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%.     

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.     

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%.     

Equation of State for Nonpolar Fluid Mixtures: Prediction from Boiling Point Constants

Our previous corresponding-states correlation for the second virial coefficient of nonpolar fluids, based on the normal boiling point parameters, has been employed to predict the equation of state of nonpolar fluid mixtures. The analytical equation of state is that of Ihm, Song, and Mason, which requires three temperature-dependent parameters, i.e., the second virial coefficient, a scaling constant for softness of repulsive forces, and a van der Waals covolume. In the previous work, we showed that the temperature-dependent parameters could be calculated by knowing the boiling point constants. In this work, it is shown that using a simple geometric mean for the boiling point temperature and an arithmetic mean for the liquid density at the normal boiling point is sufficient to determine the temperature-dependent parameters for mixtures. The equation of state has been utilized to calculate the liquid density of several nonpolar fluid mixtures. The agreement with experiment is good.     

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.     

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.     

Surface Tension of Liquid Alumina from Contactless Techniques

New data for the surface tension of liquid alumina from 2300 to 3200 K are reported. Aerodynamic levitation of CO₂ laser-heated liquid drops allowed contactless measurement of vibration frequencies directly related to surface tension. Consistent data were obtained on drops of different mass ranging from 20 to 160 mg. It was also shown that the oxydo-reducing character of the atmosphere does not modify the results within experimental uncertainty.     

Equation of State for Nonpolar Fluids: Prediction from Boiling Point Constants

A new corresponding states correlation for the second virial coefficient of nonpolar fluids in terms of the boiling point constants is presented. The scaling constants are the normal boiling point temperature, T _bp, which is used to form a dimensionless temperature and the liquid density at the normal boiling point, _bp, which is used to form a dimensionless second virial coefficient. The procedure has been examined for a large number of substances including noble gases, diatomic molecules, saturated hydrocarbons up to C₈, and a number of aliphatic, aromatic, and cyclic hydrocarbons. The resulting correlation has been applied to predict the equation of state of fluids over the range from the vapor-pressure curve to the freezing curve at various temperatures from the triple point up to the nonanalytical critical region. The equation of state has been applied to reproduce the liquid density of a great number of compounds both in the saturation and compressed states, at temperatures up to 2000 K and pressures up to 10000 bar, within an accuracy of a few percent. In particular we have shown that knowledge of two readily measurable constants is sufficient to determine the pvT surface of pure normal fluids having a variety of structural complexities.     

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.     

An Analytical Equation of State for Alkaline Earth Metals

In this work, an analytical equation of state based on statistical mechanical perturbation theory, which was initially developed for normal fluids and can be applied to predict the P–V–T data for saturated liquid alkaline earth metals, is presented. The equation of state is that of Ihm, Song, and Mason, and the temperature-dependent parameters of the equation of state are calculated from a corresponding-states correlation as functions of the reduced temperature. Two scaling constants are sufficient for this purpose, the surface tension and the liquid density at the melting point. The equation of state is used to predict the saturated liquid density of molten alkaline earth metals from the melting point up to 2000 K, for which experimental data exist, within an accuracy of 5%.     

Prediction of Critical Properties of Binary Mixtures Using the PRSV-2 Equation of State

The aim of this work is to test the value of the Peng–Robinson–Stryjek–Vera (PRSV-2) equation of state for predicting the critical behavior of binary mixtures. A procedure adopted by Heidemann and Khalil, based on the Helmholtz free energy, has been followed. The resulting two complex nonlinear equations have been solved simultaneously for the critical temperature and volume, while the critical pressure is calculated from the PRSV-2 equation of state itself. Three forms of binary-interaction parameters have been tried: the zero-type, conventional one-parameter type, and Margules two-parameter type. The optimum values of the binary interaction parameters, based on minimizing the sum of the squares of the relative errors between predicted and experimental critical temperatures, have been calculated for 20 polar and nonpolar systems. The Margules two-parameter type gives the best results, but its mathematical derivation is cumbersome and it requires more computation time. The standard and the average of the absolute relative deviations in critical properties are included. The predicted critical temperatures and pressures agree well with the experimental results, and are always better than those predicted by the group-contribution method. The deviations in the predicted critical volumes using any of the tested binary-interaction parameter types are relatively large compared to those using the group-contribution method.     

Corresponding-States Analysis of the Surface Tension of Simple, Polar, and Ionic Fluids

The liquid–vapor interfacial tension of various simple, polar, and ionic fluids is studied in a corresponding-states analysis that was originally suggested by Guggenheim. Data for real fluids are compared to results of simulations and theoretical predictions for model fluids of each of the three types (namely, the Yukawa fluid, the square-well fluid, a fluid consisting of dipolar hard spheres, and the restricted primitive model of ionic fluids). As already demonstrated by Guggenheim, the data for simple and weakly polar fluids map onto a master curve. Strongly dipolar, associating fluids, which may also exhibit hydrogen-bonding (e.g., water), show deviations from this master curve at low temperatures. In addition, the surface tension of these fluids shows a characteristic sigmoid behavior as a function of temperature. A similar behavior is found from simulations of the ionic model fluid, but not from the electrolyte theories available up to now, for which we present new results here. Exceptionally low values of the reduced surface tension are obtained for hydrogen fluoride and for the Onsager model of dipolar fluids, which, however, agree remarkably well with each other in a corresponding-states plot.     

一种测定液体表面张力系数的新方法

针对现有的测定液体表面张力系数的方法存在设备昂贵、测定结果容易受各种因素干扰而测试精度不足等问题,提出了一种使用不等内径U型管,采用局部力平衡方法结合拉普拉斯方程,以及MATLAB图像处理和LabVIEW虚拟仪器系统,测定液体表面张力系数的方法。与现有的测定液体表面张力系数的拉脱法相比,该方法具有测试精度高、样品需要量少、测试成本低等特点。     

Prediction of Critical Properties of Mixtures from the PRSV-2 Equation of State: A Correction for Predicted Critical Volumes

The predictive capability of the Peng–Robinson–Stryjek–Vera (PRSV-2) equation of state (1986) for critical properties of binary mixtures was investigated. The procedure adopted by Heidemann and Khalil (1980) and discussed by Abu-Eishah et al. (1998) was followed. An optimized value for the binary interaction parameter based on minimization of error between experimental and predicted critical temperatures was used. The standard and the average of the absolute relative deviations in critical properties are included. The predicted critical temperature and pressure for several nonpolar and polar systems agree well with experimental data and are always better than those predicted by the group-contribution method. A correction is introduced here to modify the predicted critical volume by the PRSV-2 equation of state, which makes the average deviations between predicted and experimental values very close to or even better than those predicted by the group-contribution method.     

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 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.     

Surface Tension of HFC Refrigerant Mixtures

The surface tension of refrigerant mixtures, i.e., R-410A (50 mass% R-32/50 mass% R-125), R-410B (45 mass% R-32/55 mass% R-125), R-407C (23 mass% R-32/25 mass% R-125/52 mass% R-134a), R-404A (44 mass% R-125/52 mass% R-143a/4 mass% R-134a), and R-507 (50 mass% R-125/50 mass% R-143a), has been measured and correlated in the present study. Although the first three mixtures are very important as promising replacements for R-22 in air-conditioners and heat-pumps and the last two are promising replacements for R-502, surface tension data for these mixtures were not previously available. The measurements were conducted under conditions of coexistence of the sample liquid and its saturated vapor in equilibrium. The differential capillary rise method (DCRM) was used, with two glass capillaries with inner radii of 0.3034±0.0002 and 0.5717±0.0002 mm. The temperature range covered was from 273 to 323 K, and the uncertainty of measurements for surface tensions and temperatures is estimated to be at most ±0.2 mN·m^–1 and ±20 mK, respectively. A mixing rule was selected for representing the temperature dependence of the resultant data. These data were successfully represented by a mixing rule using mass fraction based on the van der Waals correlation.     

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.     

Equation of State for Substances that Satisfy the Corresponding States Law

Universal expressions for the free energy of substances that satisfy the law of corresponding states are obtained. The equation of state is constructed using an interpolation between the high temperature Debye approximation and the ideal-gas one. Missing data needed for calculations have been found from the comparison of calculated and tabulated values of entropy and pressure along the phase coexistence curves. The thermodynamics of corresponding states, constructed in this way, allows one to find the triple- and critical-point parameters, as well as the phase coexistence curves with an accuracy that does not exceed the accuracy of the law of corresponding states itself.     

Simplified Gradient Theory Modeling of the Surface Tension for Binary Mixtures

In this work, the gradient theory was combined with the volume translation Peng-Robinson and Soave Redlich-Kwong equations of state (VTPR and VTSRK EOSs) and the influence parameter correlation to predict the surface tension of binary mixtures. The density profiles of mixtures across the interface were assumed to be linearly distributed to simplify the gradient theory model. The only two inputs of the theory are the Helmholtz free-energy density of the homogeneous fluid and the influence parameter of the inhomogeneous fluid. The VTPR and VTSRK equations of state were applied to determine the Helmholtz free-energy density and the bulk properties. The influence parameter of the inhomogeneous fluid was calculated from a correlation published previously (Lin et al. Fluid Phase Equilib 254:75, 2007). The only adjustable coefficient of the simplified gradient theory was set equal to zero, which made the theory predictive. The surface tension predicted by this model shows good agreement with experimental data for binary non-polar and polar mixtures.