Thermodynamics of Multi-component Systems Consider a binary solid solution of A and B atoms: Let n A = # of moles of A n B = # of moles of B def:mole(or atom) fraction ;

Consider n A moles of A and n B moles of B. Before they are mixed: G1G1 GAGA GBGB XBXB 0 1 Variation of the free energy before mixing with alloy composition. X A moles of A X B moles of B

The Free Energy of the system changes on mixing X A moles of A X B moles of B MIX 1 mole of solid solution fixed T

Since Lets consider each of the terms  H mix - recall that for condensed systems ΔH ≈ ΔU and so Δ H mix = heat of solution ≈ change in internal energy before and after mixing. Defining

 S mix - change in entropy due to mixing. Take the mixing to be “perfect”i.e., random solid solution  S mix = (molar) Configurational entropy Boltzmann’s Eqn. W = # of distinguishable ways of arranging the atoms → “randomness” The number of A atoms and B atoms in the mixture is: N A = n A N a N B = n B N a N a, Avogadro’s number From combinatorial mathematics:

ln W can be approximated using Stirling’s approximation,, and using, N a k = R, where R is the gas constant, we obtain, Let’s Re-examine the  H mix term: 2 models (a)Ideal solution model (b)Regular solution model Ideal Solutions  H mix = 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., Examples: (a)Solution of two isotopes of the same element (b)Low pressure gas mixtures (c)Many dilute ( x A << x B or x B << x A ) condensed phase solutions.

Recall that the total free energy of the solution G1G1 GAGA GBGB XBXB 0 1 G  G mix Low T High T XBXB 0 1  G mix -T  S mix

Regular Solutions Assume a random solid solution and consider how the A&B atoms interact. X A moles of A X B moles of B MIX fixed T 1 mole of solid solution

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)2 nd, 3 rd, … next –near-neighbor identity and distances. Assume the interatomic distance set by the lattice sites. Let : Bond energy Note that all the V’s are < 0. Bond energy The Regular Solution model assumes only nearest-neighbor interactions, pairwise. “Quasichemical” model

Consider a lattice of N sites with Z nearest-neighbors per site. : Each of the N atoms has Z bonds so that there are bonds in the lattice. Division by 2 is for double counting. Let P AA be the probability that any bond in the lattice is an A-A bond: then and so

The energy for the mixed solution is prior to mixing and

combining terms & using x A + x B = 1 where Notice the ΔH mix can be either positive or negative. ΔH mix > 0,  > 0 and V AB > 1/2 (V AA + V BB ) from ΔG mix = ΔH mix – TΔS mix At low temps clustering of As and Bs result, i.e., “phase separation”.

for ΔH mix < 0,  < 0 and V AB < 1/2 (V AA + V BB ) The A atoms are happier with B atoms as nearest-neighbors. Short range ordering i.e., P AB is increases over the random value. For a Regular Solution

Variation of ΔG mix with composition  < 0 Note2 minima change in curvature 2 pts of inflection.  > 0 T < T C x B → 01 –TΔS mix ΔG m ΔH m ΔG mix x B → 01 ΔH mix –TΔS mix ΔG mix

As T increases, the –TΔS term begins to dominate. The inflection pt. & extremum merge the critical temperature: X B → 01 T = T c ΔG mix T < T c