Single-Component Phase Diagrams (Cont.); Thermodynamics of Solutions

3.012 Fundamentals of Materials Science Fall 2005 Lecture 16: 11.04.05 Single-Component phase diagrams continued; Thermodynamics of solutions Today: LAST TIME ....................................................................................................................................................................................2 Single-component phase diagrams and the Gibbs phase rule ...............................................................................................2 Constraints on the shape of phase boundaries (coexistence curves): The Clausius-Clapeyron equation ............................2 EXAMPLE SINGLE-COMPONENT PHASE DIAGRAMS .......................................................................................................................4 Walking along lines of constant temperature or pressure in a single-component phase diagram .......................................6 GRAPHICAL CONSTRUCTIONS OF THE FREE ENERGY IN MIXTURES AND SOLUTIONS ...................................................................7 Free energy diagrams of ideal solutions ................................................................................................................................7 Mixing to form ideal solutions always occurs spontaneously ................................................................................................................8 Ideal solutions have only an entropic contribution to the free energy of mixing: intermolecular interactions are presumed to be the same in the two components ....................................................................................................................................................................9 Extracting chemical potentials from plots of the free energy ..............................................................................................10 APPLICATION OF SOLUTION FREE ENERGY ANALYSIS: MELTING POINT DEPRESSION ................................................................12 FREE ENERGY DIAGRAMS OF MULTI-PHASE SOLUTIONS(5) ........................................................................................................13 The common tangent construction and the lever rule ..........................................................................................................13 REFERENCES ..............................................................................................................................................................................15 Reading: Gaskell, Introduction to Metallurgical Thermodynamics, Ch. 11.5 ‘The free energy of solution,’ pp. 328-338 Engel and Reid 9.6, 9.7 Supplementary Reading: - 3.012 Fundamentals of Materials Science Fall 2005 Last Time P (Pressure) Single-component phase diagrams and the Gibbs phase rule Liquid Solid Gas (Vapor) T (Temperature) Figure by MIT OCW. o There is one unique value of pressure and temperature, the triple point, that can allow the 3 phases to co-exist; any change in the variables of the system causes the equilibrium to shift to one between only 1 or 2 phases. Constraints on the shape of phase boundaries (coexistence curves): The Clausius-Clapeyron equation • For 2 phases in equilibrium (let’s use the example of solid and liquid in equilibrium at the melting temperature): µS = µL P (Pressure)  To find a nearby condition of T and P where the two phases are still in equilibrium, we must have:(1) Liquid Solid Gas (Vapor) T (Temperature) Figure by MIT OCW. 3.012 Fundamentals of Materials Science • Fall 2005 The Clausius-Clapeyron equation dictates the slope of the two-phase co-existence curve for singlecomponent materials. We already know that the enthalpy change on melting is typically positive, therefore the sign of the change in volume on melting will usually dictate whether the slope of P vs. T is positive or negative. 3.012 Fundamentals of Materials Science Example single-component phase diagrams • The phase diagram of water: • (source: http:www.lsbu.ac.uk/water/phase.html) Water transitions through increasingly dense crystal structures as pressure is increased: Courtesy of London South Bank University. Used with permission. Courtesy of London South Bank University. Used with permission. Fall 2005 3.012 Fundamentals of Materials Science Fall 2005 CARBON PHASE DIAGRAM 5000 3000 Liquid Graphite o T (Temperature), K 4000 2000 Diamond 1000 0 Figure by MIT OCW. Solid m (metallic) 200 400 600 P (Pressure), kbor 800 1000 Temperature vs. pressure phase diagram for carbon: A diamond is not forever. IRON PHASE DIAGRAM 2000 Liquid iron 1600 T (Temperature) 0C Delta iron 1200 800 400 0 Figure by MIT OCW. Gamma iron Alpha iron 25 50 Epsilon iron 75 100 125 150 P (Pressure), atm Temperature vs. pressure phase diagram for iron. 175 Pressure (atm) 3.012 Fundamentals of Materials Science Fall 2005 _l ~ Liquid Cubic Tetragonal Vapor Monoclinic -273 0 2000 1000 3000 4000 Temperature (oC) Proposed diagram for system ZrO2 (Bergeron and Risbud(2)) Figure by MIT OCW. Walking along lines of constant temperature or pressure in a single-component phase diagram P (Pressure) • Consider now how the free energy varies as we move along a line of a single-component phase diagram at constant pressure or constant temperature: G Pls(TA) PB Pvl(TA) TA Tsl(PB) Tlv(PB) T (Temperature) Figure by MIT OCW. Tsl Tlv T 3.012 Fundamentals of Materials Science Fall 2005 Graphical constructions of the free energy in mixtures and solutions • KEY CONCEPTS: Ideal solutions are models for materials that have similar crystal structures and bonding. Formation of an ideal solution from its unmixed components is a lways spontaneous, because the free energy of mixing has no pena lizing enthalpy term ( there are no unfavorable bond ing interactions) and the entropy gain on mixing is always positive. • In addition to mapping out stable phases for single-component systems, phase diagrams can also be used to chart stable phases as a function of temperature (or pressure) vs. composition for binary (2component) or ternary (3-component) systems. In order to understand how the phase boundaries in systems of more than one component arise, we will first discuss another useful graphical constructions: free energy vs. composition diagrams. Free energy diagrams of ideal solutions • Earlier we introduced the general solution model for the chemical potential: • When the activities of the components are equal to their compositions, the solution is ideal: • Knowing the chemical potentials of the components, we can determine the total free energy of the solution. What does a plot of the total free energy for a binary ideal solution of two components A and B look like vs. the composition XB? o The total molar free energy for the solution is: Figure by MIT OCW. 3.012 Fundamentals of Materials Science o Before formation of the solution, (i.e., imagine a crystal of pure Si bonded to a crystal of pure Ge) the molar free energy of the system is simply the sum of the free energies per mole of each pure component multiplied the total mole fraction: Ideal solutions have only an entropic contribution to the free energy of mixing: intermolecular interactions are presumed to be the same in the two components • Fall 2005 What are the changes in enthalpy and entropy that occur on mixing to form an ideal solution? 3.012 Fundamentals of Materials Science Fall 2005 Extracting chemical potentials from plots of the free energy We can learn more from free energy diagrams than the free energy change on mixing. The diagram also provides a convenient graphical means to determine chemical potentials/partial molar free energies of the components as a function of composition. To see how this is done, we start by writing the free energy of the solution: • C G = " µ ini = µ A nA + µ B nB i=1 o ! What is the differential of the molar free energy? 3.012 Fundamentals of Materials Science  Fall 2005 These two equations tell us how to determine the chemical potentials from free energy diagrams. Returning to our diagram: Figure by MIT OCW. 3.012 Fundamentals of Materials Science Fall 2005 Free energy diagrams of multi-phase solutions(5) • We have already dealt at length with the criteria for equilibrium at constant temperature and pressure in closed systems: the Gibbs free energy must reach a minimum, and for multi-phase materials, this implies that the chemical potential of each component must be the same in every phase present. The free energy diagrams we introduced last time can conveniently be used to analyze multiphase equilibria that satisfy these conditions graphically. The common tangent construction and the lever rule Suppose we have a binary ideal solution of A and B. We showed last time the shape off the free energy curve for such a solution. The molar free energy for the solution can be diagrammed for different states of the solution- for example the liquid state and the solid state- as a function of composition: T > Tm,B > Tm,A G (J/mole) • GS GS GL GL 0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 XBB We can answer numerous questions about the thermodynamics of this system from such a diagram: o    Which phase is thermodynamically stable at each value of XB for this temperature? What is the molar free energy change to melt pure A? to melt pure B? What is the free energy change on forming a liquid solution from 0.5 moles of pure B and 0.5 moles of pure A? Suppose we lowered the temperature from the above situation. How would the two free energy curves change? Which curve will move more, considering that: o G = H " TS ! 3.012 Fundamentals of Materials Science o Fall 2005 The answer is seen in the following two diagrams: G (J/mole) T = Tm,B > Tm,A GS GS GL GL 0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 XBB G (J/mole) TT=m,BTm,B > Tm,A > T=T 1 > Tm,A GS GS GL GL XS XL 0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 XBB 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 endpoints 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 L S composition X , and the solid curve at X . 3.012 Fundamentals of Materials Science Fall 2005 References 1. 2. 3. 4. 5. Denbigh, K. 1997. The Principles of Chemical Equilibrium. Cambridge University Press, New York. Bergeron, C. G., and S. H. Risbud. 1984. Introduction to Phase Equilibria in Ceramics. American Ceramic Society, Westerville, OH. Sque, S. Dill, K., and S. Bromberg. 2003. Molecular Driving Forces, New York. Carter, W. C. 2002. 3.00 Thermodynamics of Materials Lecture Notes.