1. Notation convention Let G' stand for total free energy and in a similar fashion S', V', H', etc. Then we will let = G'/n represent the free energy per mole, n, and in a similar fashion. S', V', H', etc. Partial Molar & Excess Quantities

2. Consider a solution of volume V’ containing n, moles of a component 1, n 2 moles of a component 2 and etc. def Partial molar volume of component i

3. Where thedef the chemical potential of i A Integrating the set of equation A, B etc

4. Can also be written as Gibbs-Duhem eqns. Differentiating the set B and comparing and AB *In the derivation of the Gibbs phase rule the Gibbs-Duhem eqns. play an important rule in restricting the allowed variations.

5. Activity and the activity coefficient Def : activity Ideal Solution Let P o i correspond to the equilibrium vapor pressure of the pure component, i. aiai 1.0 xixi 1 Raoult’s Law Raoult’s law by def of a i

6. Regular Solution Henry’s law by def of a i  i is known as the activity coeff. (*  = 1 for ideal solution)  i is independent of X i only when X i is “small” aiai 1.0 xixi Henry’s law pos. and neg. deviations (  > or < 1) from ideality

7. Consider a regular solution of components A and B in a Regular Solution the solute will often follow Henry’s law and the solvent Raoults law when the solution is “dilute”.  < 0 aiai 1.0 xBxB Henry’s law Raoult’s Law

8. Partial molar Quantities in Ideal and regular solution Grouping terms in and comparing coefficients of X A and X B in and A A B Whereand Ideal Solution A We can also write the Gibbs potential as B

9. Regular Solutions: A Using the identity Again comparing and AB This can be written in the form:

10. so that in the regular solution model: Summarizing Ideal Solution Regular Solution

11. How are the partial molar quantities related to the molar quantities ? Consider : Since x A + x B = 1 A

12. and substituting for  A from A And similarly Also using

13. Graphical Interpretation Given G as a function of x B as above, the chemical potential @ a composition x B * can be obtained graphically by extrapolating the tangent of the G curve @ x B = x B * to x B = 0 and x B = 1 01 G (X B * ) ΔG mix X* B G XBXB

14. Chemical Equilibrium aA + bB + …  xX + yY + … A, B … reactants X, Y … products } standard states a, b, …., x, y,…. Stoichiometric Coefficients Equilibrium is defined by the condition Since ≣ standard free energy change /mole

15. K is the equilibrium constant for the reaction. Solving for The equilibrium constant is defined by When components are not in standard state:

16. Equilibrium in multiphase Solutions Consider 2 phases containing the same component i in α and β The component i has and activity a α i and a β i. Imagine and infinitesimal amount of i is transferred from α to β so that the compositions have not been altered. The reaction is: The chemical potential of i in each phase is

17. if => spontaneous (ΔG < 0) Equilibrium is defined by ΔG = 0 or ; In general for a multicomponent system with different phases present, equilibrium is defined by: Equal P & T for all phases … The free energy change for the reaction at const T, P is,

18. The Gibbs Phase Rule Equilibrium for different phases, , , , …p in contact with one another is given by the conditions; A B … C Equilibrium will in general not be maintained if the parameters are arbitrarily varied. The Gibbs phase rule restricts the manner in which the parameters can be varied such that equilibrium is maintained. The composition of a given phase is set by the additional condition. C - 1 composition variables per phase C components present p phases present (p-1) eqns. f or each component

19. For each phase there are (C – 1) + 2 = C + 1 independent variables or degrees of freedom. C The set of equations represent a system of coupled linear equations. There are ( P – 1) equations for each component and C components so we have (P – 1)(C+2) equations. T, P P (C +1) unknowns (P – 1)(C+2) equations In order for a solution to these equations to exist, # unknowns  # of equations P (C +1)  (P – 1)(C+2) P  C + 2 The Gibbs Phase Rule

20. P  C + 2 For a system of C independent components not more than C + 2 phases can co-exist in equilibrium. Trivial Example:C = 1, P = 3 solid, liquid vapor If P is less than C + 2 then C + 2 – P variables can take on arbitrary values (degrees of freedom) without disturbing equilibrium. def. Thermodynamic degrees of freedom, f Gibbs Phase Rule The Gibbs Phase Rule

21. Geometrical Interpretation of Equilibrium in Multi-component Systems. Consider a 2 phase binary : } Equilibrium 1 0 G x B →   The intercepts of the common tangent to the free energy curve give the chemical potentials defining the heterogeneous equilibrium. - single phase;  - 2 phase;  field @ T and P - single phase; 

22. Composition x* is in the 2 phase α + β field. Consider a simple rule of mixtures mass balance : f i is the fraction of alloy composed of i These formulas are analogous to mass balance for a lever with fulcrum @ x*. => “ Lever Rule”

23. Binary Phase Diagrams x B → AB T1T1 G S l AB T mp (A) G S l x B → AB T2T2 G S l G AB T mp (B) S l A and B complete miscibility in both solid and liquid x B → AB T3T3 G S l AB T1T1 T2T2 T3T3 x*x* b c xsxs xlxl

24. At temp T 2 x* is composed of some f l with composition c and f s at composition b: Lever Rule Ag-Au Si-Ge

25. System with a miscibility gap AB T > T c G x B →xB →xB → AB T = T c G x B → AB T << T c G gh G x B → AB T < T c e f A B T < T’ c T < <T’ c x B → TcTc ef g h  ’’

26. The miscibility gap is the region where the overall composition exceeds the solubility limit. The solid solution a is most stable as a mixture of two phases  ’ +  ’’. Usually ,  ’, and  ’’ have the same crystal structure. Cu –Pb, Au-Ni, Fe-Sn, Cr-W, NaCl-KCl, TiO 2 -SiO 2.

27. G T1T1 XBXB Solid Liquid T2T2 a b c d T3T3 f e Negative Curvature a b c d f e   ’’ T3T3 T2T2 T1T1 liquid A B XBXB Free energy curves and phase diagram for ∆H s mix > ∆H L mix = 0.

28. The A and B atoms dislike each other. Note that the melting point of the alloy is less than that of either of the pure phases.

29. xB →xB → AB ’’  liquid Phase diagram for ΔH S mix < ΔH l mix < 0 Phase diagram for ΔH S mix << ΔH l mix < < 0 Since ΔH S mix < 0, a maximum melting point mixture may appear. Ordered Alloy Formation B liquid xB →xB → A   +     Ordered Alloy Formation

30. Eutectic Alloy; ΔH S mix >> ΔH l mix > 0 A B T1T1 G solid l x B → A B T3T3 solid l l α1α1 l+  1 l+αa2l+αa2 a2a2 A B T4T4 solidl A B T5T5 l A B T3T3 T4T4 TbTb T2T2 T1T1 T5T5 x B → T TATA A B T2T2 solid l l 11 l+l+ 11 l1l1 l2l2 22 11 1+1+ 22 l 11 22 l+  1 l+  2 1+1+

31. If  H s mix >> 0, the miscibility gap can extend into the liquid phase (T 2, T 3, T 4 ) resulting in a simple eutectic phase diagram. A similar result can occur if the A and B components have a different crystal s tructure.

32. A B T2T2 l α β l α l+ α l+ β β T4T4 l β α α β α+ β AB A B T1T1 G l x B → β β α α l+ α l A B T3T3 l β α α l+ β β A B α l+ α l+ β α +βα +β β T2T2 T3T3 TbTb T1T1 T4T4 x B → T TATA l Eutectic phase diagram where each solid phase has a different crystal structure.

33. A B T1T1 liquid β α γ A B T2T2 β α γ α α +β+l l l A B T3T3 β α γ α +βα +β l l α ββ+l A B T4T4 β α α +βα +β l l+γl+γ α β γ l A B T5T5 β α α +βα +β α β β+l+ γ l

34. The derivation of a complex phase diagram showing the formation of stable intermediate phases(β). At a composition indicated by the red line, just above T 2, (green line) a solid at composition P is in equilibrium with a liquid at composition Q. At a temperature just below T 2 ( not shown ) the two phases in equilibrium are solid  and liquid. The following peritectic reaction occurs on cooling: l + α → β