A Derivation of a Quadratic Activity Coefficient vs Composition Relationship in a Quaternary System, A-B-C-D

作者：Ghosh Dinabandhu^*

来源：Metallurgical and Materials Transactions B-Process Metallurgy and Materials Processing Science, 2010, 41(6): 1274-1283.

DOI：10.1007/s11663-010-9417-x

In the literature, no direct derivation exists of the quadratic activity coefficient vs composition relationships for a quaternary system with high solute concentrations. Such relations for a ternary system (1- 2- 3) were derived by Darken by extending the results of a binary system (1- 2), introducing a new concept of `` hypothetical system'' ( 2- 3). To present a better scheme to find the activity coefficient- composition relations for multicomponent systems, derivations are made for a quaternary system A- B- C- D in the current work. Using a MacLaurin series expansion, the ( Raoultian) activity coefficient, ln ci, of each component is equated with a quadratic expression of mole fractions (x), involving the activity coefficient at zero concentration (gamma(o)(i)) and nine interaction coefficients (epsilon). Subsequently, with the help of a Gibbs- Duhem equation, followed by a comparison of coefficients, most preceding 9 x 4, i.e., 36 interaction coefficients are eliminated, leaving behind only three self- and three ternary interaction coefficients, which are enough to express the activity coefficient vs composition relationships for the solutes B, C, and D, as well as for the solvent A. Setting the mole fraction x(D) = 0, the preceding expressions establish the same relations as proposed by Darken for the ternary system A- B- C. The derivation also clarifies how the quadratic concentration terms accompany the first- order interaction coefficients, not the second- order ones. Applications of the derived relations to determine simultaneously the activity coefficients gamma(o)(i) and the interaction coefficients e in a new way in some iron- and steelmaking systems are presented. A new data on interaction coefficients in liquid iron at 1873 K ( 1600 degrees C), epsilon(V)(v) = 6: 1, has been generated through such an application.

• 出版日期2010-12