Thermodynamics
5/31/2016
CIDER/ITP Summer School
How does Earth respond?
•
To changes in energy
– Change in temperature (Heat Capacity)
– Change in Density (thermal expansivity
– Phase Transformations (Gibbs free energy)
• To hydrostatic stress
– Compression (bulk modulus)
– Adiabatic heating (Grüneisen parameter)
• To deviatoric stress
– Elastic deformation (elastic constants)
– Flow (viscosity)
– Failure (state and rate dependent friction)
• Rates of Transport of
–
–
–
–
Mass (chemical diffusivity)
Energy (thermal diffusivity)
Momentum (viscosity, attenuation)
Charge (electrical conductivity)
Mantle Phases
sp
hpcpx
plg
opx
0.8
cpx
capv
2000
gt
1900
ak
0.6
1800
mgpv
0.4
wa
ol
1700
ri
0.2
1600
fp
0.0
0
200
400
600
1500
800
Depth (km)
Wadsleyite (wa); Ringwoodite (ri); akimotoite (ak); Mg-perovskite (mgpv);
Ca-perovskite (capv); Ferropericlase (fp)
Temperature (K)
Atomic Fraction
1.0
Scope of Thermodynamics
• Applicable to any system
• Only to a certain set of
states, called equlibrium
states
• Only to certain properties of
those states: macroscopic
• No specific, quantitative
predictions, instead, limits
and relationships
Equilibrium
Spinel
Peridotite
P(1),T(1),Ni(1)
Ni(1)=Ni(2)=Ni(f)
Perovskitite
P(2),T(2),Ni(2)
Garnet
Peridotite
P(f),T(f),Ni(f)
Macroscopic Properties
•
•
•
•
•
•
•
Volume (Density)
Entropy
Energy
Proportions of Phases
Composition of Phases
…
Not
– Crystal size
– Crystal shape
– Details of arrangement
Thermodynamic Variables
• Second Order
– Heat capacity
S 
CP  T 
T P,Ni
– Thermal expansivity

– Bulk modulus
P 
K S  V  
V S,Ni
– Grüneisen parameter
V T 
    
T V S,Ni
• First Order
–
–
–
–
–

V: Volume,
S: Entropy

Ni: Amount of components
P: Pressure
T: Temperature
I : Chemical Potential
1 V 
 
V T P,Ni
Foundations 1
H. B. Callen, Thermodynamics and an Introduction to
Thermostatistics, 2ed, Wiley, 1985
• What defines the equilibrium
state of our system?
– Three quantities
• Ni (Composition)
• V (Geometry)
• U (potential and kinetic)
• Ideal Gas
– Energy is all kinetic so
– U = i mivi2
  ways to redistribute KE
among particles while leaving
U unchanged
Foundations 2
• Entropy
– S = Rln
– S = S(U,V,Ni)
• Relationship to U
– As U decreases, S decreases
– S monotonic, continuous,
differentiable: invertible
• U = U(S,V,Ni)
• Fundamental
Thermodynamic Relation
Properties of Internal Energy, U
U 
U 
U 
dU    dS    dV  
dN j



S V ,N i
V S,N i
N j V ,S,N i  j
define
U 
T   
S V ,N i
U 
P   
V S,N i
U 
 j  
N 

 j V ,S,N i  j
then
dU  TdS  PdV   j dN j
• Complete First Law
Internal Energy, U
• U = U(S,V,Ni)
• Complete
information of all
properties of all
equilibrium states
• S,V,Ni are natural
variables of U
• U = U(T,V,Ni) not
fundamental
Internal Energy, U
Fundamental Relation
Entropy, S
Temperature, T=(dU/dS) V,Ni
Legendre Transformations
y
• Two equivalent representations
– y=f(x)
 =y-px
– i.e. =g(p)

• Identify
– yU, xV,S, pP,T, G
• G = U - dU/dV V - dU/dS S or
• G(P,T,Ni) = U(V,S,Ni) + PV - TS
• G(P,T,Ni) is also fundamental!
x
Fundamental thermodynamic relation
H. B. Callen, Thermodynamics, Wiley, 1965, 1985.
Steam Tables
All thermodynamic properties of
water and steam at all P,T
Density, thermal expansivity,
compressibility, heat capacity,
entropy, saturation, enthalpy, …
pg. 1: F(V,T)
pp. 2-320: V and T derivatives of F
Thermodynamic Square
• Thermodynamic Potentials
–
–
–
–
F = Helmholtz free energy
G = Gibbs free energy
H = Enthalpy
U = Internal energy
V
F
U
S
• Surrounded by natural variables
• First derivatives
• Second derivatives (Maxwell Relations)
T
G
H
P
Summary of Properties
1
  
 
P T
  
 
T P
1
  
 
P T
G
V
V
V

KT
S 
S  
P T
  
 
T P
S
V
CP

T
Phase Equilibria
• How does G change in a spontaneous
process at constant P/T?
–
–
–
–
–
–
L1: dU = dQ - PdV
L2: dQ ≤ TdS
dG = dU + PdV + vdP - TdS - SdT
Substitute L1, take constant P,T
dG = dQ - TdS
This is always less than zero by L2.
• G is lowered by any spontaneous process
• State with the lowest G is stable
One Component Phase Equilibria
dG = VdP - SdT
G
G
A
Slope=-S
B
Slope=V
Ptr
P
Phase Diagram
T
Ttr
T
dT V

dP S
B
A
P
Two Component Phase
Equilibria
• Phase: Homogeneous in
chemical composition and
physical state
• Component: Chemically
independent constituent
• Example: (Mg,Fe)2SiO4
• Phases: olivine, wadsleyite,
ringwoodite, …
• Components:
Mg2SiO4,Fe2SiO4
• Why not Mg,Fe,Si,O?
Phase Rule
• p phases, c components
• Equilibrium: uniformity of
intensive variables across
coexisting phases:
• P(a),T(a),i(a)
• Equations
– 2(p-1)+c(p-1)
• Unknowns
– 2p+p(c-1)
• Degrees of Freedom
– f=c-p+2
c≈5,p≈4
f≈3
Properties of ideal solution 1
– N1 type 1 atoms, N2 type 2 atoms, N
total atoms
– x1=N1/N, x2=N2/N
• Volume, Internal energy: linear
– V = x1V1 + x2V2
• Entropy: non-linear
– S = x1S1 + x2S2 -R(x1lnx1 + x2lnx2)
• Sconf = Rln
  is number of possible of
arrangements.
Properties of Ideal Solution 2
• Gibbs free energy
• Re-arrange
– G = x1(G1 + RTlnx1)
+ x2(G2 + RTlnx2)
• G = x11 + x22
 i = Gi + RTlnxi
Mechanical
Mixture
Gibbs Free Energy
– G = x1G1 + x2G2 +
RT(x1lnx1 + x2lnx2)
Ideal
Solution
2
1
0
1
Composition, x 2
Gibbs Free Energy, G

Gibbs Free Energy, G


0

1
Composition, x B
0
Composition, x B


0
1
Gibbs Free Energy, G
Pressure

1


Composition, x B
0
1
Composition, x B
Non-ideal solutions
• Internal Energy a non-linear
function of composition
– Compare A-B bond energy to
average of A-A, B-B
– Tendency towards dispersal,
clustering.
• Exsolution
– cpx-opx,Mg-pv,Ca-pv
• Formally:
 I = Gi + RTlnai
– ai is the activity and may differ
from the mole fraction
Example: Thermochemistry
• Increase temperature
slightly, measure heat
evolved
• CP=(dH/dT)P
• CV3R
• Calculate S,H by
integration
• (dT/dP)eq=V/S
• Debye Model
Stixrude and Lithgow-Bertelloni (2005)
Equation of State
• Start from fundamental relation
• Helmholtz free energy
• Isotherm, fixed composition
– F=F(V)
• Taylor series expansion
• Expansion variable must be V or
function of V
– F = af2 + bf3 + …
• f = f(V) Eulerian finite strain
• a = 9K0V0
F=af
120
Pressure (GPa)
– F=F(V,T,Ni)
140
2
MgSiO3
Perovskite
300 K
100
80
60
40
20
0
0.70 0.75 0.80 0.85 0.90 0.95 1.00
Volume, V/V 0
Grüneisen Parameter
1 P  K T  ln T 
    
 

 U  C   ln  S
•Thermal pressure
•Adiabatic Gradient
•Dimensionless
•1-2 for most mantle phases
•Decreases on compression
Note: =1/V=density
UTH  3nRT
PTH   3RT
KTH    1 q
 ln 
q
 ln 
Thermal expansion
Thermal pressure