Thermodynamic modeling calculations

Thermodynamic model calculations, which take into account the current temperature gradient in the Rhine graben and the composition of the altered granite, have shown that the hydrothermal fluids present in the granite cause a new stage of alteration (Komninou Yardley 1997). Model calculations have been performed in order to predict the formation of new... 

While complexation/dissociation of chromium and natural organics has been postulated very often to explain data otherwise not in agreement with thermodynamic model calculations, there is a lack of direct evidence of its significance. Advances in understanding the biogeochemistry of chromium. 

Phase equilibria in systems containing both an anionic and a cationic amphiphile. A thermodynamic model calculation... 

Sufficiently accurate thermodynamic models used for calculating these equilibria are not available In simulation programs. It Is generally not recommended to use the models proposed. Only a specific study based on accurate experimental results and using a model adapted to the case will succeed. 

In some cases the atomic charges are chosen to reproduce thermodynamic properties calculated using a molecular dynamics or Monte Carlo simulation. A series of simulations is performed and the charge model is modified until satisfactory agreement with experiment is obtained. This approach can be quite powerful despite its apparent simplicity, but it is only really practical for small molecules or simple models. 

Thermodynamic models are widely used for the calculation of equilibrium and thermophysical properties of fluid mixtures. Two types of such models will be examined cubic equations of state and activity coefficient models. In this chapter cubic equations of state models are used. Volumetric equations of state (EoS) are employed for the calculation of fluid phase equilibrium and thermophysical properties required in the design of processes involving non-ideal fluid mixtures in the oil and gas and chemical industries. It is well known that the introduction of empirical parameters in equation of state mixing rules enhances the ability of a given EoS as a tool for process design although the number of interaction parameters should be as small as possible. In general, the phase equilibrium calculations with an EoS are very sensitive to the values of the binary interaction parameters. 

In several occasions the overall fit obtained by the CLS estimation could simply be found unacceptable despite the fact that the predictions of erroneous phase separation have been suppressed. In such cases, one should proceed and either modify the employed mixing rules or use a different EoS all together. Of course, the estimation should start from Step 1 once the new thermodynamic model has been chosen. Calculations using... 

Activity coefficient models offer an alternative approach to equations of state for the calculation of fugacities in liquid solutions (Prausnitz ct al. 1986 Tas-sios, 1993). These models are also mechanistic and contain adjustable parameters to enhance their correlational ability. The parameters are estimated by matching the thermodynamic model to available equilibrium data. In this chapter, vve consider the estimation of parameters in activity coefficient models for electrolyte and non-electrolyte solutions. 

We consider Pitzer s model for the calculation of activity coefficients in aqueous electrolyte solutions (Pitzer, 1991). It is the most widely used thermodynamic model for electrolyte solutions. 

Equation 4.26 defines the relationship between the vapor and liquid mole fractions and provides the basis for vapor-liquid equilibrium calculations on the basis of equations of state. Thermodynamic models are required for (/) and [ from an equation of state. Alternatively, Equations 4.21, 4.22 and 4.25 can be combined to give... 

Needless to say, an analysis which will finally allow one to nail down all rates, activation parameters, and equilibrium constants requires a large amount of precise and reliable kinetic data from appropriate experiments, including the determination of isotope effects and the like, as well as a rather sophisticated treatment and solution of the complete kinetic scheme. Then a comparison is necessary between various organosilanes with different types of C-H and C-Si bonds as well as the comparison between the dtbpm and the dcpm ligand systems, not to speak of model calculations in order to understand the molecular origin of the kinetic and thermodynamic numbers. We are presently in the process of solving these problems. 

It has been observed that cobalt may undergo large-scale reconstruction under a synthesis gas environment.27 Reconstruction is a thermodynamically driven process that results in the stabilization of less reactive surfaces. Recent molecular modeling calculations have shown that atomic carbon can induce the clock reconstruction of an fee cobalt (100) surface.28 It has also been postulated and shown with in situ x-ray adsorption spectroscopy (XAS) on cobalt supported on carbon nanofibers that small particles (<6 nm) undergo a reconstruction during FTS that can result in decreased activity.29... 

Considerable use has been made of the thermodynamic perturbation and thermodynamic integration methods in biochemical modelling, calculating the relative Gibbs energies of binding of inhibitors of biological macromolecules (e.g. proteins) with the aid of suitable thermodynamic cycles. Some applications to materials are described by Alfe et al. [11]. 

For the purpose of this case study we will select Isopropyl alcohol as the crystallization solvent and assume that the NRTL-SAC solubility curve for Form A has been confirmed as reasonably accurate in the laboratory. If experimental solubility data is measured in IPA then it can be fitted to a more accurate (but non predictive) thermodynamic model such as NRTL or UNIQUAC at this point, taking care with analysis of the solid phase in equilibrium. As the activity coefficient model only relates to species in the liquid phase we can use the same model with each different set of AHm and Tm data to calculate the solubility of the other polymorphs of Cimetidine, as shown in Figure 21. True polymorphs only differ from each other in the solid phase and are otherwise chemically identical. 

Equilibrium constants calculated from the composition of saturated solutions are dependent on the accuracy of the thermodynamic model for the aqueous solution. The thermodynamics of single salt solutions of KC1 or KBr are very well known and have been modeled using the virial approach of Pitzer (13-15). The thermodynamics of aqueous mixtures of KC1 and KBr have also been well studied (16-17) and may be reliably modeled using the Pitzer equations. The Pitzer equations used here to calculate the solid phase equilibrium constants from the compositions of saturated aqueous solutions are given elsewhere (13-15, 18, 19). The Pitzer model parameters applicable to KCl-KBr-l O solutions are summarized in Table II. 

Using the model parameters of Table II the calculated osmotic coefficient is within 0.15% or better for all solutions investigated. Agreement with the experimental results (17) is within 0.02% or better if ( ci.Br.K = 0.0003 (Table III) instead of zero (Table II). We may conclude from this comparison that the thermodynamic model of Pitzer (Table II) is very realistic. An uncertainty of 0.0003 in i(ic Br K leads to uncertainties of less than 0.4% in log K(x). The largest uncertainty in equilibrium constants may thus be attributed to the original analytical data (j3). 

Thermo-Calc (Sundman et al. 1985, Andersson et al. 2002). ft features a wide spectrum of thermodynamic models, databases and modules making it possible to perform calculations on most problems involving phase equilibria (phase transformation, stable and metastable equilibria, etc.). The calculations are performed using databases produced by an expert evaluation of experimental data. There are thermodynamic databases available for many different systems and applications. 

As for the thermodynamic modelling of Laves phases, this has been successfully performed for several alloy systems. Especially using the sublattice model (see 2.4.2.2) a number of phase diagrams containing Laves phases have been calculated taking into account homogeneity ranges and polytypism. 

It is tempting to relate the thermodynamics of electron-transfer between metal atoms or ions and organic substrates directly to the relevant ionization potentials and electron affinities. These quantities certainly play a role in ET-thermo-dynamics but the dominant factor in inner sphere processes in which the product of electron transfer is an ion pair is the electrostatic interaction between the product ions. Model calculations on the reduction of ethylene by alkali metal atoms, for instance [69], showed that the energy difference between the M C2H4 ground state and the electron-transfer state can be... 

Thermodynamic modelling of solution phases lies at the core of the CALPHAD method. Only rarely do calculations involve purely stoichiometric compounds. The calculation of a complex system which may have literally 100 different stoichiometric substances usually has a phase such as the gas which is a mixture of many components, and in a complex metallic system with 10 or 11 alloying elements it is not unusual for all of the phases to involve solubility of the various elements. Solution phases will be defined here as any phase in which there is solubility of more than one component and within this chapter are broken down to four types (1) random substitutional, (2) sublattice, (3) ionic and (4) aqueous. Others types of solution phase, such as exist in polymers or complex organic systems, can also be modelled, but these four represent the major types which are currently available in CALPHAD software programmes. 

Most of this book concerns the development and application of theoretical thermodynamic models, as these are the basis of the CALPHAD method. However, none of this would be possible without the existence of the computational methods and software which allow these models to be applied in practice. In essence, the issues involved in computational methods are less diverse and mainly revolve around Gibbs energy minimisation. In addition, there are optimiser codes which are used for the thermodynamic assessment of phase equilibria. The essential aim of these codes is to reduce the statistical error between calculated phase equilibria, thermodynamic properties and the equivalent experimentally measured quantities. 

Felmy, A.R. Rustad, J.R. (1998) Molecular statics calculations of proton binding to goethite surfaces Thermodynamic modeling of the surface charging and protonation of goethite in aqueous solution. Geochim. Cosmochim. Acta 62 25—31... 

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