Lecture 18Multicomponent Phase Equilibrium1 Thermodynamics of Solutions Let’s consider forming a solution. Start with X A moles of pure A and X B moles of pure B at a given P, and T and mix them to form a single phase mixture of A and B at that same P and T. The change in the G in forming the mixture is the G of the solution minus G for each of the pure components: Since Then, Using the definition for the partial molar G of mixing G’ XBXB 0 1

Lecture 18Multicomponent Phase Equilibrium2 Extensive Properties Versus Composition Diagrams G’ 0 1 XBXB V’ 0 1 XBXB G’ 0 1 XBXB 0 1 XBXB What happens if we change T?

Lecture 18Multicomponent Phase Equilibrium3 Extensive Properties Versus Composition Diagrams G’ 0 1 XBXB 0 1 XBXB 0 1 XBXB 0 1 XBXB 0 1 XBXB 0 1 XBXB

Lecture 18Multicomponent Phase Equilibrium4 Construction of Phase Diagrams GMGM 0 XBXB 1   L GMGM 0 XBXB 1  L GMGM 0 XBXB 1   L E

Lecture 18Multicomponent Phase Equilibrium5 Phase Diagram Construction G’ 0 1 XBXB L  0 1 XBXB L  0 1 XBXB L  0 1 XBXB L  0 1 XBXB T 0 1 XBXB T L  L+ 

Lecture 18Multicomponent Phase Equilibrium6 Construction of a Eutectic Phase Diagram GMGM 0 XBXB T 1 0 XBXB 1  ++   +L L  +L   L GMGM 0 XBXB 1   L GMGM 0 XBXB 1   L GMGM 0 XBXB 1   L GMGM 0 XBXB 1   at T M (A) at T M (B) Above T E and below melting points at T E Below T E

Lecture 18Multicomponent Phase Equilibrium7 Example: Construction of a Phase Diagram GMGM 0 XBXB T 1 0 XBXB 1  +   L  +L  Low T 0 XBXB 1  L  XBXB  XBXB  L  XBXB  XBXB  L    L    L   L   +   +L  + L High T 

Lecture 18Multicomponent Phase Equilibrium8 Phase Diagrams T 0 XBXB 1  +   L  +   L +   +L  + L  T 0 XBXB 1  ++  L  +L T 0 XBXB 1  +   L  +    +L  + L  L +   +  T 0 XBXB 1   +   L  +L   +   +L T 0 XBXB 1   +   L  +L   +   +L  +  T 0 XBXB 1 L  +     +L  + L  +  L + 

Lecture 18Multicomponent Phase Equilibrium9 Flawed Phase Diagrams T 0 XBXB 1  +   L  +   L +   +L  + L  T 0 XBXB 1  ++  L  +L T 0 XBXB 1  +   L  +    +L  + L  L +   +  T 0 XBXB 1   +   L  +L   +   +L T 0 XBXB 1   +   L  +L  +   +L  +  T 0 XBXB 1 L  +     +L  + L  +  L +     +    +L 

Lecture 18Multicomponent Phase Equilibrium10 Lever Rule T 0 XBXB 1  ++  L  +L The fraction of each phase present in equilibrium is determined by doing a mass balance of a component. The resulting expressions are called the lever rule:

Lecture 18Multicomponent Phase Equilibrium11 Example: Lever Rule Example T 0 XBXB 1  ++  L  +L The fraction of each phase present in equilibrium is determined by doing a mass balance of a component. The resulting expressions are called the lever rule: T 0 XBXB 1  +   L  +    +L  + L  What are the fractions of each phase present at points shown on the two phase diagrams to the right?  +L

Lecture 18Multicomponent Phase Equilibrium12 Temperature Processing T 0 XBXB 1  ++  L  +L Heating from (a) first results in less and less  until you have only the  phase. Further heating results in melting  over a range of T. During this process the additional liquid that is formed becomes richer and richer in A until all of phase  is melted. a b Heating from (b) first results in making both the  and  phases less pure, although the fraction of each phase remains relatively constant. At the eutectic T heating results in melting  and above the eutectic T there is equilibrium between the  phase and the liquid phase. Further heating results in melting  over a range of T. During this process the additional liquid that is formed becomes richer and richer in A until all of phase  is melted. c Heating from (c) first results in making both the  and  phases less pure, and the fraction of each phase remains relatively constant. At the eutectic T heating results in melting both  and . Above the eutectic T there is only the liquid phase.

Lecture 18Multicomponent Phase Equilibrium13 Two-Phase Fields in Binary Phase Equilibria GMGM 0 XBXB 1  L GMGM 0 XBXB 1  L GMGM 0 XBXB 1  L   T 0 XBXB 1  L

Lecture 18Multicomponent Phase Equilibrium14 Miscibility Gap in Binary Systems GMGM 0 XBXB 1 For systems with negative enthalpies of mixing, both the entropy of mixing and enthalpy of mixing will lead to decreases in G of mixing at intermediate compositions. This will lead to complete miscibility of A and B. SMSM 0 XBXB 1 HMHM 0 XBXB 1 Increasing T

Lecture 18Multicomponent Phase Equilibrium15 Miscibility Gap in Binary Systems GMGM 0 XBXB 1 L T 0 XBXB 1   1 +  2 22 11  For systems with positive enthalpies of mixing, the entropy of mixing will become less important as T decreases. This will lead to phase separation into two phases rich in A and B at low T. As the temperature is lowered further, the positive enthalpy of mixing dominates and less and less mixing occurs, thus leading to a larger miscibility gap at lower temperatures.

Lecture 18Multicomponent Phase Equilibrium16 Spinodal Decomposition GMGM 0 XBXB 1 T 0 XBXB 1  1 +  2 22 11  GMGM XBXB GMGM 0 XBXB 1 The spinodal compositions are defined by the inflection points in the G vs X diagram. For a G vs X diagram as shown, compositions within the spinodal range lead to uphilldiffusion, but in the spinodal region two phase equilibrium will be kinetically hindered since uphill diffusion is not favorable in this range. Note that phase separation initially leads to increases for G outside the spinodal, and decreases in G inside the spinodal.