# Lecture 17: Free Energy of Multi-phase Solutions at Equilibrium

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1 Lecture 17: Free Energy f Multi-phase Slutins at Equilibrium Tday: LAST TIME...2 FREE ENERGY DIAGRAMS OF MULTI-PHASE SOLUTIONS The cmmn tangent cnstructin and the lever rule...3 Practical surces f free energy data...8 INTRODUCTION TO BINARY PHASE DIAGRAMS...9 Phase diagram f an ideal slutin...9 Tie lines and the lever rule n a binary phase diagram...10 BINARY SOLUTIONS WITH LIMITED MISCIBILITY: MISCIBILITY GAPS...11 The Regular Slutin Mdel, part I...11 REFERENCES...14 Reading: Lupis, Chemical Thermdynamics f Materials, Ch 8 Binary Phase Diagrams, pp Supplementary Reading: - Lecture 18 Multi-phase Equilibria 1 f 14 11/6/05

2 Last time Lecture 18 Multi-phase Equilibria 2 f 14 11/6/05

3 Free energy diagrams f multi-phase slutins 1 Last lecture we examined the structure and interpretatin f free energy vs. cmpsitin diagrams fr ideal binary (tw-cmpnent) slutins. The free energy diagrams we intrduced last time can cnveniently be als used t analyze multiphase equilibria that allw us t graphically depict the requirements fr equilibrium. The cmmn tangent cnstructin and the lever rule KEY CONCEPTS: Free energy vs. cmpsitin diagrams are useful tls fr graphically analyzing phase equilibria in binary systems at cnstant pressure. Cmmn tangents between the free energy curves f different phases ccur in regins where 2 phases are in equilibrium. The pints where cmmn tangents tuch the free energy curves identify the cmpsitins f the tw phases in equilibrium. The lever rule is used t determine hw much f each phase is present in tw phase equilibrium regins. Suppse we have a binary ideal slutin f A and B. We shwed last time the shape f the free energy curve fr such a slutin. The mlar free energy fr the slutin can be diagrammed fr different phases f the slutin- fr example the liquid state and the slid state- as a functin f cmpsitin: Suppse we lwered the temperature frm the abve situatin. Hw wuld the tw free energy curves change? Which curve will mve mre, cnsidering that: G = H " TS Lecture 18 Multi-phase Equilibria 3 f 14 11/6/05

4 What is happening in the secnd figure? We have reduced the temperature t the pint where the stable state f pure B is a slid. Remember that the chemical ptential is given by the end- Lecture 18 Multi-phase Equilibria 4 f 14 11/6/05

5 pints f the tangent t the free energy curve at a given cmpsitin. But we find that at T 1, a line can be drawn tangent t bth free energy curves- a line that is tangent t the liquid curve at cmpsitin X L, and the slid curve at X S. Lwering the temperature slightly mre: We find that in the cmpsitin range frm X L t X S, the chemical ptentials f cmpnent A in the slid and liquid states are equal, and the chemical ptentials f B in the slid and liquid states are equal but this is the cnditin fr tw-phase equilibrium! Thus fr cmpsitins between the cmmn tangent pints, tw phases are present in the material, slid and liquid. Why d tw phases c-exist between X S and X L? Let s analyze the blwn-up diagram belw: Lecture 18 Multi-phase Equilibria 5 f 14 11/6/05

6 At cmpsitin X 1, cmparisn f the slid state free energy with that f the liquid shws that the liquid wuld be the frm with lwest free energy- thus the liquid slutin wuld be mre stable than the slid. Hwever, the free energy f the liquid is nt the lwest pssible free energy state. If the A and B atms in the hmgenus liquid slutin re-arrange, a prtin transfrming t a slid with cmpsitin X S and a prtin remaining in a liquid slutin with cmpsitin altered t X L, the hetergeneus slid/liquid mixture takes n the free energy G sep, which is lwer than that f the hmgeneus liquid slutin at X 1. f L and f S are the phase fractins f liquid and slid phases, respectively. Nte that because a hetergeneus (2- phase) mixture is being frmed, the free energy is determined in a manner similar t that discussed earlier fr hetergeneus mixtures (e.g. ur blck f Si in cntact with a blck f Ge)- simply a weighted average f the mlar free energies f the liquid phase (cmpsitin X L ) and the slid phase (cmpsitin X S ). Lecture 18 Multi-phase Equilibria 6 f 14 11/6/05

7 Hw much slid phase frms? Hw much liquid is present? The cmpsitin f the liquid phase is X L, and the cmpsitin f the slid phase is X S. Therefre, the amunt f each phase present can be determined simply by requiring that the average cmpsitin f the system remains X 1 : Similarly, if we write X 1 in terms f f S we btain: These tw equatins fr the fractin f slid and liquid frmed have a graphical equivalent: The mathematical and graphical cnstructin t identify the fractin f each phase is knwn as the lever rule. Lecture 18 Multi-phase Equilibria 7 f 14 11/6/05

8 Practical surces f free energy data Where d we get the infrmatin fr these diagrams? Natinal Institute f Standards and Technlgy Chemistry WebBk heat capacity, enthalpy, and entrpy data JANAF Tables Jint Army Navy Air Frce database f thermchemical data Exhaustive Cp, entrpy, enthalpy, free energy data QD511.J Selected Values f Thermdynamic Prperties f Metals and Allys R. Hultgren, R.L. Orr, P.D. Andersn, and K.K. Kelley Jhn Wiley, NY 1963 QD171.S44 Therm-Calc Sftware available n Athena fr perfrming many thermdynamic calculatins, building phase diagrams, etc. Lecture 18 Multi-phase Equilibria 8 f 14 11/6/05

9 Intrductin t binary phase diagrams KEY CONCEPTS: The phase equilibria as a functin f cmpsitin fr a fixed temperature (and fixed pressure) predicted by Free energy vs. cmpsitin diagrams can be cllated t create a binary phase diagram, which maps ut stable phases in T vs. cmpsitin space (pressure assume fixed) the binary system analg f single cmpnent phase diagrams. The Gibbs phase rule can be applied t these diagrams, accunting fr the fixed pressure (D + P = C + 1). Tie lines allw the lever rule t be directly applied t phase diagrams in rder t calculate the amunt f each phase present in multiphase equilibria. Phase diagram f an ideal slutin Frm an examinatin f free energy vs. cmpsitin diagrams, we fund that phase separatin (induced fr example by reducing the temperature and freezing a liquid slutin) prceeds by the fllwing prgressin acrss the cmpsitin windw f an ideal binary slutin (Fr a system where T m,b > T m,a ): T > T m,b > T m,a T = T m,b > T m,a T m,b > T 1 > T m,a T m,b > T 1 > T 2 > T m,a T m,b > T 2 > T 3 > T m,a T m,b > T = T m,a It wuld make sense t btain a cntinuus map f the phases present as a functin f X B and temperature fr a binary system: such a map is a key tl in materials science & engineering and is knwn as a binary phase diagram. Fr the ideal binary slutin we have been analyzing, the phase diagram lks like this: Lecture 18 Multi-phase Equilibria 9 f 14 11/6/05

10 P = cnstant T m (pure B) T = T 1 T Hmgeneus liquid mixture tw-phase regin T = T 2 T m (pure A) Hmgeneus slid mixture K B Figure by MIT OCW. This is the simplest frm a binary phase diagram can take. Tie lines and the lever rule n a binary phase diagram The lever rule that we develped using free energy vs. cmpsitin diagrams can be directly applied t a binary phase diagram (T vs. cmpsitin). This is dne using tie lines hrizntal istherms cnnecting the bundaries f a tw-phase regin: Lecture 18 Multi-phase Equilibria 10 f 14 11/6/05

11 Binary slutins with limited miscibility: Miscibility gaps The Regular Slutin Mdel, part I What happens if the mlecules in the slutin interact with a finite energy? The enthalpy f mixing will nw have a finite value, either favring ( "H mix < 0) r disfavring ( "H mix > 0) mixing f the tw cmpnents. The simplest mdel f a slutin with finite interactins is called the regular slutin mdel: Let the enthalpy f mixing take n a finite value given by: We take the entrpy f mixing t be the same as in the ideal slutin. This gives a ttal free energy f mixing which is: The regular slutin mdel describes the liquid phase f many real systems such as Pb-Sn, Ga- Sb, and Tl-Sn, and sme slid slutins. Tday we will analyze the behavir f a system with this free energy functin; in a few lectures we will shw hw the given frms f the enthalpy and entrpy f mixing arise frm cnsideratin f mlecular states (using statistical mechanics). Lecture 18 Multi-phase Equilibria 11 f 14 11/6/05

12 T=100 K " = 20,000 J/mle!H mix,rs " = 10,000 J/mle " = -10,000 J/mle XB X B The verall free energy f mixing arises frm the balance between favrable mixing entrpy and unfavrable enthalpy cntributins: Lecture 18 Multi-phase Equilibria 12 f 14 11/6/05

13 80000 T=100 K " = 20,000 J/mle #!S mix,rs !H mix,rs !G mix,rs X B The free energy f the system varies with the value f Ω and with temperature: As a functin f temperature at a fixed psitive value f Ω: Lecture 18 Multi-phase Equilibria 13 f 14 11/6/05

14 References 1. Carter, W. C Thermdynamics f Materials Lecture Ntes (2002). Lecture 18 Multi-phase Equilibria 14 f 14 11/6/05

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