Fourth Virial Coefficient Research Articles

Speed of sound measurements and derived third and fourth acoustic virial coefficients of supercritical neon

We report comprehensive and accurate measurements of the speed of sound in neon. These measurements were carried out by a double-path-length pulse-echo technique and cover the temperature range between 200 K and 420 K with pressures up to 100 MPa. The standard uncertainties are 1.9 mK in temperature, 22 parts in 106 in pressure and 35 parts in 106 in speed of sound. The third and fourth acoustic virial coefficients of neon were derived from the speed of sound data in the temperature range of the measurements by fitting a fourth-order acoustic virial expansion in pressure with the second acoustic virial coefficient constrained from first-principles calculations. To support our claimed uncertainty, we determined the ratio M/γ0 between the molar mass M and the ideal-gas heat capacity ratio γ0 of the neon sample with a relative standard uncertainty of 7.7 parts in 106 by additional speed of sound measurements using a spherical resonator at 273.16 K.

Thermodynamics of NaCl in dense water vapor via cross virial coefficients

Sodium chloride in the fluid phase is important for the chemistry of natural hydrothermal processes in the Earth’s crust and derived industrial utilizations, where thermodynamics is an essential part that can also to some extent fill in missing experimental data. The solubility of sodium chloride in crystalline form (halite) in water vapor is modeled using a novel combination of an accurate equation of state for pure water and a truncated virial equation for water mixed with sodium chloride. In the resulting Helmholtz energy expression, interactions between H2O and NaCl are incorporated through the second (B12), third (C112), and fourth (D1112) cross virial coefficients. The temperature dependence of the coefficients has been determined by regression to experimental data. Specifically, the temperature dependence of B12 is assumed to follow that of a square-well gas, whereas C112 and D1112 are calculated using purely exponential functions of inverse temperature. The temperature dependence of B12 is supported by independent calculations of ab initio interaction energies between H2O and NaCl. For water itself, all terms in the virial equation are collectively modeled by the IAPWS-95 equation of state. For sodium chloride, interactions between two or more NaCl molecules are disregarded. The model shows quantitative agreement with available solubility data in the vapor–halite coexistence region, which implies that the water vapor density is approximately less than 130kgm−3. Its theoretical basis ensures that predictions are reliable in regions where experimental data are missing. The direct calculation of cross virial coefficients is overall useful for modeling geochemical fluids with low solute concentrations, including systems with multiple solvents.

Virial coefficients of the uniform electron gas from path-integral Monte Carlo simulations.

The properties of plasmas in the low-density limit are described by virial expansions. Analytical expressions are known from Green's function approaches only for the first three virial coefficients. Accurate path-integral Monte Carlo (PIMC) simulations have recently been performed for the uniform electron gas, allowing the virial expansions to be analyzed and interpolation formulas to be derived. The exact expression for the second virial coefficient is used to test the accuracy of the PIMC simulations and the range of validity of the interpolation formula of Groth etal. [Phys. Rev. Lett. 119, 135001 (2017)0031-900710.1103/PhysRevLett.119.135001], and we discuss the fourth virial coefficient, which is not exactly known yet. Combining PIMC simulations with benchmarks from exact virial expansion results would allow us to obtain more accurate representations of the equationof state for parameter ranges of conditions which are of interest, e.g., for helioseismology.

Four-Body Nonadditive Potential Energy Surface and the Fourth Virial Coefficient of Helium.

The four-body nonadditive contribution to the energy of four helium atoms is calculated and fitted for all geometries for which the internuclear distances exceed a small minimum value. The interpolation uses an active learning approach based on Gaussian processes. Asymptotic functions are used to calculate the nonadditive energy when the four helium atoms form distinct subclusters. The resulting four-body potential is used to compute the fourth virial coefficient D(T) for helium, at temperatures from 10 to 2000 K, with a path-integral approach that fully accounts for quantum effects. The results are in reasonable agreement with the limited and scattered experimental data for D(T), but our calculated results have much smaller uncertainties.

Theoretical determination of speed of sound and fourth virial coefficient by using Kihara (12–6) potential

Abstract In this paper, a new theoretical approach is suggested for the numerical calculation algorithm of the fourth virial coefficient and its components with the Kihara potential. For the first time, this study reports the results of the fourth virial coefficient over the Kihara potential. The proposed method can be used to calculate the thermodynamics properties of gases with developed correctness. The method is tested for some molecules and as an example of application, the proposed method has been performed for gases of C3H8 and SF6. The obtained calculation results have been compared with the theoretical and experimental data and the results have been found to be consistent with the literature data. It can be seen from the application that the calculation of the fourth virial coefficient with the Kihara potential can provide important results in the calculation of the thermophysical properties of other real systems.

Influence of molecular potential type on the accuracy of virial equation of state in describing PVT relations of high temperature and high pressure gas

In order to compare the influence of two commonly used molecular potentials: L-J (Lennard-Jones) 6-12 potential and EXP-6 (exponential-6) potential on the accuracy of calculating pressure-volume-temperature (PVT) relations of high temperature and high pressure gas by high order virial equation of state (EOS). In this paper, using the Kihara’s calculation method, the third order dimensionless virial coefficients of EXP- 6 potential (α=12) and EXP-6 potential (α=13) in dimensionless temperature T* =1- 500 are calculated. Using the orthogonal integration method proposed by Barker, the fourth and fifth order dimensionless virial coefficients of L-J 6-12 potential, EXP-6 potential (α=12) and EXP-6 potential (α=13) in T* =1-500 are calculated. By fitting these theoretical values, the corresponding high order virial EOS: VLJ, VEXP612 and VEXP613, are established. Using VLJ, VEXP612 and VEXP613 EOS to calculate the PVT relations of methane, oxygen, carbon dioxide and carbon monoxide under high temperature and high pressure, the accuracy and applicability of the two molecular potentials in the description of PVT relations of high temperature and high pressure gases are analyzed and evaluated by comparison. On this basis, the characteristics of intermolecular interaction in high temperature and high pressure gas are further analyzed.

Probabilistic computations of virial coefficients of polymeric structures described by rigid configurations of spherical particles: A fundamental extension of the ZENO program.

We describe an extension of the ZENO program for polymer and nanoparticle characterization that allows for precise calculation of the virial coefficients, with uncertainty estimates, of polymeric structures described by arbitrary rigid configurations of hard spheres. The probabilistic method of virial computation used for this extension employs a previously developed Mayer-sampling Monte Carlo method with overlap sampling that allows for a reduction of bias in the Monte Carlo averaging. This capability is an extension of ZENO in the sense that the existing program is also based on probabilistic sampling methods and involves the same input file formats describing polymer and nanoparticle structures. We illustrate the extension's capabilities, demonstrate its accuracy, and quantify the efficiency of this extension of ZENO by computing the second, third, and fourth virial coefficients and metrics quantifying the difficulty of their calculation, for model polymeric structures having several different shapes. We obtain good agreement with literature estimates available for some of the model structures considered.

Quatrièmes coefficients d’amas et du viriel d’un gaz unitaire de fermions pour un rapport de masse quelconque

We calculate the fourth cluster coefficients of the homogeneous unitary spin 1/2 Fermi gas as functions of the internal-state mass ratio, over intervals constrained by the 3- or 4-body Efimov effect. For this we use our 2016 conjecture (validated for equal masses by Hou and Drut in 2020) in a numerically efficient formulation making the sum over angular momentum converge faster, which is crucial at large mass ratio. The mean cluster coefficient, relevant for equal chemical potentials, is not of constant sign and increases rapidly close to the Efimovian thresholds. We also get the fourth virial coefficients, which we find to be very poor indicators of interaction-induced 4-body correlations. We obtain analytically for all $n$ the cluster coefficients of order $n$ + 1 for an infinity-mass impurity fermion, matching the conjecture for $n=3$. Finally, in a harmonic potential, we predict a non-monotonic behavior of the 3 + 1 cluster coefficient with trapping frequency, near mass ratios where this coefficient vanishes in the homogeneous case.

The Fourth Virial Coefficient for Hard Spheres in Even Dimension

The fourth virial coefficient is calculated exactly for a fluid of hard spheres in even dimensions. For this purpose the complete star cluster integral is expressed as the sum of two three-folded integrals only involving spherical angular coordinates. These integrals are solved anallytically for any even dimension d and working with existing expressions for the other terms of the fourth cluster integral we obtain an expression for the fourth virial coefficient B4(d) for even d that sums a finite number of simple terms, with the number of terms increasing with d.

Path-integral calculation of the fourth virial coefficient of helium isotopes.

We use the path-integral Monte Carlo (PIMC) method and state-of-the-art two-body and three-body potentials to calculate the fourth virial coefficients D(T) of 4He and 3He as functions of temperature from 2.6 K to 2000 K. We derive expressions for the contributions of exchange effects due to the bosonic or fermionic nature of the helium isotope; these effects have been omitted from previous calculations. The exchange effects are relatively insignificant for 4He at the temperatures considered, but for 3He, they are necessary for quantitative accuracy below about 4 K. Our results are consistent with previous theoretical work (also with some of the limited and scattered experimental data) for 4He; for 3He, there are no experimental values, and this work provides the first values of D(T) calculated at this level. The uncertainty of the results depends on the statistical uncertainty of the PIMC calculation, the estimated effect of omitting four-body terms in the potential energy, and the uncertainty contribution propagated from the uncertainty of the potentials. At low temperatures, the uncertainty is dominated by the statistical uncertainty of the PIMC calculations, while at high temperatures, the uncertainties related to the three-body potential and omitted higher-order contributions become dominant.

Analytical evaluation of third and fourth virial coefficients for hard disk fluids in narrow channels and equation of state

The third and fourth virial coefficients of hard disks in narrow channels have been evaluated as a series in powers of the channel width deviations from the one dimensional limit. We have obtained analytically the coefficients of the power series to very high order. These virial coefficients are used in truncated virial series to provide approximate analytical equation of state (EOS). The EOS is shown to accurately describe Monte Carlo simulations over a wide range of pressures and densities near the one dimensional limit. Explicit expressions for the free energy, chemical potential, compressibility and EOS as a function of the channel widths are given. These results represent progress toward understanding the long standing problem of the thermodynamic crossover from a one dimensional to quasi one dimensional fluid. They are also useful in a number of applications. We mention the examples of thermodynamic perturbation theory and stochastic dynamic mobility in anomalous single-file diffusion.

Developing the evaluation method of heat capacity and speed of sound of real gases using fourth virial coefficient over Lennard-Jones (12-6) potential

In this study, we suggest an effective method of the evaluation of HC and SS of real gases by using the FVC over Lennard-Jones (12-6) potential. As known, the determination of the FVC is a key step to correct evaluation of the thermal properties. As an example of application, the suggested method has been performed for gases of Ar, SF6 and SiH4. The obtained results of HC at constant pressure and SS of gases Ar, SF6 and SiH4 are in good agreement with the corresponding theory and experimental data in the range of temperature from 90 to 800 K and range of pressure from 0.09 to 100.7 atm. The precision and accuracy of obtained results from the suggested method have been validated by the literature observations.

Virial expansion coefficients in the unitary Fermi gas

Virial expansion is widely used in cold atoms to analyze high temperature strongly correlated many-body systems. As the n-th order virial expansion coefficient can be accurately obtained by exactly solving up to n-body problems, the virial expansion offers a few-body approach to study strongly correlated many-body problems. In particular, the virial expansion has successfully been applied to unitary Fermi gas. We review recent progress of the virial expansion studies in the unitary Fermi gas, in particular the fourth order virial coefficient.

Anomaly in the Virial Expansion of IAPWS-95 at Low Temperatures.

In the standard reference equation of state for the thermodynamic properties of water, known as IAPWS-95, the fourth virial coefficient D(T) becomes abnormally large in magnitude at temperatures below approximately 300 K. At conditions where a virial expansion using only second and third virial coefficients should be essentially exact (such as vapors at pressures near 100 Pa or 1000 Pa), such a truncated expansion may miss on the order of 2 % of the deviation from ideal-gas behavior in the compressibility factor or the fugacity. The term in IAPWS-95 that causes this issue is identified, and suggestions are made for future equation-of-state development.

Data analysis method for estimating virial acoustic virial coefficients of Argon from speed of sound data in the pressure-temperature domain

Virial series of imperfect gases yield a way for study the reliability of potential interaction models by means of high precision the speed-of-sound data, which are measured as function of pressure and temperature. However, a theoretical study for Argon showed that the third acoustic virial coefficient estimated from by high precision velocity data for Argon, can be inconsistent with ab initio calculations in density-temperature domain. In this work we propose a data analysis method to solving this apparent contradiction. The results show that speed-of-sound data for Argon in the pressure-temperature domain can indeed yield reliable estimations of the third and fourth acoustic virial coefficients in the density-temperature domain.

Resummation of Diagrammatic Series with Zero Convergence Radius for Strongly Correlated Fermions.

We demonstrate that a summing up series of Feynman diagrams can yield unbiased accurate results for strongly correlated fermions even when the convergence radius vanishes. We consider the unitary Fermi gas, a model of nonrelativistic fermions in three-dimensional continuous space. Diagrams are built from partially dressed or fully dressed propagators of single particles and pairs. The series is resummed by a conformal-Borel transformation that incorporates the large-order behavior and the analytic structure in the Borel plane, which are found by the instanton approach. We report highly accurate numerical results for the equation of state in the normal unpolarized regime, and reconcile experimental data with the theoretically conjectured fourth virial coefficient.

The quantum equation of state until the fourth virial coefficient for many‐component plasmas

The aim of the present article is to calculate the quantum excess free energy until the fourth virial coefficient in terms of Slater sums for neutral many‐component plasmas. We used the results to calculate the equation of state; we used the thermal equilibrium plasma. Bound states are not taken into account in this work.

Hadron resonance gas model with induced surface tension

Here we present a physically transparent generalization of the multicomponent Van der Waals equation of state in the grand canonical ensemble. For the one-component case the third and fourth virial coefficients are calculated analytically. It is shown that an adjustment of a single model parameter allows us to reproduce the third and fourth virial coefficients of the gas of hard spheres with small deviations from their exact values. A thorough comparison of the compressibility factor and speed of sound of the developed model with the one and two component Carnahan-Starling equation of state is made. It is shown that the model with the induced surface tension is able to reproduce the results of the Carnahan-Starling equation of state up to the packing fractions 0.2-0.22 at which the usual Van der Waals equation of state is inapplicable. At higher packing fractions the developed equation of state is softer than the gas of hard spheres and, hence, it breaks causality in the domain where the hadronic description is expected to be inapplicable. Using this equation of state we develop an entirely new hadron resonance gas model and apply it to a description of the hadron yield ratios measured at AGS, SPS, RHIC and ALICE energies of nuclear collisions. The achieved quality of the fit per degree of freedom is about 1.08. We confirm that the strangeness enhancement factor has a peak at low AGS energies, while at and above the highest SPS energy of collisions the chemical equilibrium of strangeness is observed. We argue that the chemical equilibrium of strangeness, i.e. $\gamma_s \simeq 1$, observed above the center of mass collision energy 4.3 GeV may be related to the hadronization of quark gluon bags which have the Hagedorn mass spectrum, and, hence, it may be a new signal for the onset of deconfinement.

Intermolecular model potentials and virial coefficients from acoustic data

A simple method is presented for the determination of model two- and three-body intermolecular potentials from acoustic data. The method is applied to argon, nitrogen, methane, ethane and propane. For argon, the second, third and fourth volumetric virial coefficients from the model potentials are compared with values determined in direct experiments. Second virial coefficients are shown to be in very good agreement with experiment. For the third volumetric virial coefficients, significant discrepancies are noted in the region of the maximum in the function. There are no experimental data for a direct comparison of the fourth volumetric virial coefficients presented here; however, because in a previous work, it was shown that fourth acoustic virial coefficients calculated from a similar method agreed with values determined in direct experiment, we argue that the values presented here are accurate.Key words: Equation of state, virial coefficients, speed of sound, intermolecular potentials

The thermodynamic functions until fourth virial coefficient for many-component plasmas

This study aims to calculate the analytical form of the classical excess free energy until the fourth virial coefficient for neutral many-component plasmas, by using the Coulomb potential. We also obtain the equation of state. Our results were compared with others.