Speed of Sound in Helium-4 from Ab Initio Acoustic Virial Coefficients

Abstract

We present values for 4He of temperature-dependent acoustic virial coefficients that appear in series expansions for the speed of sound in the gas (AVEOS). Coefficients are computed from first-principles molecular models from the literature and are presented for expansions both in density and in pressure. These coefficients, labeled here as Ω n and ω n , respectively, are determined from previously reported values of the pressure virials B n as a function of temperature T. Data are provided for T from 2.6 to 1000 K and for n from 2 to 7; those for n = 2–5 include nuclear quantum effects but not exchange, and those for n = 6 and 7 are semiclassical. Analysis of experimental data for the speed of sound from the literature is performed to estimate thermodynamic temperatures, to regress coefficient values to compare to ab initio results, and to evaluate the performance of the AVEOS for pressures up to 1000 MPa. A study of the convergence of the series shows that the pressure-expanded AVEOS diverges at about 100 MPa, while the density-expanded series appears to remain convergent over the entire pressure range; at the highest pressures, the seventh-order series still yields results that are within 1% of the correct value of the speed of sound, at least when applied at temperatures above 200 K.

Speed of Sound Measurements in Helium at Pressures from 15 to 100 MPa and Temperatures from 273 to 373 K

“…The virial EOS leads in turn to a similar power-series expansion for the “acoustic compressibility factor” Z a as follows Z normala = ( M c 2 γ 0 R T ) = 1 + normalΩ 2 ρ + normalΩ 3 ρ 2 + normalΩ 4 ρ 3 + · · · Here, γ 0 = 5/3, and Ω n is the n th acoustic virial coefficient, which is related to the corresponding, and all lower order, ordinary virial coefficients B n and their first two temperature derivatives. The EOS in this form has been studied in detail by Gokul et al on the basis of the ab initio pair potential of Przybytek et al , and the non-additive three-body correction of Cencek et al Gokul et al present precise calculations of the acoustic virial coefficient up to order n = 7, estimates of the statistical uncertainty of each coefficient, and correlations of both B n and Ω n as functions of temperature.…”

“…Here, γ 0 = 5/3, and Ω n is the n th acoustic virial coefficient, which is related to the corresponding, and all lower order, ordinary virial coefficients B n and their first two temperature derivatives. The EOS in this form has been studied in detail by Gokul et al 41 on the basis of the ab initio pair potential of Przybytek et al 42,43 and the non-additive three-body correction of Cencek et al 44 Gokul et al 41 present precise calculations of the acoustic virial coefficient up to order n = 7, estimates of the To apply this EOS, we first solved eq 8 for the molar density corresponding to the experimental temperatures and pressure and then used eq 9 to evaluate the speed of sound using, in both equations, coefficients up to n = 7 as correlated by Gokul et al 41 We did not make use of the pressure-series expansion of Z a reported by Gokul et al 41 because it does not appear to converge at the highest pressures studied in this work. On the other hand, both eqs 8 and 9 appear to converge to this order to within 0.01%, as judged by the difference between truncation after n = 6 and after n = 7.…”

Origins of the Failure of the Activity Virial Series

“…The virial equation of state (VEOS) describes the thermodynamic properties of fluids in terms of a power series in a state variable, most commonly the density. − Specifically, the pressure P is given by P k normalB T = prefix∑ n = 1 ∞ B n ( T ) ρ n where T is the temperature, k B is the Boltzmann constant, and ρ ≡ N / V is the number density, with N being the number of molecules and V is the volume; according to the ideal-gas law, B 1 ≡ 1. The VEOS is an appealing framework for representing properties because the coefficients that appear in the equation, the virial coefficients, B n ( T ), can be computed rigorously from molecular models or even first-principles methods. − Other thermodynamic properties can be computed from the VEOS using standard thermodynamic manipulations, in some cases requiring temperature derivatives of the B n . The primary disadvantage of the VEOS as commonly formulated is that the power series is not convergent, or not quickly convergent, at some conditions of interest, and in particular, it is not applicable to condensed phases at all.…”

Eighth-Order Virial Equation of State for Methane from Accurate Two-Body and Nonadditive Three-Body Intermolecular Potentials

“…Jäger et al showed that their VEOS7 for the noble gas argon obtained from ab initio two-body and nonadditive three-body potentials is nearly as accurate as the best experimental data at those densities at which the virial expansion is sufficiently converged. Virial coefficients of the noble gas helium have been calculated with extremely low uncertainties by several research groups (see ref and references therein), resulting in a VEOS7 that, when converged, has an accuracy that cannot be attained by any existing experimental technique. Although our VEOS8 for carbon dioxide obtained from fine-tuned ab initio two-body and nonadditive three-body potentials cannot match the accuracy of the best experimental data, the agreement with experiment is still very satisfactory.…”