Major Concepts. Multiphase Equilibrium Stability Applications to Phase Equilibrium. Two-Phase Coexistence

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1 Major Concepts Multiphase Equilibrium Stability Applications to Phase Equilibrium Phase Rule Clausius-Clapeyron Equation Special case of Gibbs-Duhem wo-phase Coexistence Criticality Metastability Spinodal decomposition Equilibrium Stability III

2 Recall: dp d = Δ trs S m Δ trs V m Clapeyron Equation Solid-liquid boundary: Recall: dp d = Δ H fus Δ fus V Δ fus S = Δ fus H ( transition = fusion )! # d = " dp d almost does not depend on! # # " Δ fus H Δ fus V d always correct But if H and V are constant w.r.t., then P P dp Δ fus H Δ fus V ( *d ) P P Δ H fus Δ fus V ln ( * ) Recall: ln(+x) = x x / + ln(y) = (y ) (y ) / + P P + Δ H fus Δ fus V ) ) ( ( Q: Why are the other coexistence curves not linear? Q: Can * be 3? Equilibrum Stability III 3

3 Stability Criterion at Constant P Equilibrum Stability III 4

4 Stability Criterion at Constant Equilibrum Stability III 5

5 Clausius-Clapeyron Equation Recall: dp d = Δ trs S m Δ trs V m = Δ trs H m Δ trs V m Solid Liquid vapor boundary: Assumptions: ( sublimation, vaporization ) Why? Δ trs V m V m (g) Ideal gas behavior: V m (g) = R/P Ideal gas behavior: Δ trs H is weakly dependent on. dp d = Δ H trs m R P dp dp P = P = Δ H trs m d R P " ln P " = Δ H " trs # P # R # (Clausius-Clapeyron Equation) P Δ trs H R d Equilibrum Stability III 6

6 Vapor Pressure hink-pair-share, but not multiple choice! Equilibrum Stability III 7

7 Classifying Phase ransitions (ala Ehrenfest) Recall: " # µ α P " µ β # P = Δ trs S m = Δ trs H m " # H P = C P st order: µ α µ β pc µ H (continuous with kink) (discontinuous) C p solid-liquid-gas trans. µ α = µ β nd order: µ α µ but β C< p µ (continuous) H C p (continuous with kink) (discontinuos) conductingsuperconducting transition λ-transition: but C p µ α = µ β µ H C p superfluidity of He, ferromagnetism, order-disorder trans. (in alloys) (continuous) (extreme kink) Equilibrum Stability III 8

8 Multiphase Equilibrium E = E E (g) E (l) -phase system -phase -component system α labels the phases i labels components E (g) (g), E E (l) (l), E r E = E i i= surface terms are ignored since E(vol) N E(surf) N /3 E(surf) E(vol) N /3 if surface terms cannot be ignored, then they must be added as additional phases E (g) E (s) E (l) Equilibrium Stability III 9

9 Multiphase Equilibrium -phase r-component system E (g), E (g) (α E ) (α, E ) (β E ) (β, E ) E (l), E (l) Extensive variables: E = E V = V S = S n i = n i E (γ ), E (γ ) # r δe = δe = δs p δv (α + µ ) i δn i ( i= Equilibrium Stability III 30

10 Multiphase Equilibrium -phase r-component system E (g), E (g) (α E ) (α, E ) (β E ) (β, E ) E (l), E (l) Extensive variables: E = E V = V S = S n i = n i E (γ ), E (γ ) # r δe = δe = δs p δv (α + µ ) i δn i ( i= Equilibrium Stability III 3

11 Multiphase Equilibrium -phase r-component system E (g), E (g) (α E ) (α, E ) (β E ) (β, E ) E (l), E (l) Extensive variables: E = E V = V S = S n i = n i E (γ ), E (γ ) # r δe = δe = δs p δv (α + µ ) i δn i ( i= Equilibrium Stability III 3

12 Multiphase Equilibrium -phase r-component system E (g), E (g) (α E ) (α, E ) (β E ) (β, E ) E (l), E (l) Extensive variables: E = E V = V S = S n i = n i E (γ ), E (γ ) # r δe = δe = δs p δv (α + µ ) i δn i ( i= Equilibrium Stability III 33

13 Multiphase Equilibrium -phase r-component system E (g), E (g) (α E ) (α, E ) (β E ) (β, E ) E (l), E (l) Extensive variables: E = E V = V S = S n i = n i E (γ ), E (γ ) # r δe = δe = δs p δv (α + µ ) i δn i ( Equilibrium ( δe) S,V,ni 0 S, V, n i must be constant: δv i= = 0, δs = 0, δn i = 0 Equilibrium Stability III 34

14 -phase r-component system: =, i= r E (), E () E (), E () Equilibrium ( δe) S,V,ni 0 Multiphase Equilibrium 0 ( δe) S,V,ni = ( () () )δs () p () p () S, V, n i must be constant: δv = 0 δv () = δv () δs = 0 δs () = δs () δn i = 0 δn i () = δn i () " () = () (for unconstrained system) # p () = p () µ () () i = µ i ( )δv () + ( µ () () () i µ i )δn i r i= ( δe) S,V,ni = 0 Equilibrium Stability III 35

15 At equilibrium ( δe) S,V,ni = 0 Stability Mechanical analogue: H(x) unstable stable x ΔH = δh +δ H +... = dh dx δx + Equilibrium: Stability: δ H = δh = dh dx δx = 0 d H dx δx > 0 d H dx δx +... Equilibrium Stability III 36

16 At equilibrium ( δe) S,V,ni = 0 Stability ( ΔE) S,V,ni = ( δe) S,V,ni + ( δ E) +... S,V,ni ( δ E) > 0 S,V,ni ( δ E) < 0 S,V,ni stable unstable hese inequalities hold for all free energies! Equilibrium Stability III 37

17 -component system: Stability: Example de = " # ds p dv + µ dn If compartments exchange only heat, then δs = 0 = δs () + δs (), δv () = δv () = 0, δn () = δn () = 0, and () () ( δ E) = V,n " # E S V,n ( δ E) = ( δ E) () S,V,n V,n ( δs ) = + ( δ E) () = V,n " # S (" * )* # V,n S ( δs ) () V,n " + # S () + -, ( ) > 0 V,n- δs() # S ( V,n = C v 0 at equilibrium. Equilibrium Stability III 38

18 -component system: he volumes can fluctuate: ( δ A) =,n Stability: Example da = " # S d p dv + µ dn " A # V,n ( δ A) = ( δ A) (),V,n,n () () δv = 0 = δv () + δv (), δn () = δn () = 0, and ( δv ) = + ( δ A) () =,n " p # V,n )# p + ( * + V ( δv ) (),n # + p ( V (),n,. -. ( () δv ) > 0 p V ) (,n > 0 or the isothermal compressibility K = V # V p (,n > 0 Equilibrium Stability III 39

19 Phase Equilibria -phase r-component system E (g), E (g) (α E ) (α, E ) (β E ) (β, E ) E (l), E (l) (γ E ) (γ, E ) At equilibrium " () = () =... # p () = p () =... µ () i = µ () i =... ( δe) S,V,ni = 0 µ s are the functions of, p and r mole fractions for each phase α : (α µ ) (α i = µ ) (α i (, p, x ) (α, x ) r ) (α x ), x r At equilibrium (α µ ) (α i (, p, x ) (α x ) (β r ) = µ ) (β i (, p, x ) (β x ) (γ r ) = µ ) (γ i (, p, x ) (γ x ) r ) =... here are r ( ) equations and (r ) + variables. he number of degrees of freedom is f = (r )+ r( ) = + r he Gibbs phase rule Equilibrium Stability III 40