Thermodynamic properties partial derivatives, evaluation

Actually, the various equations listed in this section are insufficient to perform the complete calculation since one would first calculate the density of H2O through eq. 8.12 or 8.14. Equation 8.14 in its turn involves the partial derivative of the Helmholtz free energy function 8.15. Moreover, the evaluation of electrostatic properties of the solvent and of the Bom functions (o, Q, Y, X involve additional equations and variables not given here for the sake of brevity (eqs. 36, 40 to 44, 49 to 52 and tables 1 to 3 in Johnson et ah, 1991). In spite of this fact, the decision to outline here briefly the HKF model rests on its paramount importance in geochemistry. Moreover, most of the listed thermodynamic parameters have an intrinsic validity that transcends the model itself... 

When taking these partial derivatives it must be remembered that, in general, the molar densities, the mass transfer coefficients, and thermodynamic properties are functions of temperature, pressure, and composition. In addition, H is a function of the molar fluxes. We have ignored most of these dependencies in deriving the expressions given above. The important exception is the dependence of the K values on temperature and composition that cannot be ignored. The derivatives of the K values with respect to the vapor mole fractions are zero in this case since the model used to evaluate the K values is independent of the vapor composition. 

Newton s method requires the evaluation of the partial derivatives of all equations with respect to all variables. The partial derivatives of thermodynamic properties with respect to temperature, pressure, and composition are most awkward to obtain (and the ones that have the most influence on the rate of convergence). Since pressure is an unknown variable in this model, the derivatives of K values and enthalpies with respect to pressure must be evaluated. Neglect of these derivatives (even though they are often small) can lead to convergence difficulties. 

The total vapour pressure of selenium in equilibrium with a mixture of Au(cr) and a-AuSe was measured in the temperature range 505 to 602 K using the Knudsen effusion method in [71RAB/RAU]. The result is presented as the partial pressure of Sc2(g) at equilibrium and the enthalpy of formation and entropy at 298.15 were evaluated to be Af//°(AuSe, a, 298.15 K) = -7.9 kJ-moP and S°(AuSe, a, 298.15 K) = 80.8 J K -moP, respectively. However, Sc2(g) is not the major species in the gas phase at the temperatures and total pressures of the study, and it is not clearly stated how the partial pressure was derived from the experiments. The result is therefore questionable and impossible to re-evaluate using the selected thermodynamic properties of the gaseous selenium species. No values were selected by the review. 

The vapour pressure measurements gave an enthalpy of reaction Af// (1300 K) = (259.6 4.0) kJ-mol" for the reaction WSc2(cr) W(cr) + Sc2(g). This value does not agree with the partial pressures given in Table 2 of the paper. The review re-evaluated the data and derived the thermodynamic properties of WSe2(cr) at... 

The thermodynamic potentials, being a system s state functions of the corresponding (natural) parameters, arc of special importance in the system state description, their partial derivatives being the parameters of the system as well. The equalities between th( second mixed derivatives are a property of the state functions and lead to relation-ship.s between the system parameters (the Gibbs-Helmholtz equations). Hence, once any thermodynamic potential (usually, the Gibbs or the Helmholtz one) has been evaluated, by means of either simulation or experiment, this means the complete characterization of the thermodynamic properties of the system. 

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