3.012 Fundamentals of Materials Science Fall 2005 Lecture 17: 11.07.05 Free Energy of Multi-phase Solutions at Equilibrium Today: LAST TIME .........................................................................................................................................................................................2 FREE ENERGY DIAGRAMS OF MULTI-PHASE SOLUTIONS1 ................................................................................................................3 The common tangent construction and the lever rule ...............................................................................................................3 Practical sources of free energy data.........................................................................................................................................8 INTRODUCTION TO BINARY PHASE DIAGRAMS.................................................................................................................................9 Phase diagram of an ideal solution ............................................................................................................................................9 Tie lines and the lever rule on a binary phase diagram..........................................................................................................10 BINARY SOLUTIONS WITH LIMITED MISCIBILITY: MISCIBILITY GAPS...........................................................................................11 The Regular Solution Model, part I ..........................................................................................................................................11 REFERENCES ...................................................................................................................................................................................14 : Lupis, Chemical Thermodynamics of Materials, Ch 8 ‘Binary Phase Diagrams,’ Readingpp. 196-203 Supplementary Reading: -Lecture 18 – Multi-phase Equilibria 1 of 14 11/6/053.012 Fundamentals of Materials Science Fall 2005 Last time Lecture 18 – Multi-phase Equilibria 2 of 14 11/6/053.012 Fundamentals of Materials Science Fall 2005 Free energy diagrams of multi-phase solutions1 • Last lecture we examined the structure and interpretation of free energy vs. composition diagrams for ideal binary (two-component) solutions. The free energy diagrams we introduced last time can conveniently be also used to analyze multiphase equilibria that allow us to graphically depict the requirements for equilibrium. The common tangent construction and the lever rule • KEY CONCEPTS: Free energy vs. composition diagrams are useful tools for graphically analyzing phase equilibria in binary systems at constant pressure. Common tangents between the free energy curves of different phases occur in regions where 2 phases are in equilibrium. The points where common tangents touch the free energy curves identify the compositions of the two phases in equilibrium. The lever rule is used to determine how much of each phase is present in two phase equilibrium regions. • Suppose we have a binary ideal solution of A and B. We showed last time the shape of the free energy curve for such a solution. The molar free energy for the solution can be diagrammed for different phases of the solution-for example the liquid state and the solid state-as a function of composition: o Suppose we lowered the temperature from the above situation. How would the two free energy curves change? Which curve will move more, considering that: ! G = H " TS Lecture 18 – Multi-phase Equilibria 3 of 14 11/6/053.012 Fundamentals of Materials Science Fall 2005 o What is happening in the second figure? We have reduced the temperature to the point where the stable state of pure B is a solid. Remember that the chemical potential is given by the end-Lecture 18 – Multi-phase Equilibria 4 of 14 11/6/053.012 Fundamentals of Materials Science Fall 2005 points of the tangent to the free energy curve at a given composition. But we find that at T1, a line can be drawn tangent to both free energy curves-a line that is tangent to the liquid curve at composition XL, and the solid curve at XS . o Lowering the temperature slightly more: • We find that in the composition range from XL to XS, the chemical potentials of component A in the solid and liquid states are equal, and the chemical potentials of B in the solid and liquid states are equal …but this is the condition for two-phase equilibrium! Thus for compositions between the common tangent points, two phases are present in the material, solid and liquid. Why do two phases co-exist between XS and XL? Let’s analyze the blown-up diagram below: Lecture 18 – Multi-phase Equilibria 5 of 14 11/6/053.012 Fundamentals of Materials Science Fall 2005 • At composition X1, comparison of the solid state free energy with that of the liquid shows that the liquid would be the form with lowest free energy-thus the liquid solution would be more stable than the solid. However, the free energy of the liquid is not the lowest possible free energy state. If the A and B atoms in the homogenous liquid solution re-arrange, a portion transforming to a solid with composition XS and a portion remaining in a li! G sepquid solution with composition altered to XL, the heterogeneous solid/liquid mixture takes on the free energy , which is lower than that of the homogeneous liquid solution at X1 . fL and f S are the phase fractions of liquid and solid phases, respectively. Note that because a heterogeneous (2-phase) mixture is being formed, the free energy is determined in a manner similar to that discussed earlier for heterogeneous mixtures (e.g. our block of Si in contact with a block of Ge)-simply a weighted average of the molar free energies of the liquid phase (composition XL) and the solid phase (composition XS). Lecture 18 – Multi-phase Equilibria 6 of 14 11/6/053.012 Fundamentals of Materials Science Fall 2005 o How much solid phase forms? How much liquid is present? The composition of the liquid phase is XL, and the composition of the solid phase is XS . Therefore, the amount of each phase present can be determined simply by requiring that the average composition of the system remains X1: Similarly, if we write X1 in terms of fS we obtain: o These two equations for the fraction of solid and liquid formed have a graphical equivalent: The
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