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Thermodynamics of Multi-component Systems Consider a binary solid solution of A and B atoms: Let nA= # of moles of A nB= # of moles of B def: mole(or atom) fraction nA xA nB nA ; xB nB nB nA Consider nA moles of A and nB moles of B. Before they are mixed: G1 xAGA xBGB GB G1 GA 0 XB 1 Variation of the free energy before mixing with alloy composition. XA moles of A XB moles of B G1 X AGA X BGB The Free Energy of the system changes on mixing MIX fixed T XA moles of A XB moles of B 1 mole of solid solution G2 G1 Gmix G1 X AGA X BGB G2 G1 Gmix Since Defining G1 H1 TS1 G2 H 2 TS 2 H mix H 2 H1 Gmix H mix T S mix Smix S2 S1 Lets consider each of the terms Hmix - recall that for condensed systems ΔH ≈ ΔU and so Δ Hmix= heat of solution ≈ change in internal energy before and after mixing. Smix - change in entropy due to mixing. Take the mixing to be “perfect”i.e., random solid solution Smix = (molar) Configurational entropy Smix k ln W Boltzmann’s Eqn. W = # of distinguishable ways of arranging the atoms → “randomness” The number of A atoms and B atoms in the mixture is: NA = nANa NB = nBNa Na , Avogadro’s number From combinatorial mathematics: W N A NB N A NB ! ! ln W can be approximated using Stirling’s approximation, ln N ! N ln N N and using, Nak = R, where R is the gas constant, we obtain, S mix R X A ln X A X B ln X B Let’s Re-examine the Hmix term: 2 models (a) Ideal solution model (b) Regular solution model Ideal Solutions Hmix = 0 Physically this means the A atoms interact with the B atoms as if they are A and vice versa. , The only contribution to alterations in the Gibbs potential is in the configurational entropy i.e., Gmix T Smix RT ( X A ln X A X B ln X B ) Examples: (a) Solution of two isotopes of the same element (b) Low pressure gas mixtures (c) Many dilute ( xA << xB or xB << xA) condensed phase solutions. Recall that the total free energy of the solution G2 G G1 Gmix X AGA X BGB RT X A ln X A X B ln X B G GB G1 Low T GA 0 XB 0 1 XB -TSmix High T Gmix Gmix 1 Regular Solutions Assume a random solid solution and consider how the A&B atoms interact. MIX fixed T XA moles of A XB moles of B G1 X AGA X BGB 1 mole of solid solution G2 G1 Gmix In a general the interaction of an A atom with another A atom or a B atom depends upon (i) interatomic distance (ii) atomic identity (iii) 2nd, 3rd, … next –near-neighbor identity and distances. The Regular Solution model assumes only nearest-neighbor interactions, pairwise. “Quasichemical” model Assume the interatomic distance set by the lattice sites. Let : VAA A A Bond energy VBB B B Bond energy VAB A B Bond energy Note that all the V’s are < 0. Consider a lattice of N sites with Z nearest-neighbors per site. : Each of the N atoms has Z bonds so that there are NZ 2 bonds in the lattice. Division by 2 is for double counting. Let PAA be the probability that any bond in the lattice is an A-A bond: 1 NZPAA 2 then N AA and PAA x A2 so N AA N BB 1 NZx A2 2 1 NZxB2 2 N AB 1 1 NZx A xB NZxB x A 2 2 N AB NZxA xB The energy for the mixed solution is H mix 1 NZ x A2VAA xB2VBB 2 x A X BVAB 2 prior to mixing 1 H1 NZ xAVAA xBVBB 2 and H mix 1 NZ xA2VAA xB2VBB 2 x A xBVAB x AVAA xBVBB 2 combining terms & using xA + xB = 1 Hmix ZxA xB VAB 1 (VAA VBB ) 2 Hmix xA xB where Z[VAB 1/ 2(VAA VBB )] Notice the ΔHmix can be either positive or negative. ΔHmix > 0, > 0 and VAB > 1/2 (VAA+ VBB) from ΔGmix = ΔHmix – TΔSmix At low temps clustering of As and Bs result, i.e., “phase separation”. for ΔHmix < 0, < 0 and VAB < 1/2 (VAA+ VBB) The A atoms are happier with B atoms as nearest-neighbors. Short range ordering i.e., PAB is increases over the random value. For a Regular Solution Gmix xA xB RTxA ln xA xB ln xB H mix TSmix Variation of ΔGmix with composition < 0 0 ΔGmix xB → –TΔSmix 1 ΔHm ΔGm > 0 T < TC Note ΔHmix 0 ΔGmix ΔGmix xB → –TΔSmix 1 2 minima change in curvature 2 pts of inflection. 2Gmix 0 inf pts. 2 xB Gmix 0 extrem um pts. xB As T increases, the –TΔS term begins to dominate. 0 XB → ΔGmix 1 T < Tc T = Tc the critical temperature: Gmix extremum Gmix inflection.pts. The inflection pt. & extremum merge