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Part I Alloy Thermodynamics
Lecture Short Description
1 Introduction; Review of classical thermodynamics
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3 Phase equilibria, Classification of phase transitions
4 Thermodynamics of solutions I
5 Thermodynamics of solutions II
6 Binary Phase Diagrams I
7 Binary Phase Diagrams II
8 Binary Phase Diagrams III
9 Order –Disorder Phase Transitions
Atomic Transport & Phase Transformations
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Lecture I-2 Outline
Second Law of Thermodynamics
Entropy
Third Law of Thermodynamics
Entropy – Statistical Thermodynamic Treatement
Gibbs Energy
Maxwell Relations
Temperature Dependences
Chemical potential
Partial molar properties
Atomic Transport & Phase Transformations
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State Variables
Temperature T
Pressure p
Volume V
Adiabatic expansion of gasses
Heat transfer from hotter to colder parts of a body
Question – Are these all thermodynamic state variables?
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Second Law of Thermodynamics
Definition:
There exist a state variale, the Entropy (S), such that for all types of processes
and all systems,
DS ≥ Q/T; DS ≥ Q/T; (1)
Implications:
# For isolated systems (δQ = 0) the entropy can only increase;
dS ≥ 0
dS/dt ≥ 0
# Once (dS/dt) becomes zero, this means that S has stopped increasing and
Equilibrium is reached.
# Reversible Process: DS = δQ/T = (DU – δW)/T (1‘)
Equality in Eq. (1) is reached for reversible processes.
‘Clausius 1865-67: ἐντροπῐᾱ́ (turning to)’
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Equivalent Statements:
# “The entropy of the universe tends to a maximum” (Clausius)
# “Measure of how much energy is spread out during a process’’
# ‘’Entropy is a measure of the amount of energy that
cannot be transformed into work (1’)’’
Second Law of Thermodynamics
Entropy of open system:
DS = DSinter + DS*; DS * - Entropy transfered from the surrounding to the system
DS* > 0 or DS* < 0
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Second Law of Thermodynamics
Implications:
# since S and V are state variables: U = U(S,V)
dU = (U/S)VdS + (U/V)SdV
# if work is done only by change of volume W = -pdV;
From the 1st Law: dU = Q - W = Q + pdV;
From the 2nd Law: Q = TdS (reversible, quasi-static
processes)
dU = TdS + pdV
# T = (U/S)V; p = (U/V)S
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Relation between Heat Capacity and Entropy
V = const. (Reversible isochoric process)
dS = Q/T, but CV = (Q/dT)V ;
dS = (CV/T) dT DS = CV/T’dT’ + K
p = const. (Reversible isobaric process)
dS = Q/T, but Cp = (Q/dT)p;
dS = (Cp/T) dT DS = Cp/T’ dT’+ K
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Third Law of Thermodynamics
Definition (1):
The entropy change of a condensed-matter system,
undergoing a reversible process, approaches zero
as the temperature approaches 0 K (DS → 0 as T → 0).
Definition (2):
The entropy of a perfect crystal at T = 0 is zero.
Implications:
# All perfect crystals will have at T = 0 the same entropy (S = 0);
# Disordered crystals and amorphous materials have a residual
entropy at T = 0;
# The 3rd law provides an absolute scale for entropy:
S = S(To) + To
T(C/T) dT = S(0) + (C/T) dT =
o
T (C/T) dT
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EntropyEntropy values at T =298 oC
Species So (J/mol.K)
Diamond 2.38
Graphite 5.74
Sodium 51.2
Potassium 65.2
Sulfur (S) 31.8
Silver (Ag) 42.6
He (g) 126.0
Xe (g) 169.6
H2O (l) 69.9
H2O (g) 188.7
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Third Law of Thermodynamics
Residual entropy at 0 K
N2O ~5.8 J/K.mol
H2O ~3.37 J/K.mol
CO ~ 5.8 J/K.mol
Am-Se ~ 3.95 J/mol.K (P. Richet 2001)
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Entropy
in Statistical Thermodynamics
Implications:
# The entropy is considered as a measure of Disorder.
Perfect solid Glass or Liquid Gas
Structural Disorder
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Entropy
in Statistical Thermodynamics
Types of states:
# Thermodynamic (Macrostate): State which can be described by only
a few state variables (e.g. p, T, V, S)
# Statistical (Microstate): State, which is described by large
number of quantities for all the individual species, constituting the system.
Examples: Coordinates and velocities of atoms(molecules);
Magnetic moments (spins);
Polarization vectors (dipoles), etc.
M ~ 0 M > 0
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Entropy
in Statistical Thermodynamics
Different Microstates could lead to the same Macrostate
Boltzman‘s Equation: S = kB ln(W)
W is the number of microstates leading to a given
macrostate
Implications:
# At T = 0 (most) systems are in their ground state (W = 1) → S = 0
# Additivity of the Entropy: (S1, W1) and (S2, W2)
New system with W = W1xW2 microstates
S = kBln(W) = kBln(W1 W2) = S1 + S2;
# In condensed-matter systems: S = Sele + Svib + Sconf + ∙∙
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Entropy
in Statistical Thermodynamics
Contributions to the Entropy
# Static (Configurational) contributions:
Species Effect_____________________________
atoms Distribution of atoms over different sites
electrons Distribution over different (degenerate)
elecronic states (levels)
electron spins Different orientations of the spins
(Orientational order/disorder)
# Dynamic contributions:
Species Effect_____________________________
atoms lattice vibrations (Phonons)
electrons Excitations across the Fermi surface
electron spins spin waves (magnons)
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Entropy
in Statistical Thermodynamics
A/ Static electon entropy
# Transition elements (Ti,Fe,Mn)
have only d-electons in their valence shell
and unfilled d-orbitals;
# Transition metals easily oxidize M+p;
# dn Configuration;
n = group number – oxidation state
Ti3+; group 4; n = 4 – 3 = 1; d1 Configuration
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Octahedral
coordination
Tetrahedral
coordination
Entropy
in Statistical Thermodynamics
Ti3+ : d1 ConfigurationCrystal-field theory
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Entropy
in Statistical Thermodynamics
Definitions:
M - Number of species in the system
r Number of microstates f of a given species (f1, f2, ... fr)
A macrostate J is defined by a r-dimensional distribution function
(m1, m2, . . . , mr)J; mi – Number of species in microstate fi
belonging to macrostate J
(Occupation number of a given species state)
W = M!/ m1 !m2 ! … mr!
M = S1
rm
i;
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Entropy
in Statistical Thermodynamics
Nature fo the microstates fi:
Electonic energy levels
Vibrational energy levels
Rotational energy levels
Spin two-level systems
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Entropy
in Statistical Thermodynamics
S = kB ln (W) = kB ln (M!/ m1 !m2 ! … mr!)
= kB [ln M! –S ln(mi!)]
= kB {Mln(M) – M – S[miln(mi) – mi]} =
= kB { Mln(N) – M – S mi ln(mi) + M} =
= kB { S mi [ ln(M) – ln(mi)]}
Sterling’s approximation ln(x!) xln(x) – x
for large number of species
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S = - kB S mi ln(mi/M)
S = S(J) = S(m1,m2, …mr)
# Condition for maximum of the entropy
dS = 0
dS = = -kBS d{mi ln(mi/M)} =
= -kBS d{miln(mi) – miln(M)} = ∙∙∙ = -kBS ln(mi/M)dmi;
# Additional condition M = const → dM = S dmi = 0
# Lagrange equation (method):
dS + adM = 0 →
Entropy
in Statistical Thermodynamics
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Entropy
in Statistical Thermodynamics
In the equilibrium macrostate all
microstates are equally occupied
For a system without change of the
total number of species (M = const)
m1 = m2 = … = M/r
a = -kBln(r)
S{-kBln(mi/M) + a }dmi = 0
-kBln(mi/M) + a = 0 for every i
start
end
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In an isolated system the internal energy U is also constant.
Condition for constant internal energy: dU = 0
U ~ S eimi → dU = S eidmi = 0
Lagrange equation (method):
dS + adM + ßdU = 0
S{-kBln(mi/M) + a + ßei}dmi = 0
-kBln(mi/M) + a + ßei = 0 for every i;
mi /M = exp( - ei/kBT) / Z; Z = Sexp( - ei/kBT) Partion function
Entropy
in Statistical Thermodynamics
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Entropy
in Statistical ThermodynamicsBoltzman distribution
pi = exp( - ei/kBT) / Z; gives the probability of finding species in a given microstate
fi, characterized by energy ei.
Validity (Limits):
# species in different states are non-interacting;
# the microstates fi (ei) are not changing;
# valid for system with very large number of species (particles)
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Entropy
in Statistical Thermodynamics
S = - kB S mi ln(mi/M) = - kB S mi ln [(exp(-ei/kBT)/Z] =
= - kB S mi [-ei/kBT – ln(Z)]
= 1/T S mi ei + kBMln(Z); S = U/T + kBMln(Z) ;
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Gibbs EnergyThermodynamic Functions of State
System of constant composition:
Gibbs Energy (G): G = H – TS
Most convinient state function at constant p and/or T
Differential: dG = (∂G/∂p)Tdp + (∂G/∂T)pdT
Local Minimum: (∂G/∂p) = 0 ; T = const
(∂G/∂T) = 0 ; p = const (dG =0)dG =0
dG =0
dG =0
G
Path variable
Stable (Equilibrium) State G*
at constant p:
(∂G/∂T)p = 0
(∂2G/∂2T)p > 0
G ≥ G*
J. Gibbs
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Implications:
# The Entropy of an isolated system with fixed U has a maximum value
in the equilibrium state
G = H –TS = U + pV – TS U - TS
Path
G
Path
S
Gibbs Energy
G*
Units: J (Joule; J/mol)
V = 1 cm3 = 1x10-6 m3
p = 1.01x105 Pa = 1.01x105 kg/ms2;
pV = 0.1 J
Maximum Entropy Principle
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Helmholz Free Energy (F)
Definition: F = U – TS
Differential: dF = dU – TdS - SdT
Implications:
# for reversible processes
dF = dU – TdS – SdT = (δQ – δW) – T δQ/T – SdT
# … at constant temperature
dF = δQ – δW – δQ
= - δW; dF is equal to the total (reversible) work done on the system
H. von Helmholz
F = U – TS = U - T[U/T + kBMln(Z) ]; F = - kBMln(Z)
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Gibbs Energy
Implications:
# if there is no other types of work (δW‘ = 0)
V = (∂G/∂p)T and S = - (∂G/∂T)p;
# at constant pressure and temperature dG = – δW‘
# Differential:
dG = dH – (SdT + TdS) = d(U + pV) – (SdT + TdS) =
= dU + Vdp + pdV – SdT – TdS =
= TdS – pdV – δW‘+ Vdp + pdV – SdT – TdS
dG = Vdp – SdT – δW‘dG = (∂G/∂p)Tdp + (∂G/∂T)pdT
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From the internal energy:
(∂p/∂S)V = - (∂T/∂V)S
From the Helmholz free energy:
(∂S/∂p)T = - (∂p/∂T)V
From the Gibbs free energy:
(∂S/∂p)T = - (∂V/∂T)p
Maxwell Relations
Z = Z(X,Y)
∂(∂Z/∂X)/∂Y = ∂(∂Z/∂Y)∂X
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Gibbs Energy
# Gibbs – Helmholz Equation
G = H – TS = H + T (∂G/∂T)p;
GdT = HdT + TdG
(TdG – GdT)/ T2 = - HdT/T2;
(1/T)dG – GdT/ T2 = - HdT/T2;
d(G/T)/dT = - H/T2
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Equations-of-State (EOS)
In the Gibbs energy description the volume V is a function of p and T.
V = (∂G/∂p)T , also V = V(p,T) – Equation of state
Murnaghan EOS
V(p) = Vo [1 + p(B‘/B)] (1/B‘)
B‘ – first derivative of B with respect to p
B = Bo + B‘p (K = Ko + Ko‘p)
Diamond Anvil Cell (DAC)
Birch - Murnaghan EOS
P(V) = 3Bo/2[(Vo/V)7/3 – (Vo/V)5/3] {1 +
¾ (Bo‘ – 4)[(Vo/V)2/3 – 1]}
Bulk Modulus
B = - V(∂p/∂V)T = - V/ (∂V/∂p)T
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Equations-of-State (EOS)
Powders
Re gaskets
Ne/He gas
as pressure
medium
Synchrotron
Radiation
XRD
Dorfman et al (2012)
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Equations-of-State (EOS)
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Temperature Dependences (1)
At p = const
Cp = Cv + a2VTB(T)
T > 298 K
DH = H(T) - Ho = ∫298
TCp(T‘)dT‘ (Kirchhof‘s law)
S(T) = So + ∫0
T(Cp/T‘)dT‘ = ∫
0
T(Cp/T‘) dT‘
G(T) = H – TS = Ho + ∫298
TCp(T‘)dT‘ - T ∫
298
T[Cp(T‘)/T‘] dT‘
Standart reference state
T = 298 oC
P = 1 atm = 101325 Pa
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Temperature Dependences (2)
0 200 400 600 800 10000
5
10
15
20
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CV
(J/m
ol.K
)
Temperature (K)
a-Sn
Debey Model
QD ~ 230 K
0 200 400 600 800 10000,00
0,05
0,10
0,15
0,20
0,25
En
tro
py (
J/m
ol.K
)
Temperature (K)
S(T) = ∫0
T(Cp/T‘)dT‘
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Temperature Dependences (2)
a-Sn
Debey Model
QD ~ 230 K
G is negative, decreases with increasing
Temperature and the Slope (∂G/∂T)p = -S.
200 400 600 800 1000
-150
-100
-50
0
50
100
150
200
250
En
erg
y (
J/m
ol)
Temperature (K)
H
TS
G
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Temperature Dependences (3)
Empirical Model ( T > 300 K)
cp (J/mol.K) = a + bT + c/T2;
Species a bx103 c x 105
Al 20.7 12.3
Cu 22.6 5.6
Fe 37.12 6.17
Ag 21.3 8.5 1.5
Si 23.9 2.5 -4.1
Temperature Dependences (3)Heat capacities
DeHoff (2008)
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300 400 500 600 700 800 900
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28
30
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CP
(J/m
ol.K
)
T (K)
Si
Al
Temperature Dependences (3)Heat capacities
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Gibbs Energy
System with changing composition*:
Gibbs Energy (G): G = G(p,T,n1,n2,…)
ni – Number of moles of species i
Differential:
dG = Vdp – SdT + Si (∂G/∂ni) dni; i = 1, 2 .. , C
Chemical Potential:
µi = (∂G/∂ni) p,T, n≠ni;
* Open systems (Diffusion)
Closed systems with chemical reactions
N = S ni
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Gibbs Energy
Partial Molar Properties
System with (a possible change) of composition:
For any extensive state function A = A(p,T, n1, n2,…):
dA = (A/T)p,n dT + (A/p)T,n dp + S(A/ni)T,p,n≠ni dni;
Ai = (A/ni)T,p,n≠ni Partial molar property of A for component i.
For a system at T = const and p = const.
dA = S(A/ni)T,p,n≠ni dni = S A i dni ;
A = S A i dni = S Ā i ni ; The total property A is a weighted
sum of the partial molar properties
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Gibbs Energy
Partial Molar Properties
Gm = S µk nk; Gk = µk;
All general thermodynamic relations can be expressed
in terms of the partial molar properties;
µk = Gk = Hk - TSk;
Vm = S Vk nk; Vk = (µk/p)T,n
Sm = S Sk nk; Sk = -(µk/T)p,n
Hm = S Hk nk; Hk = µk - T(µk/T)p,n
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Gibbs Energy
Partial Molar Properties
Generalized Gibbs – Duhem Equation
dA = d(S Ā i ni) = S d(Ā ini) = S [Āidni + S nid Āi ]
= dA + S nid Ā i
S nidĀi = 0 Not all partial molar properties are
independent
System with C = 2 components:
n1 + n2 = N; Molar fractions: x1 = n1/N; x2 = n2/N; x1 + x2 = 1
dµ1 = - (x1/1-x1) dµ2 ;
A ≡ G
S ni dµi = 0