Perturbation theory based equation of state for polar molecular fluids: I. Pure fluids

doi:10.1016/S0016-7037(02)01347-9

Geochimica et Cosmochimica Acta

Volume 67, Issue 13, 1 July 2003, Pages 2397-2414

https://doi.org/10.1016/S0016-7037(02)01347-9 Get rights and content

Abstract

Based on the thermodynamic perturbation theory an equation of state (EOS) for molecular fluids has been formulated which can be used for many fluid species in geological systems. The EOS takes into account four substance specific parameters. These are the molecular dipole moment, the molar polarizability and the two parameters of the Lennard-Jones potential. For many fluids these parameters can be evaluated directly or indirectly from experimental measurements. In the absence of direct experimental determinations, as a first approximation, for a pure fluid the parameters of the Lennard-Jones potential can be evaluated using the critical temperature and the critical density if for polar molecules in addition the dipole moment is known with reasonable accuracy. The EOS with its model potential has the appropriate asymptotic behaviour at high pressures and temperatures and can be used to calculate both vapor-liquid equilibria and thermodynamic properties of single phase fluids up to at least 10 GPa and 2000 K. Currently, parameters for 98 inorganic and organic compounds are available. In this article the EOS for pure fluids is presented. In a further communication the EOS is extended to fluid mixtures (Churakov and Gottschalk, 2003).

Introduction

Fluids are important for many geological and petrological processes in the Earth’s crust and mantle. For the evaluation of such processes the thermodynamic properties of these fluids are needed. The required free energies, i.e., fugacities, of the fluid components at high pressures and temperatures are usually derived using equations of state (EOS). Several approaches are currently used to derive equations of state. For geological purposes these are sophisticated modifications to the van der Waals equation (e.g., Holloway 1976, Jacobs and Kerrick 1981, Kerrick and Jacobs 1981, Halbach and Chatterjee 1982, Holland and Powell 1991, Grevel and Chatterjee 1992, Anderko and Pitzer 1993a, Anderko and Pitzer 1993b, Duan et al 1995b, virial equations of state (e.g., Saxena and Fei 1987, Saxena and Fei 1988, Duan et al 1992a, Duan et al 1992b, Sterner and Pitzer 1994 and other equations based on the thermodynamic perturbation theory (e.g., Shmulovich et al 1982, Ree 1984. In all these EOS the required coefficients are determined primarily using experimental results (e.g., PVT-data, phase equilibria). In addition to experimental results, fluid properties can be derived or at least approximated by molecular dynamical calculations considering appropriate models or molecular interaction potentials (e.g., Brodholt and Wood 1990, Brodholt and Wood 1993a, Brodholt and Wood 1993b, Belonoshko and Saxena 1991a, Belonoshko and Saxena 1991b, Belonoshko and Saxena 1992, Duan et al 1992c, Duan et al 1995a, Duan et al 1996, Duan et al 2000, Kalinichev and Heinzinger 1995, Destrigneville et al 1996.

All types of EOS mentioned are based on physical models and require various coefficients. If the physical description used by an EOS is perfect all required coefficients will have an exact physical meaning. Molecular interactions are manifold, however. Types of interactions involve repulsion due to a finite size of the molecules, attraction between permanent and induced dipoles, quadrupole and higher multipoles, and dispersion forces. In the case of permanently charged molecules Coulomb interaction forces must be taken into account. Any EOS uses simplifications by grouping sets of similar interactions into specific terms and therefore the precise physical significance of the required coefficients fades or is even lost. In many cases fitting experimental data to an EOS reveals coefficients having complicated dependencies with respect to temperature and either pressure or volume and therefore are often described using empirical functions such as polynomials. The use of arbitrary functions with questionable or no physical meaning transforms these EOS into empirical equations. Such empirical approaches generally offer relatively simple explicit expressions which are very attractive from a computational point of view. However, the determination of the coefficients of such EOS requires a large experimental database over the entire P-T range for which the equation is intended to be used. Generally, it can not be safely extrapolated. Furthermore, because of the applied simplifications, these EOS are often unable to reproduce available experimental data with the required accuracy. The accuracy can be improved by introducing additional totally empirical parameters, which can dramatically degrade the reliability of any extrapolation, however. All EOS based on the van der Waals equation of state are therefore more or less empirical.

An alternative to EOS derived from the van der Waals equation are equations based on perturbation theory. The general idea of the thermodynamic perturbation theory can be described as follows (e.g., Gray and Gubbins, 1984). The potential of intermolecular interaction u of a real fluid can be always expressed as the sum of a model dependent interaction potential u₀ (the reference potential) and the residual potential u₁ which considers the differences of an interaction between the real fluid and the reference model. For any given u and u₀, a parametrical potential depending on a perturbation parameter λ can be formulated: $V(λ)=u_{0} + λ (u – u_{0})=u_{0} + λ u_{1}$

It follows that V(0)=u₀ and V(1)=u. The configurational part of the Helmholtz free energy A, i.e., the part of energy depending only on intermolecular or interatomic interaction forces, for a thermodynamic system of N molecules can be expressed at constant T and V by Eqn. 2 using the configurational integral z (Eqn. 3) which is the integral overall possible configurations Γ of N molecules. $A=– RT ln (z)$ $z= ∫ exp – u(Γ) RT dΓ$

If in Eqn. 3 u will be substituted by the parameterized potential V(λ) (Eqn. 1), the Helmholtz free energy A can be expressed as a power series of the perturbation parameter λ: $A(λ)=A_{λ=0} + λ ∂ A ∂λ_{λ=0} + λ^{2} 2! ∂^{2} A ∂λ^{2}_{λ=0} + λ^{3} 3! ∂^{3} A ∂λ^{3}_{λ=0} +…$

For example as a first approximation, the second and higher order terms can be neglected. Noting that u=V(λ=1) , the first order approximation of the Helmholtz free energy for the system of N molecules described by the potential u can be obtained as $A=–RT ln (z_{0})+z_{0}^{–1} ∫ u_{1} exp – u_{0} (Γ) RT dΓ=A_{0} +A_{1}$ $z_{0} = ∫ exp u_{0} (Γ) RT dΓ$

Thus the energy A of a thermodynamic system with the potential u can be expressed as the sum of the energy A₀ of the reference system of molecules with the interaction potential u₀ and the perturbation term A₁. The approximation requires, however, that u₁ is small and the reference potential u₀ is therefore close to the interaction potential in the real fluid. The accuracy can be improved, if the higher order terms in Eqn. 4 will be considered.

Strict application of the perturbation theory requires that the integral representing the term A₁ in Eqn. 5 is solved. Some approaches (e.g., Anderko and Pitzer 1993a, Anderko and Pitzer 1993b, Duan et al 1995b avoid this complication by combining A₀ for the reference system with an empirical approximation for the perturbation term A₁ without strict assumptions on the character of the intermolecular interaction in the fluid. But an EOS formulated in such a way becomes again empirical in character and the capability of extrapolation will be lost. One aim of this study is to formulate an EOS which is suitable for extrapolations to high pressures and temperatures. Therefore we insist on using a type of equation like Eqn. 5 for the formulation of the EOS which is based on the strict physical model of intermolecular interaction. Because it is an approximation we realize that this model may also deviate from the interaction of real molecules under extrapolation which will lead to deviations of the calculated thermodynamic properties. However, these deviations are expected to be smaller than in other approaches, because the model is based on physical assumptions.

Most available experimental data are for pure fluids. PVT-data for fluid mixtures are rare and extraction of fluid properties of fluid mixtures requires interpolation. As a first approximation the van der Waals one-fluid theory is commonly used and parameters of pure fluids are combined algebraically. Their application requires, however, that for each fluid component the EOS is exactly of the same type. Most published EOS are for one or only few fluid components and the EOS from different publications are rarely of the same type. Therefore these EOS can not be used to derive fluid properties in complex mixtures which have not been studied experimentally. In addition the commonly used mixing rules are not strictly in accordance with statistical mechanics. Dohrn and Prausnitz (1990) showed in a systematic study of different fluid mixtures that van der Waals mixing rules may predict an unrealistic physical behaviour such as increasing volume with pressure. Nevertheless, the other aim of this study is to formulate an EOS which is suitable for as many components as possible, so that this EOS can be applied in a further step to complex fluid mixtures (Churakov and Gottschalk, 2003).

In this paper we formulate an EOS for polar and non-polar molecules using perturbation theory with the Lennard-Jones potential as the reference system. The EOS is applied to 98 pure inorganic and organic fluid components and has the following attractive features. It needs a minimum number of fitting parameters and its asymptotic high pressure and temperature behaviour predicts the fluid properties physically correct at these conditions. The form of the EOS is identical for all 98 considered fluid components and the EOS can be easily extended to the complex fluid mixtures using mixing rules. Fluid mixtures will be covered in a second communication (Churakov and Gottschalk, 2003). An implementation of the EOS for pure fluids as well mixtures is availabe in the form of a Mathematica™ package as an electronic supplement.

Section snippets

Thermodynamic model

In this section only the principal aspects of the proposed EOS are discussed. Details for the computational implementation can be found in the Appendix. Table 1 lists the used symbols.

The main contribution to the potential interaction energy between two polar molecules i and j (Eqn. 7) is given by the sum of the Lennard-Jones potential (Eqn. 8), the potential due to dipole-dipole (Eqn. 9) and the dipole-induced dipole interaction (Eqn. 10) which is the generalized Stockmayer potential (e.g.,

Limitations of the proposed EOS

The EOS presented above includes the following simplifications. First the molecules are treated as Lennard-Jones spheres with a permanent point-dipole moment. However typical polyatomic molecules also possess quadrupole and higher order moments leading to dipole-quadrupole, quadrupole-quadrupole, induced dipole–quadrupole and other interactions. The polarizability of complex molecules depends substantially on their orientation with respect to the applied electric field, while the EOS takes only

Derivation methods of the potential parameters for pure fluids

The EOS described above requires for each pure fluid the four parameters ε, σ, μ, α to be evaluated using experimental data. If thermodynamic data (PVT-data, phase equilibria) for a fluid are available it is always advantageous to fit ε, σ, μ and α directly to these experimental results. As an alternative these potential parameters can be derived from other experimental sources. Values for the electric dipole moment μ and the polarizability α of molecules in fluids are readily available from

Results and discussion

In the following specific details and parameters required for the EOS for various molecular species are discussed. Species which might be interesting for geochemical purposes in a broad sense and for which the required critical data could be found are included. Various important species like B(OH)₃ and Si(OH)₄ are omitted because required data seem no to be available. On the other hand species like BF₃ and SiF₄ are included because they might be important for volcanic fumaroles. Some of the

Conclusions

It has been shown that even strong polar molecules like water can be quite accurately described by a four parameter equation of state based on the rigorous consideration of dipole-dipole and Lennard-Jones interactions. Particularly above the critical point the EOS shows an excellent extrapolation behaviour to high pressures and temperatures. However, at near critical conditions the description of the thermodynamic properties of highly polar fluids is relatively poor. It is obvious that in this

Acknowledgements

The authors are grateful to A. G. Kalinichev for useful discussions and kindly providing the source code for the EOS of a Lennard-Jones fluid. We also thank W. Heinrich for support and standing up for funding.

Associate editor: R. C. Burruss

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^†: Present address: Centro Swizzero di Calcolo Scientifico, Via Cantonale, CH-6928 Manno, Switzerland.

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Perturbation theory based equation of state for polar molecular fluids: I. Pure fluids

Abstract

Introduction

Section snippets

Thermodynamic model

Limitations of the proposed EOS

Derivation methods of the potential parameters for pure fluids

Results and discussion

Conclusions

Acknowledgements

Geochim. Cosmochim. Acta

Geochim. Cosmochim. Acta

Geochim. Cosmochim. Acta

Geochim. Cosmochim. Acta

Geochim. Cosmochim. Acta

Earth Planet. Sci. Lett.

Geochim. Cosmochim. Acta

Chem. Geol.

Fluid Phase Equilib

Geochim. Cosmochim. Acta

Geochim. Cosmochim. Acta

Geochim. Cosmochim. Acta

Geochim. Cosmochim. Acta

Geochim. Cosmochim. Acta

Geochim. Cosmochim. Acta

Geochim. Cosmochim. Acta

Chem. Eng. Sci.

Phys. Lett.

Geochim. Cosmochim. Acta

Geochim. Cosmochim. Acta

Fluid Phase Equilib

Geochim. Cosmochim. Acta

Fluid Phase Equilib

Fluid Phase Equilib

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Critical constants, boiling points and melting points of selected compounds

Role of repulsive forces in liquidsThe optimized random phase approximation

J. Chem. Phys.

Experimental determination of CO2-H2O activity-composition relations at 600-1000 °C and 6-14 kbar by reversed decarbonation and dehydration reactions

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Molecular dynamics simulations of the properties of CO2-H2O mixtures at high pressures and temperatures

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Equilibria in the system MgO-SiO2-H2OExperimental determination of the stability of Mg-anthopyllite

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