Lecture 17: Free Energy of Multiphase Solutions at Equilibrium


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1 Lecture 17: Free Energy f Multiphase Slutins at Equilibrium Tday: LAST TIME...2 FREE ENERGY DIAGRAMS OF MULTIPHASE 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 Multiphase Equilibria 1 f 14 11/6/05
2 Last time Lecture 18 Multiphase Equilibria 2 f 14 11/6/05
3 Free energy diagrams f multiphase slutins 1 Last lecture we examined the structure and interpretatin f free energy vs. cmpsitin diagrams fr ideal binary (twcmpnent) 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 Multiphase 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 Multiphase 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 twphase equilibrium! Thus fr cmpsitins between the cmmn tangent pints, tw phases are present in the material, slid and liquid. Why d tw phases cexist between X S and X L? Let s analyze the blwnup diagram belw: Lecture 18 Multiphase 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 rearrange, 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 Multiphase 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 Multiphase 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 ThermCalc Sftware available n Athena fr perfrming many thermdynamic calculatins, building phase diagrams, etc. Lecture 18 Multiphase 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 Multiphase Equilibria 9 f 14 11/6/05
10 P = cnstant T m (pure B) T = T 1 T Hmgeneus liquid mixture twphase 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 twphase regin: Lecture 18 Multiphase 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 PbSn, Ga Sb, and TlSn, 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 Multiphase 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 Multiphase 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 Multiphase Equilibria 13 f 14 11/6/05
14 References 1. Carter, W. C Thermdynamics f Materials Lecture Ntes (2002). Lecture 18 Multiphase Equilibria 14 f 14 11/6/05
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