Partial Molar & Excess Quantities Notation convention Let G' stand for total free energy and in a similar fashion S', V', H', etc. Then we will let G = G'/n represent the free energy per mole, n, and in a similar fashion. S', V', H', etc. Consider a solution of volume V containing n, moles of a component 1, n 2 moles of a component 2 and etc. V V n1, n2, V V dV dn1 dn2 n1 T , P ,n2 ,n3 n2 T , P ,n2 ,n3 def V V Partial molar volume of component i

ni T , P , n j dV V1dn1 V2 dn 2 dH H1dn1 H 2 dn 2 dG G1dn1 G2 dn 2 A dG 1dn1 2 dn 2 Where the Gi i def the chemical potential of i Integrating the set of equation A , G 1n1 2 n2 V V1n1 V2 n2 etc B Differentiating the set B dG 1dn1 n1d1 2 dn2 n2 d 2

and comparing A and B n1d1 n2 d 2 0 Can also be written as etc Gibbs-Duhem eqns. x1d1 x2 d 2 0 *In the derivation of the Gibbs phase rule the Gibbs-Duhem eqns. play an important rule in restricting the allowed variations. Activity and the activity coefficient Let Poi correspond to the equilibrium vapor pressure of the pure component, i. Def : activity pi ai 0 pi Ideal Solution pi xi pi

Raoults law 1.0 ai xi by def of ai ai Raoults Law xi 1 Regular Solution pi bxi Henrys law bxi b ai xi gi xi pi pi by def of ai

gi is known as the activity coeff. (* g = 1 for ideal solution) gi is independent of Xi only when Xi is small 1.0 ai pos. and neg. deviations (g > or < 1) from ideality Henrys law xi 1.0 Consider a regular solution of components A and B 1.0 Raoults Law ai <0 Henrys law xB 1.0

in a Regular Solution the solute will often follow Henrys law and the solvent Raoults law when the solution is dilute. Partial molar Quantities in Ideal and regular solution Ideal Solution A G x AGA xB GB RT x A ln x A xB ln xB We can also write the Gibbs potential as B G A x A B xB Grouping terms in A and comparing coefficients of X A and XB in A and B A A0 RT ln x A 0 B B RT ln xB Where A0 G A and

B0 GB Regular Solutions: A G x A A0 xB B0 x A xB RT x A ln x A xB ln xB Using the identity Again comparing x A xB x A xB x A xB x 2A xB xB2 x A A and B 0 A 2 A 1 x A RT ln x A 0 B 2 B 1 xB RT ln xB This can be written in the form: A A0 RT ln a A

B B0 RT ln aB so that in the regular solution model: pA 2 1 x A a A 0 x A exp pA RT pB 1 xB 2 aB 0 xB exp pB RT Summarizing Ideal Solution ai xi Regular Solution 2 1 xi

ai xi exp RT How are the partial molar quantities related to the molar quantities ? Consider : dG A dx A B dxB dx A dxB Since xA + xB = 1 dG B A dxB A dG B A dxB Also using G A x A B xB G B xB

A 1 xB and substituting for A from A dG B G 1 xB dxB And similarly dG A G xB dx B Graphical Interpretation A A GGmix B G (XB )

* G B 0 X*B dG dx B X B* 1 XB dG A ( x ) G ( x ) x dX B x x*B * B * B

* B dG B ( x ) G ( x ) (1 x ) dxB x x*B * B * B * B Given G as a function of xB as above, the chemical potential @ a composition xB* can be obtained graphically by extrapolating the tangent of the G curve @ xB= xB* to xB= 0 and xB= 1 Chemical Equilibrium aA + bB + xX + yY + A, B reactants } standard states X, Y products a, b, ., x, y,. Stoichiometric Coefficients Equilibrium is defined by the condition

dG x x y y a A b B 0 Since standard free energy change /mole i i0 RT ln ai dG x x0 y 0y a 0A b B0 RT ( x ln ax y ln a y ) RT (a ln a A b ln a B ) 0 Solving for x y a 0 X aY RT ln a b RT ln K a A aB

K is the equilibrium constant for the reaction. The equilibrium constant is defined by K exp( ) RT When components are not in standard state: x y a 0 X aY RT ln a b a A aB Equilibrium in multiphase Solutions Consider 2 phases containing the same component i in and The component i has and activity ai and ai . Imagine and infinitesimal amount of i is transferred from to so that the compositions have not been altered. The reaction is: i (ai ) i (ai ) The chemical potential of i in each phase is i io RT ln ai i io RT ln ai The free energy change for the reaction at const T, P is,

ai G RT ln ai if ai ai => spontaneous (GG < 0) Equilibrium is defined by GG = 0 or ai ai ; i i In general for a multicomponent system with different phases present, equilibrium is defined by: A A Ag Ap B B Bg Bp

C C Cg Cp Equal P & T for all phases The Gibbs Phase Rule Equilibrium for different phases, , , g, p in contact with one another is given by the conditions; A p p pg p p T T Tg Tp C A A Ag Ap B B Bg Bp p phases present (p-1) eqns. for each component C components present C

C g C p C B 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. x AP xBP ... xCP 1 C - 1 composition variables per phase The Gibbs Phase Rule For each phase there are (C 1) + 2 = C + 1 independent variables or degrees of freedom. T, P The set of equations C 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.

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) PC+2 The Gibbs Phase Rule PC+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 f C 2 P 0 Gibbs Phase Rule

Geometrical Interpretation of Equilibrium in Multi-component Systems. G G ( x A , x B , T , p) ; G Consider a 2 phase binary : G A A A A 0 B x x x x B 1

B } Equilibrium B B The intercepts of the common tangent to the free energy curve give the chemical potentials defining the heterogeneous equilibrium. 0 xB x - single phase; x xB x - 2 phase; + + field x xB 1 - single phase; @ T and P Composition x* is in the 2 phase + field. Consider a simple rule of mixtures mass balance : x* f x f x

fi is the fraction of alloy composed of i f f 1 x* f x (1 f ) x * x x f x x x* f x x x These formulas are analogous to mass balance for a lever with fulcrum @ x*. => Lever Rule Binary Phase Diagrams A and B complete miscibility in both solid and liquid T1 Tmp(A) S S G

G l A xB B G xB T3 l l G xB B A G xB

S B A l B xB T1 T2 S S A l A Tmp(B) T2 b T3

B A c xs x * xl xB B At temp T2 x* is composed of some fl with composition c and fs at composition b: x* f s xs fl xl : b xs c xl Lever Rule xl x* x * xs fs ; fl

xl xs xl xs Ag-Au Si-Ge System with a miscibility gap T > Tc T = Tc G T < Tc G G f e A B A xB xB

B A B xB T << Tc Tc G g A Tc T < Tc h xB 2R B T <

A g e f h xB B 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, FeSn, Cr-W, NaCl-KCl, TiO2-SiO2. Free energy curves and phase diagram for Hsmix> HLmix= 0. 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. 1

T T2 G XB b a c d Solid Liquid T3 liquid T1 T2 b a

c f e Negative Curvature e T3 A XB d Negative Curvature f B Ordered Alloy Formation liquid liquid Ordered Alloy Formation

g + g A xB B Phase diagram for GHSmix < GHlmix < 0 A xB Phase diagram for GHSmix << GHlmix < < 0 Since HHSmix < 0, a maximum melting point mixture may appear.

B Eutectic Alloy; GHSmix >> GHlmix > 0. Components have same crystal structure If Hsmix >> 0, the miscibility gap can extend into the liquid phase (T 2, T3, T4) resulting in a simple eutectic phase diagram. A similar result can occur if the A and B components have a different crystal Eutectic phase diagram where each solid phase has a different crystal structure. The derivation of a complex phase diagram showing the formation of stable intermediate phases(). At a composition indicated by the red line, at a temperature just above T2, (green line) a solid at composition p is in equilibrium with a liquid at composition q. At a temperature T3 the two phases in equilibrium are solid and liquid . The following peritectic reaction occurs on cooling: l + p q