Gibbs energy function

It is commonplace to assume a form of the Gibbs energy function which excludes the pressure variable for solid-state phase transformations, as the magnitude of the PAV term is small at atmospheric pressures. This is of course not the case in geological systems, or if laboratory experiments are deliberately geared to high-pressure environments. Klement and Jayaraman (1966) provide a good review of the data available at the time when some of the earliest CALPHAD-type calculations were made (Kaufman and Bernstein 1970, Kaufman 1974). Much work was also carried out on specific alloy systems such as Fe-C (Hilliard 1963) and the Tl-In system (Meyerhoff and Smith 1963). 

Statistical thermodynamics can provide explicit expressions for the phenomenological Gibbs energy functions discussed in the previous section. The statistical theory of point defects has been well covered in the literature [A. R. Allnatt, A. B. Lidiard (1993)]. Therefore, we introduce its basic framework essentially for completeness, for a better atomic understanding of the driving forces in kinetic theory, and also in order to point out the subtleties arising from the constraints due to the structural conditions of crystallography. 

At constant T and P (easy to experimentally control), the condition for equilibrium is AG = 0. The Gibbs energy function (Eq. 2.11) is a generally more useful function than the internal energy function (Eq. 2.6) that requires constant S and V at equilibrium (AU = 0). 

The AG of the reaction is then calculated in one of two ways (1) the appropriate addition and substraction of AGf for reactants and products, or (2) the calculation of Gibbs energy functions for reactants and products from enthalpy and entropy increments... 

We can define the Gibbs energy function, G in thermodynamics by the equation ... 

Variation of Gibbs Energy Function, G Versus Temperature, T for Solid, Liquid and Gaseous Phases... 

Oxides. Decomposition pressure measurements on the TbO system by Eyring and his collaborators (64) have been supplemented by similar and related studies on the PrO system (46) and on other lanthanide-oxygen systems (43, 44). Extensive and systematic studies of vaporization processes in lanthanide-oxide systems have been undertaken by White, et al. (6, 188,196) using conventional Knudsen effusion measurements of the rates of vaporization of the oxides into high vacuum. Combination of these data with information on the entropies and Gibbs energy functions of reactants and products of the reaction yields enthalpies of reaction. In favorable instances i.e., if spectroscopic data on the gaseous species are available), the enthalpies of formation and the stabilities of previously undetermined individual species are also derived. The rates of vaporization of 17 lanthanide-oxide systems (196) and the vaporization of lanthanum, neodymium, and yttrium oxides at temperatures between 22° and 2700°K. have been reported (188). 

Key words critically evaluated data enthalpy enthalpy of formation entropy equilibrium constant of formation Gibbs energy function Gibbs energy of formation heat capacity thermochemical tables. 

The third-law method is based on a knowledge of the absolute entropy of the reactants and products. It allows the calculation of a reaction enthalpy from each data point when the change in the Gibbs energy function for the reaction is known. The Gibbs energy function used here is defined as... 

In the JANAF Thermochemical Tables, Gibbs energy functions are based on an enthalpy reference temperature of... 

K regardless of its stability. For example, if the vapor pressure over liquid copper is analyzed using Cu(/) Gibbs energy functions, the result is the enthalpy of vaporization of the liquid at 298.15 K. To calculate the enthalpy of sublimation of Cu(cr) it is necessary to add the enthalpy of fusion at... 

K, which is the difference in the enthalpy of formation of Cu(/) and Cu(cr) at 298.15 K. It should also be noted that Gibbs energy functions are always negative, thus the negative of the function is usually tabulated and the proper sign must be remembered when using these functions. 

Calculation of the contributions of rotation and translation involves the use of quantum statistics, but to obtain a numerical solution the quantum statistics are usually replaced by classical statistics at temperatures above about 10 K below 10 K this classical approximation no longer holds. For this reason the equations presented here fail in the vicinity of 0 K. In agreement with the third-law concept, C° and are zero at 0 K. For a reference element, log A f is zero at 0 K, while for compounds the absolute values of the Gibbs energy function and log become infinite at 0 K, for the choice of the enthalpy reference temperature of 298.15 K. 

Their partial pressure (J ) should be free of large error from Error does arise from uncertainties in the evaporation coefficients, ionization cross sections and total flux of species ( ). An extreme example of this error is the formation of o-AlgOg, which is biased by -9 kcal mol" if calculated from the reported ( ) pressures of Al and 0. Our assigned uncertainty includes contributions from this error and that of the Gibbs-energy function. 

Gurvich et al. (9) calculated functions that differ by -1.3 cal K mol" in entropy and Gibbs energy function. 

Values of Gibbs energy function from Gurvlch et al. (22) are 1.0 cal K mol larger than ours at T>1400 K. 

The level at 15.254 cm" has a large effect on the heat capacity and entropy below 100 K. The heat capacity effect decreases to zero above 600 K where the 15.254 cm" level Is fully populated. The higher excited states affect the heat capacity values above 3000 K. The Gibbs energy function values up to 6000 K are essentially Independent of the cut-off procedure, the inclusion of levels for n>2, and the estimated missing levels (for n<39). 

Asub (298.15 K) is calculated from Gibbs energy functions and vapor pressure reported by Purukawa and Park (3) ideal gas table for details. 

BaClg (9). Our Gibbs energy functions differ by roughly 2 cal K mol in the range 298-2000 K from those given by Brewer et However, their values are based on a linear structure for the bromide which now appears to be Incorrect. 

We have adjusted the latter value to mak it roughly consistent with our Gibbs energy functions. Less reliable data were reviewed by Schofield ( 9) Our analysis sumiorts the conclusions of Brewer and Rosenblatt (20). 

Our analyses of equilibrium studies of Okuno (5), Culver and Hamdorf (6), Nikonov (7), Schenck and Hammerschmldt (8), and Colin et al. (9), are listed below. The calculated 3rd law ApH (298.15 K) may have an uncertainty of 0.5 kcal mol" since the JANAF Gibbs energy functions are partially based on the estimated C data (above 300 K). The enthalpy of formation of BaS(cr) derived from equilibrium studies Is In good agreement with that derived from enthalpy of solution studies (1, 4). We discount... 

Brewer et al. (8) have tabulated Gibbs energy functions for BeBr, up to 1500 K. Their values are consistently lower than -1-1 1 ours by about 2.5 cal K mol. These differences are due entirely to the higher bending frequency ( 2 ) which was... 

The vapor pressure data over the crystal was assumed to be for the e-form. The 2nd and 3rd law analyses of the data are summarized below after conversion to a common process. The data were also analyzed using gaseous Gibbs energy functions based on a bending frequency of 170 cm" but no significant improvement of the results was noted. 

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