameter dedicated equations of state from experimental data.     

Equation of State for Normal Liquid Helium-3 from 0.1 to 3.3157 K

An equation of state for normal liquid ³He has been constructed in the form of Helmholtz free energy as a function of independent parameters—temperature, and density. The equation was fitted simultaneously to the collected experimental p-ρ-T, specific heat, sound velocity, isobaric expansion coefficient and isothermal compressibility coefficient from the world’s literature to accuracies comparable with reasonable experimental errors in the measured quantities. Extensive comparisons between the equation of state and experimental data have been made by a set of deviation plots. The state equation is valid in the region for temperatures from 0.1 K to T _c = 3.3157 K, and for pressures from vapor pressures to melting pressures.     

An Equation of State for High-Temperature Superconductor

In the present paper the expression of cohesive energy and the bulk modulus as a function of volume are formulated for high-T _c copper oxide superconductors. The model employed consists of long-range electrostatic Coulomb interaction and short-range overlap repulsion. The short-range overlap potential is considered in the Born–Landé inverse power form. The model parameters of the Born–Landé model are calculated from the equilibrium condition and data of bulk modulus at room temperature. The computed values of pressure derivatives of bulk modulus at P=0 and the values of bulk modulus are found to be in very close agreement with experimental values for high-T _c copper oxide and their nonsuperconducting parent compounds. It is also found that the quantity U/U(V ₀) of these compounds increases with increasing hydrostistic pressure.     

Multiphase Equation of State for Carbon over Wide Range of Temperatures and Pressures

A semiempirical equation-of-state model, which takes into account the effects of polymorphic phase transformation and melting, is proposed. An equation of state is developed for graphite, diamond, and liquid phases of carbon, and a critical analysis of calculated results in comparison with available high-temperature, high-pressure experimental data is made.Paper presented at the Sixteenth European Conference on Thermophysical Properties, September 1–4, 2002, London, United Kingdom.     

An Equation of State for 1,1,1-Trifluoroethane (R-143a)

A fundamental equation of state has been developed for 1,1,1-trifluoroethane (R-143a) using the dimensionless Helmholtz energy. The experimental thermodynamic property data, which cover temperatures from the triple point (161 K) to 433 K and pressures up to 35 MPa, are used to develop the present equation. These data are represented by the present equation within their reported experimental uncertainties: ±0.1% in density for both vapor and liquid phase P––T data, ±1% in isochoric specific heat capacities, and ±0.02% in the vapor phase speed-of-sound data. The extended range of validity of the present model covers temperatures from 160 to 650 K and pressures up to 50 MPa as verified by the thermodynamic behavior of the isobaric heat-capacity values over the entire fluid phase.     

The Law of Corresponding States and Surface Tension of Liquid Metals

Empirical relationships for the surface tension of liquid metals (LM) are shown to follow from the principle of corresponding states. In order to relate the surface tension of LM to its bulk properties, a formula is derived by scaling with the melting point T _m (0) at the atmospheric pressure, p = 0 and the atomic volume _m (0) at the melting point as macroscopic parameters for scaling and a characterizing the interatomic potential (r)= *(r/a). Correlation rules, derived for the surface tension and its temperature coefficient, are discussed and compared with experimental data.     

Fusion Equation of Metals

An expression of initial slope of meltingcurve of pure metals was obtained as follows:(dT_m/dP)_o=T_(mo)/c, where c=1.09 (N_(at))~(5/3)z~(-1/3),the unit of c is GPa, N_(at) is the atomic concentra-tion (in 10~(28) m~(-3)), z is the valence, T_(mo)is the melting temperature (in k) of metal underone atmosphere. The calculated results forthirty-one metals agree well with experiments.It has also been proved that by using the freeelectron model of melting, the fusion equationof metals is Simon equation (T_m/T_(mo))q=1+(p/d).Two parameters q and d, which have to fit withexperiments in Simon's empirical equation, nowcan be predicted theoretically, e.g. for Mg,giving q=1.56, d=7.88GPa, the calculated meltingcurve in a fairly wide pressure range (0～60GPa)is shown to be close to the experimental one.