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Equilibrium and Stability Criteria: A Comprehensive Overview

Explore the principles of equilibrium and stability in unit processes and operations like chemical reactions, distillation, absorption, and more. Understand phases, potential energy, and the Second Law for system stability.

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Equilibrium and Stability Criteria: A Comprehensive Overview

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  1. Ch. 5 Criteria for Equilibrium & Stability • 1. Unit process: • Chemical reactions • 2. Unit operation: mass transfer • Distillation • Absorption & Adsorption • Extraction & leaching • Membrane separation

  2. Multiple phase, multiple component systems 1. Solubility (i) complete miscible : single homogeneous phase (ii) partial miscible : multiple phases (iii) immiscible: multiple phases 2. Types of phase (i) Liquid –liquid (extraction) (ii) Vapor – liquid (distillation; gas absorption) (iii) Solid –liquid (leaching; crystallization) (iv) Solids in solids (zone melting; crystallization)

  3. Limit of Intrinsic stability

  4. Classification of equilibrium • In tem of potential energy Unstable equilibrium Neutral stable Metastable equilibrium Potential energy Stable equilibrium barrier

  5. Postulate II & The Second Law ☆ Postulate II: In processes for which there is no net effect on the environment, all the systems with a given internal constraints will change in such a way to approach one and only one stable equilibrium state ☆ The Second Law ΔSUniv. (= ΔSisolated ) = ΔSsys + ΔSHRs = ΔSU, V, N≧ 0 (4-31) for all natural processes, or unstable state systems Since an isolated system, Q = 0, W = - P ΔV = 0, V = const, Further using the first law for close system, ΔU = Q + W, then ΔU = 0, or U = const If the stable processes, or stable state system, ΔSU, V, N < 0

  6. Criteria for stable state system ΔSU, V,N < 0 equilibrium ΔS = 0 S ΔS<0 ΔS>0 unstable stable One of the properties of system

  7. ΔSU,V,N < 0

  8. Perturbation If minor perturbations leave the system unchanged, we define the original state as a stable equilibrium state. So(Z1o,) S S S Zio - δZi Zio + δZi Zio Zi ΔS = S - So =δS + (1/2!) δ2S + (1/3!) δ3S + ….. < 0

  9. ΔS = S - So = δ S + (1/2!) δ2S + (1/3!) δ3S + ….. < 0 (6-3) where δ S = Σ (∂S/∂Zi) δZi δ2 S = ΣΣ(∂2S / ∂Zi ∂Zj)δZiδZj δ3 S =ΣΣΣ (∂3S / ∂Zi ∂Zj ∂Zk )δZiδZjδZk (i) δU = 0 (ii) δV = 0 (iii) δNi = 0, i = 1, 2, ….., i, …., n

  10. (1) The criterion of equilibrium • δ S = Σ δ S (k) = O (6-6) • (2) The criterion of stability • δ2S =Σ δ2S (k) ≦ 0, if = 0 • δ3S =Σ δ3S (k) ≦ 0 ≦ 0, if = 0, (6-7) • δjS < 0 • (3) The constrains (Isolated system) • (i) δU =ΣδU (k) = 0 • (ii) δV =ΣδV (k) = 0 • (iii) δNi =ΣδNi(k) = 0, i = 1, 2, ….., i, …., n

  11. Equilibrium Criteria Derived from the combination of the first and second laws ΔSU,V,N = ΔSsystem+ ΔSheat resv. > 0 (1) for irrversible spontanoeus processes ΔSheat resv = Q heat resv /To ( To = Theat resv ) and Q heat resv =-Q system =Q ΔS - Q/ To. < 0 (2) Applying the1st law for the system, ΔU = Q + W Q = ΔU - W (3) Combine (2) and (3), ΔS – (ΔU - W )/ To < 0 ) ΔU - To.ΔS < W = - WwR Only consider P-V work, WwR = [ (- PoΔVo )], and ΔVo = -ΔVsys

  12. ΔU + Po ΔV – To ΔS <0 Equilibrium criterion for natural processes (1) ΔV=0 , i.e., V = const., ΔS=0 , S = const ΔUS,V, N < 0 (2) ΔV=0 , V = constant, To.= T = constant ΔU - TΔS < 0 , ΔU - Δ (TS) < 0 Δ(U - TS) < 0 The Helmholtz free energy, A, is defined as A = U - TS ΔA < 0, ΔAT, V, N < 0 (6-28)

  13. (3) ΔS=0 , T = constant, ΔS=0 , S = constant, ΔU + PΔV < 0, Δ[U + PV ] < 0, ΔH < 0, ΔHS, P, N < 0 (6-27) (4) Po = P = constant, To.= T = constant • ΔU + Δ(PV) -Δ(TS) <0 H = U + PV Δ (U + PV -TS) < 0 The Gibbs-free energy is defined as G = U + PV –TS ΔG < 0 ΔGT, P, N < 0

  14. Criterion based on H, A and G ΔUΣ= Δ(Usys + UHR + UWR ) > 0 (i) ΔS Σ= Δ(Ssys + SHR)= ΔSWR = 0 (ii) ΔVΣ = Δ(V + VWR)= ΔVHR = 0 (iii) ΔN Σ = ΔNsys = ΔN HR= ΔN WR = 0 j

  15. 1. Enthalpy, lock the thermal gate, ΔUHR = 0 ΔUΣ = Δ(Usys + UWR ) > 0 (i) ΔSΣ = ΔSsys = ΔSHR = ΔSHR = 0; S = constant (ii) ΔVΣ = Δ(Vsys + VWR) = ΔVHR = 0 (iii) ΔNiΣ = ΔNisys= 0; Ni = constant, i = 1, 2,…., n Apply the first law for the work reservoir (WR) ΔUWR = - P WR ΔVWR = P ΔV (let PWR = Psys = P ) ΔU Σ= ΔU + P ΔV > 0, If P = constant ΔU Σ= ΔU +Δ(P V) > 0 ΔU Σ= Δ(U +P V) > 0 Since H = U + P V , ΔH S, P, N > 0

  16. 2. Helmholtz free energy, lock the piston, ΔUWR = 0 ΔUΣ = Δ(U + UHR ) > 0 (i) ΔSΣ = ΔS + ΔSHR = 0 (ii) ΔVΣ = ΔV = ΔVHR = ΔVWR = 0; V = constant (iii) ΔN Σ = ΔN = ΔNHR= ΔN WR = 0; N = constant Apply the first law for the heat reservoir (HR) ΔUHR = THR ΔSHR = - TΔS (If TWR = T = constant) ΔUΣ = ΔU + [ -Δ(TS)]= Δ(U - TS) > 0 A= U - TS ΔAT, V, N > 0

  17. 3. Gibbs free energy, open both the piston and the thermal gate, ΔUΣ= Δ(U + UHR + UWR ) > 0 (i) ΔSΣ = ΔS + ΔSHR = ΔSWR = 0 (ii) ΔVΣ = (ΔV + ΔVWR ) = ΔVHR = 0 (iii) ΔNΣ = ΔN = ΔNHR = ΔNWR = 0 Apply the first law for the HR and WR ΔUHR = THR ΔSHR = - T ΔS (If TWR = T = constant) ΔUWR = - PWR ΔVWR = P ΔV (If PWR = P = constant ΔUΣ= Δ(U + P ΔV - T ΔS) = Δ(U + PV - TS) > 0 G = U + PV - TS ΔGT, P, N > 0

  18. The criterion of stability δ2S = (∂2S/∂U2)VNδU2 + 2 (∂2S/∂U∂V)N[i]δUδV + (∂2S/∂V2)UNδV2 + 2Σ[ (∂2S/∂U∂Ni),V,N[i]δU+(∂2S/∂V∂Ni),U, N[i] δVi] δNi + ΣΣ(∂2S/∂Ni∂Nj)U,VδNiδNj < 0 δ2S =(SUUδU2 + 2 SUVδUδV + SVVδV2 + 2Σ[SU N[i]δU +SU N[i]δV]δNi+ ΣΣSNiNjδNiδNj < 0 The constraints are δU = 0; δV = 0; δNi = 0, i= 1. 2,……, n

  19. Taking ΔUS, V, N < 0 as the criterion ΔUS,V, N (= U -Uo ) = δU + (1/2!) δ2U + (1/3!) δ3U + …> 0 (1)The criterion of equilibrium δUS,V,N = δU = Σ(∂U/ ∂Zi) δZi = O (2)The criterion of stability δ2U = ΣΣ(∂2U/ ∂Zj ∂Zj)δZiδZj = ≧ 0, if = 0 δ3U = ΣΣΣ (∂3U/ ∂Zi ∂Zj ∂Zk)δZiδZjδZk≧ 0, if = 0, ……………….. If = 0 δjU > 0 The constraints (i) δS = 0 • (ii) δV = 0 • (iii) δNi = 0, i = 1, 2, ….., i, …., n

  20. Consider the system is perturbed to become two phase α and β, δ2S = δ2Sα + δ2S β = {(∂2S/∂U2)VNδU2 + 2 Σ(∂2S/∂U∂V )NδUδV + (∂2S/∂V2)UNδV2 + 2Σ [(∂2S/∂U∂Ni)V,[Ni]δU+(∂2S/∂V∂Ni)U, N[i]δVi ]δNi + 2 Σ(∂2S/∂Ni∂Nj)U,V,N[i,j]δNiδNj }α + the similar terms of the β phase < 0 δ2S= {(SUUδU2 + 2 SUVδUδV + SVVδV2 + 2ΣSUN[i]δUδNi+ 2ΣSVN[i]δVδNi] + ΣΣSNiNjδNiδNj }α + the similar terms of the β phase < 0 < 0(7-1)

  21. The equations for criterion of stability based on δ2S or δ2U δ2S= (N/Nβ ){(SUUδU2 + 2 SUVδUδV + SVVδV2 + 2ΣSUN[i]δUδNi+ 2ΣSVN[i]δVδNi] + ΣΣSNiNjδNiδNj }α < 0 ( 7-5) The similar procedure is used for the criterion of the internal energy, δ2U= (N/Nβ ){(USSδS2 + 2 USVδSδV + UVVδV2 + 2ΣUSN[i]δSδNi+ 2ΣUVN[i]δVδNi] + ΣΣUNiNjδNiδNj }α > 0 ( 7-6) The criterion of stability for the pure component system, for instance, δ2U =(N/Nβ) (USS δZ12 + A22 δZ22+ G33 δZ3 2) α > 0 , then USS> 0 , AVV> 0 , G33 > 0

  22. The mathematic treatment for obtaining the criterion of stability U = y (o) = U(S, V, N1, N2, ……, Ni, ……. Nn) y (o) = f (x1, x2, ……, xi, ……. xm) m = n + 2 δ2U= (N/Nβ ){(USSδS2 + 2 USVδSδV + UVVδV2 + 2ΣUSN[i]δSδNi+ 2ΣUVN[i]δVδNi] + ΣΣUNiNjδNiδNj }α > 0 (7-6) δ2y(o) = K ΣΣ y (o)ijδxiδxj > 0 (7-8) = (7-6) ΣΣ y (o)ijδxiδxj = Σ ykk(k-1)δZk2 > 0 k = 1, 2,….., m (7-9) δZk = {δxk + (y(o)23/ y(o)11)δx2+ (y(o)33/ y(o)11))δx3} k = 1, 2, ……,m δZm= δxm for k = m ykk(k-1) Zk2 > 0, and Zk2 > 0 ykk(k-1) > 0, k = 1, 2,….., m = n + 2 y11(o) > 0, y22(1) > > 0, y33(2) > 0, …………, ymm(m-1) (= y(n+2)(n+2)(n+1) ) > 0

  23. The Legendre transform, y(o) = f (x1, x2, ……, xi, ……. xm) d y(o) = Σ {(∂ y (o)/∂xi)x[i] dxi = Σ ξi dxi And ξi = {(∂ y (o)/∂xi)x[i] = y i(o) y (j) = f (ξ1, ξ2, … ξj , xj+1, ……. xm) = y(o) - Σξi xi d y (j) = - Σxi dξi + Σξi dxi y i(j) = {(∂ y (j)/∂ xi)ξ,x[i] = ξi = y i(o) , i > j y i(j) = {(∂ y (j)/∂ξi )ξ[j],x = - xi, i ≦ j y (m-1) = f (ξ1, ξ2, … ξj , …..ξm-1, xm) , m = n+2 y (n+1) = f (ξ1, ξ2, … ξj , …..ξn+1, xm) j j j+1

  24. ymm(m-1) = y (n+2)(n+2)(n+1) = (∂2 y (n+1) /∂xn+2 2)ξ • = (∂/∂xn+2 )(∂y (n+1)/ ∂xn+2 )ξ • yx(n+2)(n+1) =(∂y (n+1)/ ∂xn+2 )= y(n+2)(o) = ξn+2 • y (n+2)(n+2)(n+1) = (∂ ξn+2 /∂xn+2 )ξ • Since ξn+2 = f(ξ1 ,ξ2 , ………., ξn+1) • An intensive property,ξn+2, is determined by the other (n + 1) intensive properties, ξi , Therefore, as other (n + 1) intensive properties are fixed, ξn+2 should be constant, • ymm(m-1) = y (n+2)(n+2)(n+1) = (∂ ξn+2 /∂xn+2 )ξ= 0

  25. ymm(m-1) = y (n+2)(n+2)(n+1) = (∂2 y (n+1) /∂xn+2 2)ξ • = (∂/∂xn+2 )(∂y (n+1)/ ∂xn+2 )ξ • yx(n+2)(n+1) =(∂y (n+1)/ ∂xn+2 )= y(n+2)(o) = ξn+2 • y (n+2)(n+2)(n+1) = (∂ ξn+2 /∂xn+2 )ξ • ξn+2 = f(ξ1 ,ξ2 , ………., ξn+1) • An intensive property,ξn+2, is determined by the other (n + 1) intensive properties, ξi , Therefore, as other (n + 1) intensive properties are fixed, ξn+2 should be constant, • ymm(m-1) = y(n+2)(n+2)(n+1) = (∂ ξn+2 /∂xn+2 )ξ= 0

  26. y11(o) > 0, y22(1) > 0, … , y(m-1)(m-1)(m-2) > 0, ymm(m-1) (= y(n+2)(n+2)(n+1) ) = 0 The number of criterion becomes y11(o) > 0, y22(1) > 0, … , y(m-1)(m-1)(m-2) > 0, ykk(k-1) > 0, k = 1, 2, …….., m-1 y(k-2) = f(ξ1 ,…,ξ(k-2) , x(k-1) ,…….., xm ) y(k-1) = f(ξ1 ,…………,ξ(k-1) , xk ,…….., xm ) By the step down procedure, ykk(k-1) = ykk(k-2) - [yk(k-1) (k-2) ]2/ y(k-1)(k-1) (k-2) (1) ykk(k-2) > 0, y(k-1)(k-1) (k-2) > 0 (2) Decrease y(k-1)(k-1) (k-2), increase [yk(k-1) (k-2) ]2/ y(k-1)(k-1) (k-2), As continue to decrease y(k-1)(k-1) (k-2) , [yk(k-1) (k-2) ]2/ y(k-1)(k-1) (k-2) will be increased, and making{ ykk(k-2) - [yk(k-1) (k-2) ]2/ y(k-1)(k-1) (k-2)} approach to 0. Further. decrease y(k-1)(k-1) (k-2), { ykk(k-2) - [yk(k-1) (k-2) ]2/ y(k-1)(k-1) (k-2)} becomes smaller than negative, i.e., ykk(k-1) < 0

  27. It indicates the term,, ykk(k-1) becomes negative before y(k-1)(k-1) (k-2)keeps positive does, in other words, only if y(k-1)(k-1) (k-2) > 0, ykk(k-1) should be positive. Consequently, the necessary and sufficient condition criterion of stability, y(m-1)(m-1) (m-2) > 0 (7-15) The limit of stability or spinnodal condition is y(m-1)(m-1) (m-2) = 0 (7-16) y(m-1)(m-1) (m-2)= £i /Π y(r +1-)(r+1) (r)0 ≦i ≦ m -2 (7-20) For a stable system, £i > 0 (7-17) At the limit of stability, £i= 0 (7-18) m = 3 r = i

  28. y (i+1) (i+1)(i), y (i+1) (i+2)(i), …….. …. y(i+1) (m-1)(i) y (i+2) (i+1)(i), y (i+2) (i+2)(i), …….. …. y(i+2) (m-1)(i) y (i+3) (i+1)(i), y (i+3 )(i+2)(i), …….. …. y(i+3) (m-1)(i) ……………………………………. y (m-1) (i+1)(i), y (m-1) (i+2)(i), …….. …. y(m-1) (m-1)(i) • £i = 0 ≦i ≦ m -2 £3 = y(m-1)(m-1) (m-2)= y44(3)= £2 /y33(2)= £1 /y33(2)y22(1)= £o /y33(2)y22(1)y11(o)

  29. Multiple Phases multiple components equilibrium Π • A multi-phases (Π), multi-component (n) system • δS = ΣδS (s) = Σ {[1/T (s)] δ U(s) + [P (s)/T (s)] δ V(s) –Σ[μi(s)/T (s)] δNi(s) }= 0 δU = 0 = ΣδU(s) δV = 0 = ΣδV(s) δNi= 0 = Σδ Ni(s) ,i = 1, 2,…., j,…., n • δS = Σ [1/T(k) - 1/T(1) ] δ U(k) + [P (k)/T (k) - P (1)/T(1) ] δ V(k) – Σ • [μi(k)/T (k) - μi(1)/T(1) ] δNi(k) = 0 • T(k) = T(1) • P(k) = P(1) k = 1, 2,…., Π • μi(k) =μi(1) , i=1,2,…., n n Π Π Π Π n k ≠1

  30. Chemical Reaction equilibria • ν1C1 + ν2C2 + ….. + νiCi =0 • ΣνjCj = 0 • δN1/ν1 = δN2/ν2 = ………= δNi/νi = δξ • δNj/νj = δξ , j = 1, 2,…i • ΣδNj (s) = νjδξ • For the inert components, ΣδNj (s) = 0, j = i + 1, …n • δS = Σ [1/T(k) - 1/T (1) ] δ U(k) + [P (k)/T (k) - P (1)/T(1) ] δ V(k) – (1/T (1))Σνjμjδξ -Σ[μi(k)/T (k) - μi(1)/T(1) ] δNi(k) = 0 T(k) = T (1); P(k) = P (1); Σνjμj = 0; μi(k) =μi(1) , i = i+1,….., n

  31. Membrane Equilibrium

  32. The internal wall is movable, diathermal, and permeable to both A and B δU = 0 = δU(1) + δU (2) δV = 0 = δV(1) + δV (2) δNA = 0 = δ NA(1) + δ NA(2) δNB = 0 = δ NB(1) + δ NB(2) T(1) = T(2) ; P(1) = P(2) ; μA(1) = μA(2) ; μB(1) = μB(2)

  33. Case (a) The internal boundary is permeable only to B, diathermal and movable δNA = 0 = δ NA(1) = δ NA(2) δNB = 0 = δ NB(1) + δ NB(2) • δS = [1/T(1) - 1/T(2) ] δ U(1) + [P (1)/T (1) - P (2)/T(2) ] δ V(1) – • [μB(1)/T (1) - μB(2)/T(2) ] δNB(1) = 0 • T(1) = T(2) • P(1) = P(2) • μB(1) = μB(2)

  34. Case (b) The internal boundary is rigid, diathermal, and permeable to both A and B, δU = 0 = δU(1) + δU (2) δV = 0 = δV(1) = δV (2) δNA = 0 = δ NA(1) + δ NA(2) δNB = 0 = δ NB(1) + δ NB(2) • δS = [1/T(1) - 1/T(2) ] δ U(1) - [μA(1)/T (1) - μA(2)/T(2) ] δNA(1) – • [μB(1)/T (1) - μB(2)/T(2) ] δNB(1) = 0 • T(1) = T(2) • μA(1) = μA(2) • μB(1) = μB(2)

  35. Case (b) The internal boundary is movable, adiabatic, and permeable to both A and B δU = 0 = δU(1) = δU (2) δV = 0 = δV(1) + δV (2) δNA = 0 = δ NA(1) + δ NA(2) δNB = 0 = δ NB(1) + δ NB(2) • However, mass interchange between the subsystems, can vary the energy of each compartment; thus in reality, no additional restrains, and at equilibrium, • T(1) = T(2) ; P(1) = P(2) ; μA(1) = μA(2) ; μB(1) = μB(2)

  36. Derivation for Equations (7-5), (7-6) and (7-9) δ2S= (N/Nβ ){(SUUδU2 + 2 SUVδUδV + SVVδV2 + 2ΣSUN[i]δUδNi +2ΣSVN[i]δVδNi] + ΣΣSNiNjδNiδNj }α < 0 ( 7-5) The similar procedure is used for the criterion of the internal energy, δ2U= (N/Nβ ){(USSδS2 + 2 USVδSδV + UVVδV2 + 2ΣUSN[i]δSδNi +2ΣUVN[i]δVδNi] + ΣΣUNiNjδNiδNj }α > 0 ( 7-6) The mathematic expression, δ2y(o) = K ΣΣ y (o)ijδxiδxj > 0 ΣΣ y (o)ijδxiδxj = Σ ykk(k-1)δZk2 > 0 k = 1, 2,….., m

  37. Consider the system is perturbed to become two phase α and β, δ2S = δ2Sα + δ2S β = {(∂2S/∂U2)VNδU2 + 2 Σ(∂2S/∂U∂V )NδUδV + (∂2S/∂V2)UNδV2 + 2Σ [(∂2S/∂U∂Ni)V,[Ni]δU+(∂2S/∂V∂Ni)U, N[i]δVi ]δNi + 2 Σ(∂2S/∂Ni∂Nj)U,V,N[i,j]δNiδNj }α + the similar terms of the β phase < 0 δ2S= {(SUUδU2 + 2 SUVδUδV + SVVδV2 + 2ΣSUN[i]δUδNi+ 2ΣSVN[i]δVδNi] + ΣΣSNiNjδNiδNj }α + the similar terms of the β phase < 0 < 0(7-1)

  38. (1)ΣδU = δUα + δUβ = 0; δUα = - δUβ ; (δUα)2 = (δUβ)2 ΣδV = δVα + δVβ = 0; δVα = - δVβ ; (δVα)2 = (δVβ)2 ; ΣδN = δNα + δNβ = 0; δNα = - δNβ; (δNα)2 = (δNβ)2 δUαδVα = ( - δUβ)( -δVβ ) • δNα δVα =( - δNβ)( -δVβ ) • δUα δNα =( - δUβ)( -δNβ ) (2) SUU = (∂2S/∂U2)VN=[(∂S/∂U)(∂S/∂U)VN]VN = [(∂S/∂U )(1/T)]VN = (-1/T 2)(∂T/∂U)VN = (-1/T 2N) (∂T/∂U)VN SUUα + SUUβ = [( -1/T 2N)α + ( -1/T 2N)β](∂T/∂U )VNα Since (∂T/∂U ) VNα= (∂T/∂U ) VNβ; Tα = Tβ = (-1/T 2 )[Nα + Nβ) /NαNβ](∂T/∂U )VNα = (-1/T 2 )[N/ Nβ][(∂T/∂(NU )]VNα = (-1/T 2 )[N/ Nβ](∂T/∂U )VNα = [N/ Nβ] SUUα

  39. (3) SVV = (∂2S/∂V2)UN (∂S/∂V) UN = - (∂U/∂V) SN / (∂U/∂S)VN = - (-P)/T SVV = (∂2S/∂V2)UN = [(∂ (P/T) /∂V]UN = [T( ∂P/∂V )UN - P( ∂T/∂V )UN]/T2 = (1/N) [T( ∂P/∂V )U - P( ∂T/∂V )U]/T2 SVVα+SVVβ = {(1/N) [T( ∂P/∂V )U - P( ∂T/∂V )U]/T2}α+ {(1/N) [T( ∂P/∂V )U - P( ∂T/∂V )U]/T2}β At equilibrium, (∂P/∂V)Uα= (-∂P/∂V)Uβ and (∂T/∂V)Uα =(∂T/∂V)Uβ SVVα+SVVβ= {(1/Nα + 1/Nβ) [T( ∂P/∂V )U - P( ∂T/∂V )U]/T2}α = [(Nα+ Nβ)/NαNβ] [T( ∂P/∂V )U - P( ∂T/∂V )U]/T2}α = (N/Nβ {[T[(∂P/∂(NV)]UN – P[∂T/∂(NV )U]/T2}α = (N/Nβ ) SVVα

  40. (4) SUV = (∂2S/ ∂U∂V)N = [(∂/∂U )(∂S/∂V)UN]VN (∂S/∂V) UN = P/T SUV = [(∂ (P/T) /∂U]UN = [T( ∂P/∂U )VN - P( ∂T/∂U )VN]/T2 = (1/N) [T( ∂P/∂U )V - P( ∂T/∂U )V]/T2 SUVα+SUVβ = {(1/N) [T( ∂P/∂U )V - P( ∂T/∂U )V]/T2}α+ {(1/N) [T( ∂P/∂U)V - P( ∂T/∂U )V]/T2}β At equilibrium, (∂P/∂U)Vα= (∂P/∂U)Vβ and (∂T/∂U)Vα =(∂T/∂U)Vβ SUVα+SUVβ= {(1/Nα + 1/Nβ) [T( ∂P/∂U)V - P( ∂T/∂U)V]/T2}α = [N/NαNβ] [T( ∂P/∂U)V - P( ∂T/∂U)V]/T2}α = (N/Nβ {[T[(∂P/∂(NU)]VN – P[∂T/∂(NU )VN]/T2}α = (N/Nβ ) SUVα

  41. (5) xα = Nα/(Nα +Nβ ) = Nα/N ; xβ = Nβ/N SNN= (∂2S/∂N2)UV= [(∂/∂N) (∂S/∂N)]UV (∂S/∂N)UV= - (∂U/∂N)SV /(∂U/∂S)NV = - μ/T (Triple-product rule) [(∂/∂N) (∂S/∂N)]UV = [(∂ (-μ/T) /∂N)] UV = (1/N) [(∂ (-μ/T) /∂x)]UV = (1/N) [(-∂ (μ/T) /∂x)]UV = (1/N) [- T (∂μ/∂x) UV -μ(∂ T/∂x ) UV/T2] (SNNα + UNNβ) = {(1/N) [ - T (∂μ/∂x) UV -μ(∂ T/∂x ) UV/T2]}α + {(1/N)[ - T (∂μ/∂x) UV -μ(∂ T/∂x ) UV/T2]}β (∂ μ/∂x)UVα =(∂ μ/∂x)UVβ ; (∂T/∂x)vnα =(∂T/∂x)UVβ; Tα = Tβ (SNNα + UNNβ) = [ (1/Nα)+ (1/Nβ)] [ - T (∂μ/∂x) UV -μ(∂ T/∂x ) UV/T2]}α = (N/NβNα) {[ - T (∂μ/∂x) UV -μ(∂ T/∂x ) UV/T2]}α = (N/Nβ){ [ - T (∂μ/(N∂x)) UV -μ(∂ T/∂(Nx)]UV/T2]}α = (N/Nβ ) SUVα

  42. δ2S= (N/Nβ ){(SUUδU2 + 2 SUVδUδV + SVVδV2 + 2SUNδUδN+ 2 SVNδVδN + SNNδN2}α < 0 ( 7-5) The result is extended to the equations for criterion of stability of multiple components system δ2S= (N/Nβ ){(SUUδU2 + 2 SUVδUδV + SVVδV2 + 2ΣSUN[i]δUδNi+ 2ΣSVN[i]δVδNi] + ΣΣSNiNjδNiδNj }α < 0 ( 7-5) The similar procedure is used for the criterion of the internal energy, δ2U= (N/Nβ ){(USSδS2 + 2 USVδSδV + UVVδV2 + 2ΣUSN[i]δSδNi+ 2ΣUVN[i]δVδNi] + ΣΣUNiNjδNiδNj }α < 0 ( 7-6)

  43. Derivation for the mathematic expression for the criterion of stability δ2U= (N/Nβ ){(USSδS2 + 2 USVδSδV + UVVδV2 + 2ΣUSN[i]δSδNi+ 2ΣUVN[i]δVδNi] + ΣΣUNiNjδNiδNj }α < 0 ( 7-6) U = U(S, V, N1, N2, ……, Ni, ……. Nn) Let y (o) = U, x1 = S, x2= V, x3= N1, ……….., xm= Nn y (o) = f (x1, x2, ……, xi, ……. xm) m = n + 2 δ2y(o) = K ΣΣ y (o)ijδxiδxj > 0 δ2y(o) = K Σ ykk(k-1)δZk2 > 0 k = 1, 2,….., m (7-9) δZk = {δxk + (y(o)23/ y(o)11)δx2+ (y(o)33/ y(o)11))δx3} k = 1, 2, ……,m δZm= δxm for k = m

  44. Consider a single component system, δ2U = δ2Uα + δ2U β = (N/Nβ) {USSδS2 + 2USVδSδV +2 + 2USNδS δN + 2 UVNδVδN + UNNδN2}α < 0 Use the square method for the equation “y” δ2U = USS{δS2 + 2 (USV/USS )δSδV + (UVV/USS )δV2 + 2(USN /USS)δS δN +2 (UVN/USS )δVδN+ (UNN/USS )δN2} = USS{δS2 +2 (USV/USS )δSδV+ 2(USN /USS )δSδN + 2 (USVUSN/USS2) δVδN+ (USV/USS )2δV 2 + (UNN /USS )2δN2} + [UVVδV2–(USV2/USS) δV 2] +[2 UVNδVδN -2 (USVUSN/USS2)δVδN] +[ UNNδN2 –(UNN2/USS )δN2] =USS{δS + (USV/USS )δV+ (USN /USS)δN}2 + ( UVV- USV2/USS)δV 2 + 2 (UVN - USVUSN/USS2)δVδN+ (UNN - USN2/USS )δN2 Let δZ12 = {δS2 + 2 (USV/USS )δSδV + 2(USN /USS )δSδN + 2 (USVUSN/USS2)δVδN+ (USV/USS )2δV 2 + (UNN /USS )2δN2} ={δS + (USV/USS )δV + (UNN /USS )δN} 2 -

  45. δZ1 ={δS + (USV/USS )δV + (UNN /USS )δN} δ2U =USSδZ12 + (UVV- USV2/USS) {δV 2 + 2 [(UVN - USVUSN/USS2 )/(UVV- USV2/USS) ]δVδN + [(USV - USN2/USS ) /(UVV- USV2/USS) ]2 δN2 } + {(UNN - USN2/USS )- [(USV - USN2/USS )2 /(UVV- USV2/USS) ]}δN2 Let δZ22= {δV 2 + 2 [(UVN - USVUSN/USS2 )/(UVV- USV2/USS) ]δVδN + [(USV - USN2/USS )/(UVV- USV2/USS) ]2 δN2 } = {δV +[(USV - USN2/USS )/(UVV- USV2/USS) }2 δZ2= {δV +[(USV - USN2/USS )/(UVV- USV2/USS) } δZ3 = δN δ2U =USSδZ12 + (UVV- USV2/USS) δZ22+ {(UNN - USN2/USS )- [(USV - USN2/USS )2 /(UVV- USV2/USS) ]} δZ3 2

  46. Use the Legendre transform and Table 5-3 U = U(S, V, N) y(o) = y( x1, x2, x2) A = A(T, V, N) y(1) = y(ξ1, x2, x2) G = G(T, P, N) y(2) = y(ξ1, ξ2, x2) y(o)11 = USS ,y(o)22 = UVV ,y(o)12 = USV ,y(o)13 = USN ,y(o)23 = UVN ,y(o)33 = UNN y(1)11 = ATT, y(1)22 = AVV, y(1)12 = ATV, y(1)13 = ATN, y(1)23 = AVN , y(1)33 = ANN y(2)33 = GNN {(Table 5-3-3 y(1)ij = y(o)ij - y(o)1iy(o)1j/ y(o)11 , i >1, j > 1} y(1)ii = y(o)ii - y(o)1i2/ y(o)11 , i >1 UVV - (USV2/USS) = y(o)22 - y(o)12 2 / y(o)11 = y(1)22 =AVV Ψ= {[UNN– (USN2 /USS )] – [(UVN - USVUSN/USS)] /(UVV – (USV2/USS)] } = [y(o)33 – y(o)132 / y(o)11] –[y(o)23 – y(o)12 y(o)13/ y(o)11]/ [y(o)22 – y(o)122 / y(o)11]

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