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

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Stability,   phase transitions,   phase diagrams

following Callen 1985 book

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8—Stability of thermodynamic systems

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Intrinsic stability

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Basic extremum principle: dS = 0 and d2S < 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 ΔU from (1) and transfer it to (2)
Change in total entropy is:

2S(U, V, N) ⟶ S(U + ΔU, V, N) + S(U + ΔU, V, N) > 2S(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

S(U + ΔU, V, N) + S(U + ΔU, V, N) ≤ 2S(U, V, N) ∀ΔU For ΔU → 0 : ∂2S ∂U2

V,N

≤ 0 global condition local condition

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Intrinsic stability

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If we were to “transfer” volume, condition of stability:

S(U, V + ΔV, N) + S(U, V − ΔV, N) ≤ 2S(U, V, N) ∀ΔV For ΔV → 0 : ∂2S ∂V2

U,N

≤ 0

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. global condition local condition S(U + ΔU, V + ΔV, N) + S(U − ΔU, V − ΔV, N) ≤ 2S(U, V, N) ∀ΔU, ∀ΔV

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Intrinsic stability

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In the 3-dimensional S–U–V subspace, the global condition of stability:

S(U + ΔU, V + ΔV, N) + S(U − ΔU, V − ΔV, N) ≤ 2S(U, V, N) ∀ΔU, ∀ΔV Entropy surface must lie below its tangent planes. For ΔU → 0 and ΔU → 0 : ∂2S ∂U2

V,N

≤ 0 & ∂2S ∂V2

U,N

≤ 0 & ∂2S ∂U2 ∂2S ∂V2 − ( ∂2S ∂U∂V)

2

≥ 0

Mathematically: d2S(U,V) < 0 … negatively definite form … Hessian matrix …

Conditions of stability for thermodynamic potentials

∂2U ∂S2

V,N

≥ 0 & ∂2U ∂V2

S,N

≥ 0 & ∂2U ∂S2 ∂2U ∂V2 − ( ∂2U ∂S∂V)

2

≥ 0 ∂2F ∂T2

V,N

≤ 0 & ∂2F ∂V2

T,N

≥ 0 & ∂2F ∂T2 ∂2F ∂V2 − ( ∂2F ∂T∂V )

2

≥ 0 ∂2H ∂S2

P,N

≥ 0 & ∂2H ∂P2

S,N

≤ 0 & ∂2H ∂S2 ∂2H ∂P2 − ( ∂2H ∂S∂P)

2

≥ 0 ∂2G ∂T2

P,N

≤ 0 & ∂2G ∂P2

T,N

≤ 0 & ∂2G ∂T2 ∂2G ∂P2 − ( ∂2G ∂T∂P)

2

≥ 0

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Physical consequences of stability

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∂2S ∂U2

V,N

= ∂ ∂U ( 1 T)V,N = − 1 T2 ∂T ∂U

V,N

= − 1 T2NCV ≤ 0 ⇔ CV ≥ 0 ∂2F ∂V2

T,N

= − ∂P ∂V

T,N

= 1 VκT = KT V ≥ 0 ⇔ κT ≥ 0 (KT ≥ 0) CP − CV = Tvα2 κT ⇒ CP ≥ CV ≥ 0

Recall:

κS κT = CV CP ⇒ κT ≥ κS ≥ 0

Recall:

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9—First order phase transitions

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Phase diagram

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

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First order phase transition

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G

low density phase (vap.) high density phase (liq.)

V

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First order phase transition

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A,B,C … first-order transition  The two phases inhabit different regions in the "thermodynamic space"   D … second-order transition

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Discontinuity in entropy: latent heat

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g … (molar Gibbs potential) equal in the two phases u, f, h, v, s … discontinuous across the transition solid solid + liquid liquid time T steady supply of heat … latent heat of fusion … latent heat of first-order transition

lLS ≡ lf = T(sL − sS) l = TΔs = Δh h = Ts + g

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Slope of coexistence curves: Clapeyron equation

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Phase equilibrium: At any point along coexistence curve g is the same in both phases (and in single-component system, g = μ) Clapeyron slope … along coexistence curve 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 Δv<0, Δs can go both ways.

[figure source]

μA = μA′ μB = μB′ dμ = dμ′ −sdT + vdP = − s′dT + v′dP dP dT = s′− s v′− v = Δs Δv = Δh TΔv = l TΔv

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24 THERh4ODYNAhIICS

Table 5. Enthalpy, Entropy and Volume Changes for High Pressure Phase Transitions AH’ (kJ/mol) AS” (J/m01 K) AV“ (cm3/mol)

Mg2Si04(a= P>

30.0 f 2.8a f2]

Mg2SiO4(a=Y)

39.1 f 2.d21 Fe2Si04(a=P) 9.6 + 1 .3L21 I%2Si04(a=y) 3.8 f 2.4L2J MgSi03 (px = il) 59.1 f 4.3131 MgSi03 (px = gt) 35.7 f 3.d91 MgSi03 (il = pv) 51.1 + 6.d171 Ws 2siWYk MgSiWpv)+MgO 96.8 + 5.8/l 71 SiO 2 (q = co) 2.1+ 0.5[11 SiO 2 (co = st) 49.0 f 1 .l[lJ

77*19a[21

. .

15.0

t 2.4L21

10.9

AZ 0.8[21

14.0

f 1.912J

15.5

f 2.0[3J

2.0 + 0.5

L91 +5 + 4[171 +4 f 41171

5.0

* 0.4[1J

4.2

f l.l[IJ

3.16a

12]

4.14[21
3.20[21
4.24

L2J

4.94

131

2.83

[9J

1.89[171
3.19[17J
2.05

[I]

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

r 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” (Wmol) (J/K*mol) (cm 3/mol> SiO 2 ( o-quartz = p-quartz) Si02 ( fi-quartz = cristobalite)

GeO2 (Wile = quartz)

CaSiO3 (wollastonite = pseudowollastonite) Al2SiO5 (andalusite = sillimanite) Al2SiO5 (sillimanite = kyanite) MgSi03 (ortho = clino) MgSi03 (ortho = proto) FeSi03 (ortho = clino) MnSi03 (rhodonite = pyroxmangite) 0.25 Lz5J MnSi03 (pyroxmangite = pyroxene) NaAlSi308 (low albite = high albite) 13.51311 KAlSi 309 (microcline = sanidine)

0.47a.b 0.35 2.94151 1.93 561231 5:0[311 4.0 3.6 3.8815j 4.50

8.13L51
13.5
0.37[51

0.16 1.59[51 1.21

0.11~2~~
0.03
1.03
0.39

0.88[251

2.66

14.0

0.40

11.1[51

15.0

0.101

0.318

11.51 0.12

0.164
0.511
0.002

0.109

0.06
0.39
0.3

0.40 0.027

a Treated as though allfirst

rder, though a strong higher order component

bAH and AY are values near 1000 K, AV is AV ‘298 for all listings in table.

Navrotsky, A., Thermodynamic properties of minerals, in Mineral Physics & Crystallography: A Hand- book of Physical Constants, AGU Reference Shelf, vol. 2, edited by T. J. Ahrens, pp. 18–28, American Geophysical Union, Washington DC, doi:10.1029/RF002p0018, 1995.

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Unstable (van der Waals) isotherms

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P = RT v − b − a v2 high T low T

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dµ = −sdT + vdP

Gibbs-Duhem: Integrate at T = const:

µ = Z vdP + φ(T)

Assign μ(A) ≡ μA:

µB − µA = Z B

A

v(P)dP

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for a given (low) T μ–T–P

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µD = µO ⇔ Z O

D

v(P)dP = 0

Unstable isotherm ~ underlying fundamental relation Physical isotherm ~ thermodynamic fundamental rel.

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First-order phase transitions in multi-component systems

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Single-component U = U(S, V, N) u = u(s, v) Multi-component U = U(S, V, N1, …, Nc) u = U(s, v, x1, …, xc−1) r components xj = Nj N N = N1 + … + Nc where

At equilibrium, U, H, F

, G are convex functions of mole fractions x1,…,xr

If stability criteria not satisfied, a phase transition occurs.

Mole fractions (xj), molar entropies (s), molar volumes (v) differ 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) 1 )

μ(I)

2 (T, P, x(I) 1 ) = μ(II) 2 (T, P, x(II) 1 )

… condition of equilibrium coexistence w.r.t. transfer of component 1 … condition of equilibrium coexistence w.r.t. transfer of component 2

Two equations to solve for x1(I) and x1(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) 1

(T, P, x(III)

1

) μ(I)

2 (T, P, x(I) 1 ) = μ(II) 2 (T, P, x(II) 1 ) = μ(III) 2

(T, P, x(III)

1

) Four equations to solve for only three variables x1(I), x1(II), x1(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) …

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Gibbs phase rule

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f = c − p + 2

# degrees of freedom = # components - # phases + 2

Example 1: pure H2O, 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 H2O, coexistence of ice, water, steam … f = 1 - 3 + 2 = 0 … triple point

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?Phase diagram of mineral mixture?

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Phase diagram of a binary system

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Phase diagram of a binary system

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Phase diagram of a binary system

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Phase diagram of a binary system

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29 https://www.sciencedirect.com/referencework/9780444538031/treatise-on-geophysics

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Mineral phases in Earth’s mantle

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Bellini et al. 2013

P T P T

Clapeyron slope (phase equilibrium)

at 410 km depth exothermic transition at 660 km depth endothermic transition

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2000 4000 6000 Depth (km) Temperature (K) 1000 2000 3000 4000 5000 solidus t e m p e r a t u r e inner core

uter

core lower mantle

670 400 L A D" 2891 6371 5150

Depth (km)

L = lithosphere (0–80 km) A = asthenosphere (80–220 km) D" = lower-mantle D" layer 400, 670 = phase transitions

1000 2000 3000 4000 5000 Temperature (°C)

Fig. 4.14 Variations of estimated temperature and melting point with

410 km depth 660 km depth z

m

i n

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Anchor points for geotherm

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Dynamic consequences of phase transitions

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Exothermic phase transition at 410 km facilitates sinking of subducting plate Endothermic phase transition at 660 km hinders sinking of subducting plate

Subducting slabs are colder than ambient mantle Phase transition takes place and different depth Resulting density anomaly may facilitate or hinder flow

Phase equilibrium Phase equilibrium Tmantle Tplate Tmantle Tplate

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Ideal solutions

Consider a system with >1 components
Measure of concentration of components: mole fraction (alternative: molality, molarity,

weight fraction as %, ppm, ppb, …)

Ideal gaseous solution … ideal gas of vanishingly small non-interacting particles of several

types, PV = nRT (no particle feels any other)

Ideal liquid solution … all two particle interactions identical (A–A, B–B, A–B interactions

indistinguishable)

Ideal solid solutions … rigid structure, molecules/ions/atoms confined to structural sites

(crystal lattice, glass), component-dependent two-particle interactions (Si–Si, Si–O, Si–Mg O–O, Mg–O), substitutions of some species possible and indistinguishable (Mg⟷Fe)

Two kinds of ideality (for liquids and solids):
all of A–A, A–B, B–B interactions … Raoult’s law
A–A and A–B but no B–B interactions … limit of infinitely dilute B … Henry’s law
Goes back to 19th century investigations of gas/vapor pressure associated with solutions
Dalton 1811: Ptot V = n1RT + n2RT

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Henry’s law

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Raoult’s law

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Ideal Gibbs free energy of mixing

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