Physical and chemical properties of natural fluids are used to understand geological processes in crustal and mantel rock. The fluid phase plays an important role in processes in diagenesis, metamorphism, deformation, magmatism, and ore formation. The environment of these processes reaches depths of maximally 5 km in oceanic crusts, and 65 km in continental crusts, e.g. [1, 2], which corresponds to pressures and temperatures up to 2 GPa and 1000 ˚C, respectively. Although in deep environments the low porosity in solid rock does not allow the presence of large amounts of fluid phases, fluids may be entrapped in crystals as fluid inclusions, i.e. nm to µm sized cavities, e.g. , and fluid components may be present within the crystal lattice, e.g. . The properties of the fluid phase can be approximated with equations of state (Eq. 1), which are mathematical formula that describe the relation between intensive properties of the fluid phase, such as pressure (
This pressure equation can be transformed according to thermodynamic principles , to calculate a variety of extensive properties, such as entropy, internal energy, enthalpy, Helmholtz energy, Gibbs energy, et al., as well as liquid-vapour equilibria and homogenization conditions of fluid inclusions, i.e. dew point curve, bubble point curve, and critical points, e.g. . The partial derivative of Eq. 1 with respect to temperature is used to calculate total entropy change (
The Gibbs energy (
The chemical potential (
2. Two-constant cubic equation of state
The general formulation that summarizes two-constant cubic equations of state according to van der Waals , Redlich and Kwong , Soave , and Peng and Robinson  is illustrated in Eq. 13 and 14, see also . In the following paragraphs, these equations are abbreviated with
The advantage of a cubic equation is the possibility to have multiple solutions (maximally three) for volume at specific temperature and pressure conditions, which may reflect coexisting liquid and vapour phases. Liquid-vapour equilibria can only be calculated from the same equation of state if multiple solution of volume can be calculated at the same temperature and pressure. The calculation of thermodynamic properties with this type of equation of state is based on splitting Eq. 14 in two parts (Eq. 25), i.e. an ideal pressure (from the ideal gas law) and a departure (or residual) pressure, see also .
Eqs. 31 and 32 already include the assumption that the parameters
3. Thermodynamic parameters
The limits of integration are defined as a reference ideal gas at
The entropy change that is caused by a volume change of ideal gases corresponds to the second term on the right-hand side of Eqs. 36 and 37. This term can be used to express the behaviour of an ideal mixture of perfected gases. Each individual gas in a mixture expands from their partial volume (
Finally, the entropy of fluid phases containing gas mixtures at any temperature and total volume according to the two-constant cubic equation of state is given by Eq. 44 for
The subscripts "1" for the upper limit of integration is eliminated to present a pronounced equation. The standard state entropy (
where si0 is the molar entropy of a pure component
Enthalpy (Eq. 52 for
The Helmholtz energy equation (Eqs. 55, 56, and 57) is used for the definition of chemical potential (
The fugacity coefficient (
The stability limit of a fluid mixture can be calculated with two-constant cubic equations of state, e.g. see . This limit is defined by the spinodal line, i.e. the locus of points on the surface of the Helmholtz energy or Gibbs energy functions that are inflection points, e.g. see  and references therein. The stability limit occurs at conditions where phase separation into a liquid and vapour phase should take place, which is defined by the binodal. Metastability is directly related to spinodal conditions, for example, nucleation of a vapour bubble in a cooling liquid phase within small constant volume cavities, such as fluid inclusions in minerals (< 100 µm diameter) occurs at conditions well below homogenization conditions of these phases in a heating experiment. The maximum temperature difference of nucleation and homogenization is defined by the spinodal. In multi-component fluid systems, the partial derivatives of the Helmholtz energy with respect to volume and amount of substance of each component can be arranged in a matrix that has a determinant (
This matrix is square and contains a specific number of columns that is defined by the number of differentiation variables, i.e. volume and number of components in the fluid mixture minus 1. The individual components of this matrix are defined according to Eqs. 69, 70, 71, 72, 73, and 74. The exact definition of these components according to two-constant cubic equations of state can be obtained from the web site http://fluids.unileoben.ac.at (see also ).
The determinant in Eq. 68 is calculated with the Laplacian expansion that contains "
The spinodal curve, binodal curve and critical point of a binary CO2-CH4 mixture with
5. Pseudo critical point
The pseudo critical point is defined according to the first and second partial derivatives of pressure with respect to volume (Eqs. 31 and 32). This point is defined in a
The solution of this cubic equation can be obtained from its reduced form, see page 9 in :
The values of
Any temperature dependency of the
6. Critical point and curve
The critical point is the highest temperature and pressure in a pure gas system where boiling may occur, i.e. where a distinction can be made between a liquid and vapour phase at constant temperature and pressure. At temperatures and pressures higher than the critical point the pure fluid is in a homogeneous supercritical state. The critical point of pure gases and multi-component fluid mixtures can be calculated exactly with the Helmholtz energy equation (Eqs. 55-57) that is obtained from two-constant cubic equations of state, e.g. see [17, 18], and it marks that part of the surface described with a Helmholtz energy function where two inflection points of the spinodal coincide. Therefore, the conditions of the spinodal are also applied to the critical point. In addition, the critical curve is defined by the determinant (
The number of rows in Eq.97 is defined by the differentiation variables volume and number of components minus 2. The last row is reserved for the partial derivatives of the determinant
The derivatives of the spinodal determinant (Eqs. 98-100) are calculated from the sum of the element-by-element products of the matrix of "cofactors" (or adjoint matrix) of the spinodal (Eq. 101) and the matrix of the third derivatives of the Helmholtz energy function (Eq. 102).
An example of a calculated critical curve, i.e. critical points for a variety of compositions in a binary fluid system, is illustrated in Figure 2. The prediction of critical temperatures of fluid mixtures corresponds to experimental data [16, 19], whereas calculated critical pressures are slightly overestimated at higher fraction of CH4. This example illustrates that the
7. Mixing rules and definitions of ζ1 and ζ2
All modifications of the van-der-Waals two-constant cubic equation of state  have an empirical character. The main modifications are defined by Redlich and Kwong, Soave and Peng and Robinson (see Table 1), and all modification can by summarized by specific adaptations of the values of
The two-constant cubic equation of state can be applied to determine the properties of fluid mixtures by using "
These mixing rules have been subject to a variety of modifications, in order to predict fluid properties of newly available experimental data of mixtures. Soave  and Peng and Robinson  modified Eq. 108 by adding an extra correction factor (Eq. 109).
8. Experimental data
As mentioned before, modifications of two-constant cubic equation of state was mainly performed to obtain a better fit with experimental data for a multitude of possible gas mixtures and pure gases. Two types of experimental data of fluid properties were used: 1. homogeneous fluid mixtures at supercritical conditions; and 2. immiscible two-fluid systems at subcritical conditions (mainly in petroleum fluid research). The experimental data consist mainly of pressure, temperature, density (or molar volume) and compositional data, but can also include less parameters. Figure 3 gives an example of the misfit between the first type of experimental data for binary CO2-CH4 mixtures  and calculated fluid properties with
Experimental data of homogeneous supercritical gas mixtures in the ternary CO2-CH4-N2 system  are compared with the two-constant cubic equations of state in Table 4. The
|0.8||0.1||0.1||56.64||64.61 (14.1%)||54.90 (-3.1%)||57.94 (2.3%)||53.59 (-5.4%)|
|0.8||0.2||0.2||58.92||65.81 (11.7%)||56.61 (-3.9%)||59.61 (1.2%)||56.93 (-6.1%)|
|0.4||0.3||0.3||61.08||67.08 (9.6%)||58.27 (-4.6%)||61.12 (0.1%)||56.93 (-6.8%)|
|0.2||0.4||0.4||62.90||68.28 (8.6%)||59.83 (-4.9%)||62.42 (-0.8%)||58.28 (-7.3%)|
Figure 3 and Table 4 illustrate that these modified two-constant cubic equations of state still need to be modified again to obtain a better model to reproduce fluid properties at sub- and supercritical conditions.
9. Modifications of modified equations of state
The number of publications that have modified the previously mentioned two-constant cubic equations of state are numerous, see also , and they developed highly complex, but purely empirical equations to define the parameters
9.1. Chueh and Prausnitz 
The constant values in the definition of
The mixing rules in Eqs. 106-108 were further refined by arbitrary definitions of critical temperature, pressure, volume and compressibility for fluid mixtures.
The prediction of the properties of homogeneous fluids at supercritical conditions (Table 5) is only slightly improved compared to
|0.8||0.1||0.1||56.64||56.42 (-0.4%)||55.96 (-0.6%)||56.84 (0.4%)||56.53 (-0.2%)|
|0.8||0.2||0.2||58.92||57.85 (-1.8%)||57.68 (-2.1%)||59.43 (0.9%)||58.81 (-0.2%)|
|0.4||0.3||0.3||61.08||59.21 (-3.1%)||59.17 (-3.1%)||61.67 (1.0%)||60.79 (-0.5%)|
|0.2||0.4||0.4||62.90||60.44 (-3.9%)||60.38 (-4.0%)||63.45 (0.9%)||62.40 (-0.8%)|
The equation of Holloway [25, 26] is another modification of the
Table 5 illustrates that the equation of Holloway  is not improving the accuracy of predicted properties of supercritical CO2-CH4-N2 fluids, compared to Chueh-Prausnitz  or
Experimental data, including molar volumes of binary H2O-CO2 fluid mixtures at supercritical conditions [30, 31, 32] are used to estimate fugacities of H2O and CO2 according to Eq. 118 (compare Eq. 10).
The dashed line in Figure 6 is calculated with another type of equation of state: a modification of the Lee-Kesler equation of state  that is not treated in this manuscript because it is not a two-constant cubic equation of state. Fugacity estimations of H2O are similar according to both equations, and reveal only a minor improvement for the two-constant cubic equation of state . The experimental data to determine fugacity of CO2 in this fluid mixture is inconsistent at relative low pressures (< 100 MPa). The calculated fugacity  is approximately compatible with the experimental data from [31, 32].
9.3. Bowers and Helgeson  and Bakker 
Most natural occurring fluid phases in rock contain variable amounts of NaCl, which have an important influence on the fluid properties. Bowers and Helgeson  modified the
10. Application to fluid inclusion research
Knowledge of the properties of fluid phases is of major importance in geological sciences. The interaction between rock and a fluid phase plays a role in many geological processes, such as development of magma , metamorphic reactions  and ore formation processes . The fluid that is involved in these processes can be entrapped within single crystal of many minerals (e.g. quartz), which may be preserved over millions of years. The information obtained from fluid inclusions includes 1. fluid composition; 2. fluid density; 3. temperature and pressure condition of entrapment; and 4. a temporal evolution of the rock can be reconstructed from presence of various generation of fluid inclusions. An equation of state of fluid phases is the major tool to obtain this information. Microthermometry  is an analytical technique that directly uses equations of state to obtain fluid composition and density of fluid inclusions. For example, cooling and heating experiment may reveal fluid phase changes at specific temperatures, such as dissolution and homogenization, which can be transformed in composition and density by using the proper equations of state.
The calculation method of fluid properties is extensive and is susceptible to errors, which is obvious from the mathematics presented in the previous paragraphs. The computer package FLUIDS [6, 40, 41] was developed to facilitate calculations of fluid properties in fluid inclusions, and fluids in general. This package includes the group "Loners" that handles a large variety of equations of state according to individual publications. This group allows researchers to perform mathematical experiments with equations of state and to test the accuracy by comparison with experimental data.
The equations of state handled in this study can be downloaded from the web site http://fluids.unileoben.ac.at and include 1. "LonerW" ; 2. "LonerRK" ; 3. "LonerS" ; 4. "LonerPR" ; 5. "LonerCP" ; 6. "LonerH" [25, 26, 27]; and 7. "LonerB" [28, 34]. Each program has to possibility to calculate a variety of fluid properties, including pressure, temperature, molar volume, fugacity, activity, liquid-vapour equilibria, homogenization conditions, spinodal, critical point, entropy, internal energy, enthalpy, Helmholtz energy, Gibbs energy, chemical potentials of pure gases and fluid mixtures. In addition, isochores can be calculated and exported in a text file. The diagrams and tables presented in this study are all calculated with these programs.
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