An equation of state based on the intermolecular potential and the radial distribution function to estimate the virial coefficients by using PvT chaotic data

Highlights

• Approximate radial distribution function using the Ornstein-Zernike equation.

• General equation of state which allows analyzing any intermolecular potential.

• Thermodynamic model with chaotic behavior and stabilized temperature.

• Estimation of virial coefficients through chaotic PvT oscillations.

• Very good agreement between the simulated values and the experimental ones.

Abstract

We derive a new equation of state (EOS) for any non-polar, polar or quantum gas and any intermolecular potential through an approximation of the radial distribution function based on the Ornstein-Zernike equation. Such EOS is applied to estimate the virial coefficients B and C using two approaches. On one hand, B and C are directly computed from the proposed EOS considering the Mie potential for neon and water steam, which leads to values in very good agreement with the experimental data. On the other hand, the proposed EOS is applied in the context of a thermodynamic model to estimate B and C by fitting the simulated chaotic PvT data at an almost constant temperature. Such estimations of B and C are compared with the experimental values, which allows elucidating if B and C have a significant contribution on the virial equation at the considered pressures and temperatures. This methodology is applied to nitrogen and helium4 considering the intermolecular potentials of Mie, Morse, Kihara and Buckingham. For the considered gases and intermolecular potentials, the calculated values of B and C are in very good agreement with experimental data in a wide range of pressures and temperatures.

Introduction

The interacting theory of real gases based on an intermolecular potential with a hard core and a weak attractive region is widely recognized as one of the great successes of Statistical Mechanics. Two examples of such intermolecular potentials are the Mie (n,m) potential and the Lennard-Jones (12.6) potential (LJ), which can be considered as a particular case of the Mie potential [1,2]. These potentials have been used to determine thermodynamic quantities of real gases, such as, the virial coefficients, internal energy, enthalpy, density, Helmholtz free energy and heat capacity [3,4].

Besides the LJ potential, the Stockmayer potential (ST) has been used for polar fluids [5,6] whereas the Morse potential (MO) was initially used to determine the oscillating states and energies of a quantum oscillator [7]. In fact, there is a close relationship between the LJ and MO potentials as shown in Ref [8]. On the other hand, the Kihara potential (KI) modifies the hard core of the LJ potential and can be applied to non-spherical molecules that can be treated as convex rigid bodies [8,9]. Nonetheless, the KI potential can also be used assuming that the molecules have a spherical nucleus (as it will be done in this paper) [10]. Another extension of the LJ potential is the Buckingham potential (BU or Exp-6), which has been used with non-polar gases but modifying the molecule hard core [10,11].

In the study of intermolecular potentials, it is assumed that the potential energy is a pairwise sum, i.e., the potential is equal to the sum of the pair of molecule potentials V(r_ij), being r_ij the distance between molecules i and j [2,12]. The radial (or pair) distribution function (RDF) g(r,ρ,T) is obtained by dividing the local densities ρ(r) at various distances r of the molecule j by the bulk average density ρ at a given temperature T. The experimental determination of g(r,ρ,T) is rather cumbersome and has been achieved from X-ray and neutron scattering experiments [13]. The pairwise sum assumption together with the knowledge of g(r,ρ,T) allows to establish the equations for the pressure and internal energy, from which all the thermodynamic properties of a given gas can be deduced [14,15].

In this paper we derive theoretically an approximate RDF to obtain a general equation of state (EOS) that can be used with any real gas and intermolecular potential. Such general EOS is applied to estimate the virial coefficients B and C using two approaches. On one hand, B and C are directly computed from the proposed EOS for neon and water steam considering the Mie potential, which leads to results in very good agreement with the experimental data. On the other hand, we consider a thermodynamic model aimed to achieve a chaotic gas dynamic to estimate B and C [16], which will be applied to nitrogen and helium4 using the Mie, Morse, Kihara and Buckingham potentials. The parameters of the thermodynamic model are defined in agreement with well stablished commercial or industrial devices [17], [18], [19]. The values of B and C estimated from the model are compared with the experimental data, which allows elucidating if B and C have a significant contribution on the virial equation at the considered pressures and temperatures. The estimated values of B and C are in very good agreement with experimental data in wide ranges of pressures and temperatures.

The thermodynamic model considers a mechanical subsystem which includes an input valve [20], [21], a flow controller and pressure probe as well as a thermal subsystem formed by another flow controller, a heating-cooling coil and an accumulator vessel. The gas flow rate in the input valve is assumed to be isentropic, whereas the density, the heat capacity at constant volume and the pressure are calculated from the internal energy equations and the proposed equation of state, which depend on V(r_ij) and the RDF [16,22,23].

For the calculation of the second virial coefficient B, the quantum corrections will be included for neon and helium4, whereas dipole-dipole and quadrupole-quadrupole interactions will be considered for water steam and nitrogen respectively [24,25]. In addition, the gas polarization will be included by considering an approximation of the Axilrod-Teller-Muto potential [26,27] (which accounts three-body interactions) in which the molecules are at the vertices of an equilateral triangle. Other treatments for the polarization can be found in Refs [2,15] and [28], [29], [30], whereas the background of three-body interactions between dipoles and quadrupoles can be found in Refs [29,30] and references contained therein.

The effect of the intermolecular potential in the calculation of C could be also analyzed from the perturbation theory by using a hard-sphere system as a reference system (unperturbed system) and an additional Lennard-Jones potential (perturbed system) to obtain an equivalent potential formed by a hard-sphere potential with a temperature dependent diameter. For details see Ref [31] and the papers cited therein. In this paper we use an approximation of the radial distribution function based on the Ornstein-Zernike equation [32], which allows using the internal energy and pressure equations for the calculation of the third virial coefficient.