An equation of state for gases and liquids

We present an equation of state that can represent within experimental error most individual sets of published PVT data for most fluids, whether in the range of vapor at moderate pressures, or compressed liquids, or gases at very high temperatures and densities, any region in fact except the vicinity of the critical point. In terms of pressure the equation is P = DRT [1 + (D/T) (c₁T + c₂D - 1) / (c₃ + c₄T^{$\text{sol1}\text{2}$} + c₅D + c₆D²)] where D = 1/V, the density in mole 1^?1. The coefficients are readily determined by a least squares fit of the data. An additional term is sometimes needed if the D range is very wide, say several times D_c. Different fluids can be simultaneously represented over a limited range, such as the compressed liquid region, by a single reduced form of the equation in which all but three of the constants are the same for all, and these three (a reducing T, 1/c₁, a reducing D, 1 / c₂, and a dimensionless parameter) are characteristic of each individual fluid. The equation can also simultaneously represent many data sets for a single fluid from many labs and covering various T and D ranges. From this, a consistent representation of its thermodynamic properties can be derived.     

New regularities and an equation of state for liquids

Three regularities have been introduced for liquids (T < T_C and ρ > ρ_C) based on average potential energy. The experimental data have been used to show the validity of the regularities. First, there exists near-linearity relation between and ρ for all isotherms of a liquid, where P_i and ρ are internal pressure and density, respectively. Second, changes linearly with ρ for each isotherm of any liquid, where Z and V_m are compressibility factor and molar volume, respectively. Third, a new regularity using the definition of bulk modulus and our new equation of state between reduced bulk modulus and density has been introduced, that is versus ρ must be linear for all isotherms of a liquid where B_r is the reduced bulk modulus.

A new equation of state has been also derived. The density of some liquids in the extensive ranges of temperature and pressure has been calculated using the new equation of state. The densities calculated from this equation agree with experiment to better than 0.3%. The new equation of state can predict internal pressure, thermal expansion coefficient, and isothermal compressibility of liquids within experimental error.     

The Ornstein-Zernike equation, critical phenomena, and the equation of state for liquids and gases

It is shown that the Ornstein-Zernike (OZ) equation has two solutions: the standard one, which depends explicitly on the interaction potential, and a second universal one, resulting from the infinity point of the partition function. It is stressed that there are two pressure components: the standard one and a universal one that is valid over the whole of the phase plane. It is concluded that the universal solution parameters depend in general on definite integrals of functions dependent on the interaction potential. In the vicinity of the critical point, however, the dependence on the interaction potential vanishes; i.e., the solution becomes fully universal. It is shown that in this range of the phase diagram, all results of the theory of critical phenomena (scaling theory) follow from the OZ equation.     

A new equation of state for metal alloys

A perturbed hard-trimer (PHT) equation of state (EOS) has been employed to model volumetric properties of molten metals and their alloys considering a trimer expression obtained from the statistical associating fluid theory. The van der Waals dispersion forces were applied as perturbation term. Two parameters appeared in the PHT EOS, reflecting the dispersive energy among trimers, ε and the hard-core diameter σ were determined based on the molecular scaling parameters. The performance of the proposed PHT EOS has been evaluated by predicting the saturated and isobaric densities of 16 molten metals including alkali metals, alkali earth and refractory metals over the temperature range within 234–7400 K and pressures up to 436 MPa. From 677 data points examined, the average absolute deviation (AAD) of the predicted densities from the experimental ones was found to be 1.60%. Furthermore, the estimated uncertainties of predicted densities of alloys were ±3.00%.     

Quantum mechanical law of corresponding states from ISM equation of state: compressed liquids

In this paper, we explore the theory of the equation of state from the view point of Ihm–Song–Mason (ISM) equation of state, which has been derived on the basis of statistical mechanical perturbation theory, and is characterized by three temperature dependent parameters, α, b, B₂, and a free parameter Γ. This equation is applied well to non-polar fluids in subcritical and supercritical regions and to molten alkali metals. We present results that show Γ varies slightly with temperature. Among the nobles group, Γ values are quite the same and are correlated except He, which deviates so much even no moderate correlation is seen. In the alkali metals group, Γ values are roughly the same for K, Rb, and Cs but are different for Li and Na. We have previously shown that Γ conforms to B₂, the second virial coefficient, and thus to the nature of the particular fluid system. These observations plus the discussion on quantum mechanical law of corresponding states suggest that the ISM equation of state stands as an analytical equation of state which explicitly incorporates quantum effects by the parameter Γ. Then, we suggest a law of corresponding states as p*=p*(v*, T*, Γ) where, asterisks stand for reduced pressure, volume, and temperature, respectively.     

A new cubic equation of state for reservoir fluids

A new three-parameter cubic equation of state is developed with special attention to the application for reservoir fluids. One parameter is taken temperature dependent and others are held constant. The EOS parameters were evaluated by minimizing saturated liquid density deviation from experimental values and satisfying the equilibrium condition of equality of fugacities simultaneously. Then, these parameters were fitted against reduced temperature and Pitzer acentric factor. For calculating the thermodynamic properties of a pure component, this equation of state requires the critical temperature, the critical pressure, the acentric factor and the experimental critical compressibility of the substance. Using this equation of state, saturated liquid density, saturated vapor density and vapor pressure of pure components, especially near the critical point, are calculated accurately. The average absolute deviations of the predicted saturated liquid density, saturated vapor density and vapor pressure of pure components are 1.4%, 1.19% and 2.11%, respectively. Some thermodynamic properties of substances have also been predicted in this work.     

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%.     

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%.     

A new non-cubic equation of state

A new non-cubic equation of state is presented in this work. This expression is obtained from the original Redlich—Kwong equation of state by assuming that the attraction parameter depends not only on temperature, but also on density. Vapour—liquid equilibria in the coexistence region and PVT properties for the liquid, gas and supercritical fluid phases are accurately calculated with four parameters per isotherm. The generalization of this equation by a corresponding states correlation enables it to be applied over wide ranges of temperature, pressure and hydrocarbon molecular weight.     

Modelling the volumetric properties of ionic liquids using a modified perturbed hard-sphere-chain equation of state

In this paper, a modified perturbed hard-sphere-chain equation of state (EOS) by Eslami [H. Eslami, Fluid Phase Equilib. 216 (2004) 21–26], is applied for modelling the thermodynamic properties of some ionic liquids (ILs). Two reliable scaling constants are used to determine two temperature-dependent parameters in the proposed EOS. The unique adjustable parameter that is reflecting the number of segments per molecule, r, compensates the uncertainties in the calculated temperature-dependent parameters. The reliability of the proposed EOS has been checked by comparing the results with 1561 experimental data points for 18 ILs over a broad range of pressures and temperatures. The overall average absolute deviation is 0.35%. A comparison of the predicted results, using the present EOS with the results of some previous models, indicates that the determined results of this EOS are in more accordance with experimental data than those.     

Thermodynamic scaling of diffusion in supercooled Lennard-Jones liquids

The manner in which the intermolecular potential u(r) governs structural relaxation in liquids is a long standing problem in condensed matter physics. Herein, we show, in agreement with recent experimental results, that diffusion coefficients for simulated Lennard-Jones m-6 liquids (8 < or = m < or = 36) in normal and moderately supercooled states are a unique function of the variable rhogamma/T, where rho is density and T is temperature. The scaling exponent gamma is a material specific constant whose magnitude is related to the steepness of the repulsive part of u(r), evaluated around the distance of closest approach between particles probed in the supercooled regime. Approximations of u(r) in terms of inverse power laws are also discussed.     

Voids, generic van der Waals equation of state, and transport coefficients of liquids

In this Perspective, we discuss the role of voids in transport processes in liquids and the manner in which the concept of voids enters the generic van der Waals equation of state and the modified free volume theory. The density fluctuation theory is then discussed and we show how the density fluctuation theory can be made a molecular theory with the help of the modified free volume theory and the generic van der Waals equation of state. The confluence of the aforementioned three theories makes it possible to calculate the transport coefficients of liquids by using the information on the equilibrium pair correlation function, which can be calculated either by an integral equation theory or Monte Carlo simulations. A number of relations between transport coefficients are also presented, which are derived on the basis of the density fluctuation theory. Since they can be used to obtain one transport coefficient from another they can be very useful in handling experimental and theoretical data. An application of the modified free volume theory to polymer melts is discussed as an example for a theory of transport properties of complex liquids.     

Generalization of Kelvin's equation for compressible liquids in nanoconfinement

The Kelvin equation for a compressible liquid in nanoconfinement is written in a form that takes into account not only Laplace's pressure, but also the oscillatory compression pressure. This leads to a simple analytical equation for pressure in nanocapillaries. The corrected equation is used to analyze properties of aqueous systems, including the oscillatory structural forces between attractive surfaces and inert surfaces, repulsive "hydration" forces between hydrophilic surfaces, and attractive "hydrophobic" forces between hydrophobic surfaces. Relative vapor pressure in a nanocapillary also is discussed.     

Nonclassical critical indices in the statistical theory of liquids and equations of state with regular and scaling components

Nonclassical critical indices γ = 9/8 and β = 3/8 obtained in works using the statistical theory of liquids are used to approximate data on the isothermal compressibility of ⁴He and SF₆ on critical isochores and binodals, and on the shape of a binodal. The ratio of asymptotic compressibility coefficients Γ ₀ ⁺ /Γ ₀ ^? on critical isochores and binodals is determined from experiments using statistical theory and it is shown that Γ ₀ ⁺ /Γ ₀ ^? coincides with the magnitude of this ratio for linear scaling model with γ = 9/8 and β = 3/8 indices. Highly accurate P-ρ-T data on ⁴He and SF₆ are approximated at these values of the indices using new combined equations of state that include regular and scaling components. The advantage of describing of P-ρ-T data in this manner, compared to describing them with the same equations of state using the critical indices of the three-dimensional Ising model, is demonstrated.     

A quartic hard chain equation of state for normal fluids

A new empirical equation of state is proposed which is applicable to mixtures of chain-like molecules. The equation is based on a simple model which uses an approximate chain theory and an approximation of the Carnahan—Starling equation. The results for pure component properties of normal fluids are comparable to the common cubic equations of state. For mixtures of chain-like molecules, the new equation is as good or better than the Peng—Robinson equation.     

Correlation of the viscosity of pure liquids at high pressures based on an equation of state

A novel model has been presented for correlating the dynamic viscosity of Newtonian liquids at high pressures. The proposed model was started with the activation volume, which could simultaneously be influenced by temperature and pressure. The core of the model was based on Hu and Liu's work to calculate the compressibility factor. The final expression may contain two adjustable parameters, namely κ₁ and κ₂, which had been determined by fitting literature viscosity data. The results show that the agreement between experimental data of viscosity and the calculated ones with the proposed model was reasonably good for the selected systems. It was found that the logarithm of parameter κ₁ was a linear function of the reciprocal of the temperature, and κ₁ was approximately equal to viscosity of liquid at 0.1 MPa. Besides, for linear chain hydrocarbons, the logarithm of the parameter κ₁ was completely a linear function of the number of the carbon atoms under certain temperature. A comparison between model with two-parameter and one with one-parameter had been given to show their applicability. The results calculated by not only two-parameter but also one-parameter were superior to corresponding ones by previous pressure equation for estimating viscosity.     

A crossover lattice fluid equation of state for pure fluids

A lattice fluid model is one of the most versatile, molecular-based engineering equations of state (EOS) but, in common with all analytic equations of state, the lattice fluid (LF) EOS exhibits classical behaviour in the critical region rather than the non-analytical, singular behaviour seen in real fluids. In this research, we use the LF EOS and develop a crossover lattice fluid (xLF) equation of state near to and far from the critical region which incorporates the scaling laws valid asymptotically close to the critical point while reducing to the original classical LF EOS far from the critical point. We show that, over a wide range of states, the xLF EOS yields the saturated vapour pressure data and the density data with much better accuracy than the classical LF EOS.     

A new scaled crossover parametric equation of state for water

In the present work, the asymmetric nature of water coexistence curve has been studied by investigating a new scaled crossover parametric equation of state. To do so, the concept of complete scaling [Fisher and Orkoulas, Phys Rev Lett 696, 85 (2000)] has been applied and the critical amplitudes near and far from the critical point have been derived. Also two mixing parameters $ a_{3} $ and $ b_{2} $ in the definition of scaling fields in terms of physical fields have been obtained for water. We have shown that mixing of the complete scaling theory and parametric equation of state can explain this nature quite carefully.     

A cubic perturbed hard-chain equation of state

A cubic equation of state is developed on the basis of perturbation theory. The equation is an association of three segments: the hard-sphere, the hard-chain, and the attraction. The expression for each segment was invoked from approximations of computer simulations of rigorous molecular theories of fluids, but compromised to some extent accuracy and theory for simplicity. This model equation is shown to be potentially capable of describing the PVT behavior of real fluids. As limiting cases, the new equation is reduced to expressions for the hard-sphere and the hard-body fluids. It also represents square-well fluids when the hard-chain contribution is eliminated. The square-well equation was found satisfactory in conforming with the molecular simulation results for square-well fluids and their mixtures.