1. Standard molar thermodynamic functions of substances (dependent components)
1.1. Cross-Checking Procedure for G^{o}_{Tr} , H^{o}_{Tr} and S^{o}_{Tr}
2. Temperature and pressure corrections (DComp data format)
2.1. Common techniques for condensed substances and ideal gases
3. Calculation of molar properties of substances via reactions (ReacDC data format)
4. Calculation of T,P corrected molar properties for ReacDC -defined chemical species
The GEM algorithms [Karpov et al., 1997; 2001] require at least two kinds of input data for each dependent component included into the definition of chemical system. (i) Main input is the formal (elemental) stoichiometry, expressed as a formula in either DCform or RDform fields of DComp or ReacDC window, respectively. (ii) The molar (partial molal) apparent Gibbs energy function g_{(T,P) }at temperature and pressure of interest. Optionally, the molar (partial molal) volume V_{(T,P) } is needed at input because, if provided for all DCs, it allows to calculate the total volume of the system and each phase at equilibrium.
Other molar thermodynamic functions of DCs such as the molar (partial molal) apparent enthalpy function i_{(T,P)}, entropy S_{(T,P)}, internal energy U_{(T,P)}, Helmholtz energy F_{(T,P)} are needed for minimization of five other thermodynamic potentials than Gibbs energy [cf. Karpov et al., 2002]. This parametric minimization is not available in GEM-Selektor version 2-PSI, but we expect implementing it in the forthcoming major versions.
The molar entropy S^{o}_{298} (or enthalpy H^{o}_{298}) and isobaric heat capacity Cp_{298} functions of a substance at reference temperature Tr (25 ^{o}C or 298.15 K) and pressure Pr (1 bar) are in any case needed for performing temperature corrections for the g_{(T,P)} function; the standard molar volume V^{o}_{298} and its change with temperature and pressure is needed for pressure correction of g_{(T,P)}. Three "energy" standard molar functions G^{o}_{298}, H^{o}_{298} and S^{o}_{298} depend on each other (see below), so, in fact, it is enough to provide any two of them.
The same holds for the apparent molar (partial molal) ehthalpy i^{o}_{Tr} and entahlpy of formation H^{o}_{f,Tr}. However, the molar (partial molal) entropy s^{o}_{Tr} is taken here numerically equal to the absolute, or third-law entropy S^{o}_{a,Tr}, whereas the entropy of formation S^{o}_{f,Tr} is
G^{o}_{f,Tr} = H^{o}_{f,Tr} - Tr S^{o}_{f,Tr} (1).
Absolute entropy is related to the entropy of formation by
S^{o}_{f,Tr} = S^{o}_{a,Tr} - sum_{i}(ai S^{o}_{a,i}) , i = 1,...,N (2),
where S^{o}_{a,i} stands for the absolute entropy of
i-th chemical element in standard state at Tr, ai is a stoichiometry
coefficient of i-th element in the formula of the compound (i.e. ai is
the number of moles of element per one mole of the compound). Equations
(1) and (2) can be combined, permitting calculation of any one of
values G^{o}_{Tr}, H^{o}_{Tr} and S^{o}_{Tr}
from the two
other known (S^{o}_{a,i} values for elements are
provided
in the IComp records). For example, the
molar
enthalpy is calculated as
H^{o}_{Tr} = G^{o}_{Tr} + Tr (S^{o}_{Tr} - sum_{i}(ai S^{o}_{a,i}) ), i = 1, ... ,N (3).
This equation is also used in cross-checking procedure that can be executed in the GEMS program when thermodynamic data for gases, solids, or aqueous species are entered or imported into DComp data records. In the case when all three values are entered, the enthalpy is calculated from the above equation (3) and compared to the input H^{o}_{Tr} value:
dev(H^{o}) = | H^{o}_{Tr}(input) - H^{o}_{Tr}(calculated) | (4).
The program will issue a warning message if dev(H^{o}) > 57 J mol^{-1} and give the user a choice - either to accept the H^{o}_{Tr}(calculated) value or to check other input parameters. In this way, the internal data consistency is maintained.
The standard molar properties of dependent components must be corrected from reference (Tr, Pr) to temperature and pressure of interest (T,P) before using them in GEM calculations of chemical equilibria. Depending on the phase state and nature of dependent components, such corrections can be performed using different techniques, each with its own additional input data. Some common methods are implemented and incorporated into the GEMS code. They will be selected and executed automatically according to the method codes set during creation or re-making of a DComp record.
In the majority of GEM-based modelling codes or programs for plotting phase equilibrium diagrams, temperature corrections for standard molar properties of minerals, melt components, non-electrolyte liquids and gases are performed using the integration of heat capacity function Cp = f(T) over one or more temperature intervals. The Cp = f(T) function is represented by several empirically fitted coefficients at different powers of absolute temperature T. One should be aware that, in general, such coefficients are not applicable outside of their temperature intervals and therefore, the extrapolated values of g^{o}_{T} should not be used in calculation of chemical equilibria as large errors may be introduced. The GEMS code will automatically mark such "extrapolated" data with an 'e' character, visible in the TPmark column on the "Thermodynamic Data" window, which is accessible through the "Single Equilibrium System" dialog.
Details on temperature corrections using Cp = f(T) dependences can be found in the file T-corrections.pdf.
At elevated pressures, thermodynamic data (g^{o}_{T} values) must be corrected from the reference pressure Pr = 1 bar to pressure of interest. For condensed substances (solids, liquids), the minimum required information is the standard molar (partial molal) volume at Pr, Tr; over moderate temperature and pressure intervals, the molar volume can be assumed constant. However, at wider intervals, the thermal expansion and compressibility of substances must be considered. There is a variety of approaches for pressure corrections using variable molar volume V_{(T,P)} = f(T,P). Simpler techniques, based on empirical coefficients of the V = f(T,P) dependence, are being replaced at present by an extrapolatable method based on the Birch-Murnaghan equation of state.
In the file P-corrections.pdf, you can find details on pressure corrections as implemented in the GEMS code. The pressure corrections related to data for minerals from a companion M.Gottschalk's database will be described in a separate document P-corrections-MGD.pdf (under construction).
Molar thermodynamic properties of gases and partial molal properties of aqueous ions and complexes strongly depend on both temperature and pressure. Temperature corrections for pure ideal gases are made using the Maier-Kelly coefficients or any similar form of Cp = f(T) dependence (see above). It is also possible to regress formal Cp = f(T) dependences for aqueous and complexes, as it has been done in some early versions of Selektor code [Dorogokupets et al., 1988]. This approach has a disadvantage that thermodynamic data can be corrected only along the so-called "saturated vapor curve" for pure H_{2}O that corresponds to equilibrium partial pressures of water vapor at temperatures up to critical point of H_{2}O (Tcr = 374 ^{o}C, Pcr = 221 bar). At this curve, the value of pressure P_{SAT} can be calculated for any temperature of interest T.
At present, the most widespread and practical method of temperature/pressure corrections of partial molal properties of aqueous species is based on the revised Helgeson-Kirkham-Flowers equation of state (HKF EoS) and the database of its coefficients, provided in the SLOP98 data file [1997SHO/SAS and references therein]. All the necessary subroutines from the SUPCRT92 code [1992JOH/OEL] have been added to the GEM-Selektor code to perform T,P corrections using the HKF EoS coefficients kept in the DComp record format. Details on these calculations can be found in a separate document TP-corrections-HKF.pdf (to be provided) or in papers by H.Helgeson and co-workers.
Pressure corrections for molar thermodynamic properties of non-ideal gases require knowledge of their EoS parameters. The molar volume of pure ideal gas can be found at any temperature of interest from the ideal gas law: PV = RT, where R = 8.31451 J/K/mol is the universal gas constant. As the standard state for gases is defined at Pr = 1 bar at any T, it follows the pressure correction for the molar properties of any pure ideal gas will be g_{(T,P)} = g^{o}_{T} + RT ln P. The RT ln P term is actually included into the expressions for primal chemical potentials calculated in the GEM algorithm (cf. [Karpov et al., 1997; 2002].
However, especially at moderate temperatures and elevated pressures, gases show strongly non-ideal behavior. Molar volumes of pure gases deviate from the ideal gas lew; there are also complex non-ideal effects of mixing in multi-component fluids . The non-ideality of pure gases can be expressed using the concept of fugacity via the fugacity coefficient fi: g_{(T,P)} = g^{o}_{T} + RT(ln P + ln fi). To calculate the fugacity or the fugacity coefficient, one must know the functional form of fi = f(T,P) for each gas, determined from the experimental compressibility data or predicted theoretically. Many such dependencies follow from various EoS for non-ideal gases. In the past, the fugacities of pure gases were often calculated using the "correspondence states" theory (cf. [Nordstrom and Munos, 1994]), using critical parameters of gases Tcr (temperature at critical point), Pcr (pressure), Vcr (critical volume), and Zcr (compressibility factor). This calculation has also been implemented in the FGL module of Selektor codes (details in a separate document TP-corrections-FGL.pdf, to be provided). Note that the FGL module is disabled in this version of GEM-Selektor code.
Recently, new predictive and extrapolative EoS for gases (fluids) have been developed, also capable of predicting mixing properties of multi-component fluids. Such a "perturbation theory based" EoS [Churakov and Gottschalk, 2003a,b] is implemented in the GEM-Selektor code as a third-party contribution in connection with the companion chemical thermodynamic database of M.Gottshalk (http://www.gfz-potsdam.de/pb4/pg1/dataset/index.html). Details about these calculations are given in a separate document TPX-corrections-CG-EoS.pdf .
This technique is indispensable for surface- or aqueous complexes, amorphous minerals or solid solution end members, isotopic forms of substances that cannot be described using the HKF EoS or for which the full set of standard molar properties is not known.
Usually, such dependent components form the so-called "application-specific" extensions of thermodynamic database. By constructing the so-called isoelectric or isocoulombic reactions, reasonable temperature and pressure extrapolations of molar properties of such DCs can be obtained even in cases when only logK of reaction is known at 25 ^{o}C.
Simple algebraic calculations required for this purpose are described in the file T-corrections-Reac.pdf. The ReacDC format can also be used for the fast retrieval of properies of arbitrary reactions (in 3-term extrapolation) between chemical species already presented in GEMS database as DComp and/or ReacDC records.
Details on temperature corrections for thermodynamic properties of reactions and molar properties of the "new" reaction-defined species, as well as simple pressure corrections, can be found in the file T-corrections-Reac.pdf. This way of representing thermodynamic properties is especially useful for some aqueous complexes and for surface species, which cannot be described using conventional thermochemical or EoS based methods. However, pressure corrections in this case are limited to the simplest constant-molar-volume case; for surface species, molar volumes are not known, thus the pressure correction techniques are not available and still have to be developed.
HKF EoS (see Shock et al., Geochimica et
Cosmochimica
Acta 1989, 1992, 1997).
SUPCRT92 [Johnson et al.,1992]
SPRONS92.DAT file and later extensions including SLOP98.DAT file.
To be completed in Version 3. 2
Last change: 30.04.2013
Copyright (c) 2003-2013
GEMS Development Team.