Thermodynamic consistency of the ternary functions

A ternary thermodynamic function has been developed based on statistico-thermodynamic considerations, with a particular emphasis on the higher-order terms indicating the effects of truncation at the various stages of the treatment. Although the truncation of a series involved in the equation introduces inconsistency, the latter may be removed by imposing various thermodynamic boundary conditions. These conditions are discussed in the paper. The present equation with higher-order terms shows that the α function of a component reduces to a quadratic function of composition at constant compositional paths involving the other two components in the system. The form of the function has been found to be representative of various experimental observations.     

Thermodynamic consistency of the interaction parameter formalism

The apparent contradiction between the exact nature of the interaction parameter formalism as presented by Lupis and Elliott and the inconsistencies discussed recently by Pelton and Bale arise from the truncation of the Maclaurin series in the latter treatment. The truncation removes the exactness of the expression for the logarithm of the activity coefficient of a solute in a multi-component system. The integrals are therefore path dependent. Formulae for integration along paths of constant X_i,or X _i/X_j are presented. The expression for In γ_solvent given by Pelton and Bale is valid only in the limit that the mole fraction of solvent tends to one. The truncation also destroys the general relations between interaction parameters derived by Lupis and Elliott. For each specific choice of parameters special relationships are obtained between interaction parameters.     

Thermodynamic consistency of the interaction parameter formalism

The apparent contradiction between the exact nature of the interaction parameter formalism as presented by Lupis and Elliott and the inconsistencies discussed recently by Pelton and Bale arise from the truncation of the Maclaurin series in the latter treatment. The truncation removes the exactness of the expression for the logarithm of the activity coefficient of a solute in a multi-component system. The integrals are therefore path dependent. Formulae for integration along paths of constant X_i,or X _i/X_j are presented. The expression for In γ_solvent given by Pelton and Bale is valid only in the limit that the mole fraction of solvent tends to one. The truncation also destroys the general relations between interaction parameters derived by Lupis and Elliott. For each specific choice of parameters special relationships are obtained between interaction parameters.     

Distribution coefficients in dilute binary and ternary alloys

In most treatments of the solidification of dilute multicomponent alloys, the distribution coefficients (k ₀) are assumed to retain their binary values. Thermodynamic calculations indicate that for alloys containing up to a few percent solute,k ₀ will be approximately constant. However, large solute interaction effects have been reported for certain tin-based ternary alloys, which would make the above procedure of doubtful validity. In this paper, a brief survey pointing out the limitations and inaccuracies inherent in standard methods for the measurement of distribution coefficients is given. The determination ofk ₀ for several tin-based ternary alloys is then described. With careful technique, the large interaction effects reported earlier are not confirmed. Maximum variations ink ₀ of 20 pct within the concentration range 0 to 2 wt pct were found, compared with up to 300 pct variation reported elsewhere. It is concluded that the use of binary distribution coefficients for dilute multicomponent alloys is a reasonable approximation.     

Coefficient alpha and related internal consistency reliability coefficients.

The author studied the conditions under which coefficient alpha and 10 related internal consistency reliability coefficients underestimate the reliability of a measure. Simulated data showed that alpha, though reasonably robust when computed on n components in moderately heterogeneous data, can under certain conditions seriously underestimate the reliability of a measure. Consequently, alpha, when used in corrections for attenuation, can result in nontrivial overestimation of the corrected correlation. Most of the coefficients studied, including lambda2, did not improve the estimate to any great extent when the data were heterogeneous. The exceptions were stratified alpha and maximal reliability, which performed well when the components were grouped into two subsets, each measuring a different factor, and maximized lambda4, which provided the most consistently accurate estimate of the reliability in all simulations studied. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

On the relationship between interaction coefficients

The widely used relationship between Wagner coefficients,ɛi(j) = ɛj(i) has been analyzed. Based on various representations of the independent composition variables, the validity of the relationship has strictly been limited to the cases when the ratio ln(γj/γi), i, j = 2,…,s is constant at the pointx ₁ = 1 and in the neighborhood of this point, provided that the derivatives of the activity coefficients have finite values within the specified limit. The clarification of misconceptions about the Wagner coefficients, frequently encountered in the literature, has been attempted.     

Evaluation of an empirical representation of ternary diffusion paths in a system with constant diffusion coefficients

An empirical two-parameter representation of ternary diffusion paths has been proposed by Dayananda and Kim. Their proposal has been tested on a hypothetical ternary diffusing system with constant diffusion coefficients for which the exact diffusion paths can be calculated analytically. It is found that the error in using the two-parameter representation rather than the true diffusion path is strongly dependent on the value of the ratio of the terminal relative concentration differences of the diffusion couple selected for analysis.     

Experimental determination of ternary partition coefficients in Fe-Ni-X alloys

A directional solidification experiment to measure partition coefficients of ternary additions in Fe-Ni alloys is described. A model is developed to calculate the extent of solid state diffusion effects during and after solidification. Ternary diffusivities of Ge and Au in Fe-Ni were measured for use in the calculation procedure. The model is used to correct the measured partition coefficients for solid state diffusion effects. The resulting equilibrium partition coefficients, K _x ^o , for elementX in Fe-Ni-X alloys are 0.43 for Au, 0.58 for Ge, 0.12 for P, 0.87 for Ni, and 1.45 for Pt. These studies demonstrated that solid state diffusion effects are significant when the diffusivity of the third elementX, in Fe-Ni, is higher than 1 × 10⁻¹⁰ cm²/s at 1300 °C. The partition coefficient data is of particular interest in understanding the fractionation behavior of trace elements in iron meteorites.     

Psychometric inferences from a meta-analysis of reliability and internal consistency coefficients.

A meta-analysis of the reliability of the scores from a specific test, also called reliability generalization, allows the quantitative synthesis of its properties from a set of studies. It is usually assumed that part of the variation in the reliability coefficients is due to some unknown and implicit mechanism that restricts and biases the selection of participants in the studies' samples. Sometimes this variation has been reduced by adjusting the coefficients by a formula associated with range restrictions. We propose a framework in which that variation is included (instead of adjusted) in the models intended to explain the variability and in which parallel analyses of the studies' means and variances are performed. Furthermore, the analysis of the residuals enables inferences to be made about the nature of the variability accounted for by moderator variables. The meta-analysis of the 3 studies' statistics—reliability coefficient, mean, and variance—allows psychometric inferences about the test scores. A numerical example illustrates the proposed framework. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Thermodynamic analysis of the Zn−Pb−Cd system and interaction studies in ternary liquid dilute solutions

Dilute zinc solutions in the Zn−Pb−Cd system were investigated by an emf method with concentration cells to show the influence of variation in lead and cadmium concentrations on the zinc activity coefficients. Zinc concentrations were held in the mole fraction range 0.03≤X _Zn≤0.1 with measurements in the temperature range 714<T<877°K. The interaction parameters, ∈ _Zn ^Pb and ∈ _Cd ^Zn , were deduced from plots of (ln γ_Zn) _{X Zn→0} vsX _Pb. 3 The limiting values of zinc activity coefficients for other ternary systems previously investigated were analyzed using Krupkowski's method and the random solution model of Alcock and Richardson. Experimental data for ∈ ₁ ² and ∈ ₁ ³ (where 1=Zn and 2, 3 are components of investigated ternaries) are compared also with the values obtained from limiting activity coefficients of the particular binary systems.     

The thermodynamics of the Ni-Co-S ternary system

The thermodynamic properties of the Ni-Co-S ternary system were determined by equilibrating the system with a known sulfur pressure established by using a gas mixture of H₂S/H₂ at 1373, 1473, and 1573 K. The isoactivity contours for Ni and Co were determined by application of the Gibbs-Duhem equation using Schuhmann’s method for ternary systems. For this purpose, a computer program was developed to calculate the activities of nickel and cobalt. The cobalt activity coefficients in dilute solutions of cobalt in the nickel-cobalt-sulfur system were evaluated and found to be 0.33, 0.50, and 0.55 at 1373, 1473, and 1573 K, respectively.     

Thermodynamic investigations of the ternary Au-Sn-Zn system

The thermodynamic properties of liquid ternary Au-Sn-Zn alloys were measured with an electromotive force (emf) method at four cross sections with a constant Au to Sn ratio of 9:1, 3:1, 1:1, and 1:3, respectively. At the last three cross sections, calorimetric measurements were also carried out in order to compare the integral enthalpy of mixing from the direct measurements with the data of the emf measurements which were derived by Gibbs-Duhem integration.     

The unified interaction parameter formalism: Thermodynamic consistency and applications

In a recent article, we proposed modifications to the standard interaction parameter formalism. The modified formalism, known as the “Unified Interaction Parameter Formalism,” is discussed in the present article with respect to thermodynamic consistency at finite concentrations in binary, ternary, and multicomponent systems. A new method, which is independent of integration paths, is proposed to derive the equations of the formalism by differentiation of the integral Gibbs energy expression. It is shown that the formalism is thermodynamically exact in both dilute and nondilute composition regions. It is also shown that the formalism reduces to Wagner’s formalism at infinite dilution and to Darken’s quadratic formalism in dilute solutions. Examples are presented and methods are discussed for determining the parameters of the formalism from thermodynamic data.     

Thermodynamic analysis of the liquidus in ternary systems involving small contents of one component

If the liquidus of a binary system 1–2 for coexistence of liquid alloy with virtually pure solid 1 has a distinct flat section, small additions of a third component may enlarge or narrow the two-phase field liquid alloy + solid 1 in a rather spectacular manner,e.g. in the systems Cu-Fe-S and Bi-Ag-Zn. Pertinent thermodynamic equations are derived.