Stability, phase transitions, phase diagrams following Callen - PowerPoint PPT Presentation

Stability, 
 phase transitions, 
 phase diagrams following Callen 1985 book

8—Stability of thermodynamic systems 2

Intrinsic stability • Basic extremum principle: dS = 0 and d 2 S < 0 or extremum and maximum • Stable vs. unstable equilibrium (analogy in mechanics) • Considerations of stability … most interesting and significant predictions of thermodynamics • Consider two identical subsystems, fundamental relation S (1) = S (2) = S ( U,V,N ) • Suppose we remove Δ U from (1) and transfer it to (2) • Change in total entropy is: 2 S ( U , V , N ) ⟶ S ( U + Δ U , V , N ) + S ( U + Δ U , V , N ) > 2 S ( U , V , N ) ! • That is, if adiabatic restraint were removed, energy would flow spontaneously 
 across the wall, U (1) would increase and U (1) would decrease • Even within one subsystem, internal inhomogeneities would tend to develop • Loss of homogeneity … hallmark of a phase transition • Condition of stability: concavity of the entropy global condition S ( U + Δ U , V , N ) + S ( U + Δ U , V , N ) ≤ 2 S ( U , V , N ) ∀Δ U ∂ 2 S For Δ U → 0 : ≤ 0 local condition ∂ U 2 V , N 3

Intrinsic stability • If we were to “transfer” volume, condition of stability: global condition S ( U , V + Δ V , N ) + S ( U , V − Δ V , N ) ≤ 2 S ( U , V , N ) ∀Δ V ∂ 2 S For Δ V → 0 : ≤ 0 local condition ∂ V 2 U , N • A fundamental relation may be obtained somehow (e.g., extrapolation of experimental data or statistical mechanical calculation) that looks, for example: • The “underlying” F .R. above does not satisfy the concavity condition. • The proper thermodynamic F .R. is constructed as the envelope of the superior tangent lines. • A point on the straight line BHF (e.g., H) corresponds to a phase separation. Part of the system is in state B and the rest in state F . • In the 3-dimensional S–U–V subspace, the global condition of stability : 
 Entropy surface must lie below its tangent planes. S ( U + Δ U , V + Δ V , N ) + S ( U − Δ U , V − Δ V , N ) ≤ 2 S ( U , V , N ) ∀Δ U , ∀Δ V 4

Intrinsic stability • In the 3-dimensional S–U–V subspace, the global condition of stability: Entropy surface must lie below its tangent planes. S ( U + Δ U , V + Δ V , N ) + S ( U − Δ U , V − Δ V , N ) ≤ 2 S ( U , V , N ) ∀Δ U , ∀Δ V ∂ V 2 − ( 2 ∂ U ∂ V ) ∂ 2 S ∂ 2 S ∂ 2 S ∂ 2 S ∂ 2 S For Δ U → 0 and Δ U → 0 : ≤ 0 & ≤ 0 & ≥ 0 ∂ U 2 ∂ V 2 ∂ U 2 V , N U , N • Mathematically: d 2 S ( U , V ) < 0 … negatively definite form … Hessian matrix … Conditions of stability for thermodynamic potentials ∂ V 2 − ( 2 ∂ S ∂ V ) ∂ 2 U ∂ 2 U ∂ 2 U ∂ 2 U ∂ 2 U ≥ 0 & ≥ 0 & ≥ 0 ∂ S 2 ∂ V 2 ∂ S 2 V , N S , N ∂ V 2 − ( 2 ∂ T ∂ V ) ∂ 2 F ∂ 2 F ∂ 2 F ∂ 2 F ∂ 2 F ≤ 0 & ≥ 0 & ≥ 0 ∂ T 2 ∂ V 2 ∂ T 2 V , N T , N ∂ P 2 − ( 2 ∂ S ∂ P ) ∂ 2 H ∂ 2 H ∂ 2 H ∂ 2 H ∂ 2 H ≥ 0 & ≤ 0 & ≥ 0 ∂ S 2 ∂ P 2 ∂ S 2 P , N S , N ∂ P 2 − ( 2 ∂ T ∂ P ) ∂ 2 G ∂ 2 G ∂ 2 G ∂ 2 G ∂ 2 G ≤ 0 & ≤ 0 & ≥ 0 ∂ T 2 ∂ P 2 ∂ T 2 P , N T , N 5

Physical consequences of stability ∂ 2 S ∂ U ( T ) V , N 1 = − 1 ∂ T 1 = ∂ = − ≤ 0 ⇔ C V ≥ 0 ∂ U 2 T 2 T 2 NC V ∂ U V , N V , N ∂ 2 F = K T = − ∂ P 1 = V ≥ 0 ⇔ κ T ≥ 0 ( K T ≥ 0) ∂ V 2 ∂ V V κ T T , N T , N C P − C V = Tv α 2 Recall: ⇒ C P ≥ C V ≥ 0 κ T = C V κ S Recall: ⇒ κ T ≥ κ S ≥ 0 C P κ T 6

9—First order phase transitions 7

Phase diagram A chart showing conditions (P ,T,V,…) at which thermodynamic phases occur and coexist 8

First order phase transition G high density phase (liq.) low density phase (vap.) V 9


 First order phase transition A,B,C … first-order transition 
 The two phases inhabit di ff erent regions in the "thermodynamic space" D … second-order transition 10

Discontinuity in entropy: latent heat g … (molar Gibbs potential) equal in the two phases u, f, h, v, s … discontinuous across the transition T liquid solid + liquid steady supply of heat solid time l LS ≡ l f = T ( s L − s S ) … latent heat of fusion … latent heat of first-order transition l = T Δ s = Δ h h = Ts + g 11

Slope of coexistence curves: Clapeyron equation Phase equilibrium: At any point along coexistence curve g is the same in both phases (and in single-component system, g = μ ) μ A = μ A ′ � μ B = μ B ′ � … along coexistence curve d μ = d μ ′ � − sdT + vdP = − s ′ � dT + v ′ � dP dP dT = s ′ � − s v ′ � − v = Δ s Δ v = Δ h l T Δ v = T Δ v Clapeyron slope [figure source] Solid–solid phase transitions of minerals in the Earth's: some dP/dT>0, some <0 While the low-P phase → high-P phase transition always has Δ v<0, Δ s can go both ways. 12

Navrotsky, A., Thermodynamic properties of minerals, in Mineral Physics & Crystallography: A Hand- book of Physical Constants, AGU 24 THERh4ODYNAhIICS Reference Shelf, vol. 2, edited by T. J. Ahrens, pp. 18–28, American Geophysical Union, Washington DC, doi:10.1029/RF002p0018, 1995. Table 5. Enthalpy, Entropy and Volume Changes for High Pressure Phase Transitions AH’ (kJ/mol) AS” (J/m01 K) AV“ (cm3/mol) 30.0 f 2.8a f2] -77*19a[21 . . -3.16a 12] Mg2Si04(a= P> 39.1 f 2.d21 -15.0 t 2.4L21 -4.14[21 Mg2SiO4(a=Y) Fe2Si04(a=P) 9.6 + 1 .3L21 -10.9 AZ 0.8[21 -3.20[21 I%2Si04(a=y) 3.8 f 2.4L2J -14.0 f 1.912J -4.24 L2J MgSi03 (px = il) -15.5 -4.94 59.1 f 4.3131 f 2.0[3J 131 [9J MgSi03 (px = gt) -2.0 + 0.5 L91 35.7 f 3.d91 -2.83 MgSi03 (il = pv) 51.1 + 6.d171 +5 + 4[171 -1.89[171 +4 f 41171 96.8 + 5.8/l 71 Ws 2siWYk MgSiWpv)+MgO -3.19[17J SiO 2 (q = co) 2.1+ 0.5[11 -5.0 -2.05 [I] * 0.4[1J SiO 2 (co = st) -4.2 f l.l[IJ 49.0 f 1 .l[lJ -6.63 [II a AH and AS are values at I atm near 1000 K, AV is AV”298, for all listings in table, a = olivine, /I = spinelloid or wadsleyite, y = spinel, px = pyrox-ene, il = ilmenite, gt = garnet, pv = perovskite, q = quartz, co = coesite, st = stishovite Table 6. Thermodynamic Parameters for Other Phase Transitions Transition AH” AS” AV” (J/K*mol) (Wmol) (cm 3/mol> SiO 2 ( o-quartz = p-quartz) 0.101 0.47a.b 0.35 Si02 ( fi-quartz = cristobalite) 2.94151 1.93 0.318 GeO2 (Wile = quartz) 561231 11.51 4.0 CaSiO3 (wollastonite = pseudowollastonite) 5:0[311 0.12 3.6 Al2SiO5 (andalusite = sillimanite) -0.164 3.8815j 4.50 Al2SiO5 (sillimanite = kyanite) -0.511 -8.13L51 -13.5 MgSi03 (ortho = clino) -0.002 -0.37[51 0.16 MgSi03 (ortho = proto) 0.109 1.59[51 1.21 FeSi03 (ortho = clino) -0.11~2~~ -0.06 -0.03 MnSi03 (rhodonite = pyroxmangite) 0.25 Lz5J -1.03 -0.39 -0.39 MnSi03 (pyroxmangite = pyroxene) 0.88[251 -0.3 -2.66 NaAlSi308 (low albite = high albite) 13.51311 14.0 0.40 0.40 KAlSi 309 (microcline = sanidine) 11.1[51 0.027 15.0 a Treated as though allfirst order, though a strong higher order component bAH and AY are values near 1000 K, AV is AV ‘298 for all listings in table. 13

Unstable (van der Waals) isotherms P = RT v − b − a v 2 high T low T 14

Gibbs-Duhem: dµ = − sdT + vdP Z Integrate at T = const: µ = vdP + φ ( T ) Z B Assign μ (A) ≡ μ A : µ B − µ A = v ( P ) dP A 15

for a given (low) T μ –T–P 16

Unstable isotherm ~ underlying fundamental relation Physical isotherm ~ thermodynamic fundamental rel. Z O µ D = µ O v ( P ) dP = 0 ⇔ D 17

First-order phase transitions in multi-component systems Single-component Multi-component r components U = U ( S , V , N ) U = U ( S , V , N 1 , …, N c ) N j where u = u ( s , v ) u = U ( s , v , x 1 , …, x c − 1 ) x j = N = N 1 + … + N c N • At equilibrium, U, H, F , G are convex functions of mole fractions x 1 ,…, x r • If stability criteria not satisfied, a phase transition occurs. 
 Mole fractions ( x j ), molar entropies ( s ), molar volumes ( v ) di ff er in each phase. • Example 1 : a two-component system (1,2) at a given T , P with two coexisting phases (I,II) μ ( I ) 1 ( T , P , x ( I ) 1 ) = μ ( II ) 1 ( T , P , x ( II ) … condition of equilibrium coexistence w.r.t. transfer of component 1 1 ) μ ( I ) 2 ( T , P , x ( I ) 1 ) = μ ( II ) 2 ( T , P , x ( II ) … condition of equilibrium coexistence w.r.t. transfer of component 2 1 ) Two equations to solve for x 1(I) and x 1(II) . • Example 2 : a two-component system (1,2) at a given T , P with three coexisting phases (I,II,III) μ ( I ) 1 ( T , P , x ( I ) 1 ) = μ ( II ) 1 ( T , P , x ( II ) 1 ) = μ ( III ) ( T , P , x ( III ) ) 1 1 μ ( I ) 2 ( T , P , x ( I ) 1 ) = μ ( II ) 2 ( T , P , x ( II ) 1 ) = μ ( III ) ( T , P , x ( III ) ) 2 1 Four equations to solve for only three variables x 1(I) , x 1(II), x 1(III) . ?? ⟶ Cannot independently specify T and P . Set one, the other is given by equilibrium coexistence. • Example 3 : a two-component system (1,2) at a given T , P with four coexisting phases (I,II,III) … 18

Gibbs phase rule f = c − p + 2 # degrees of freedom = # components - # phases + 2 • Example 1 : pure H 2 O, coexistence of two phases … f = 1 - 2 + 2 = 1 … coexistence curves in the P–T diagram (s–l, l–g, s–g) • Example 2 : pure H 2 O, coexistence of ice, water, steam … f = 1 - 3 + 2 = 0 … triple point 19

?Phase diagram of mineral mixture? 20