Transcript Thermodynamic Systems

Learning Objectives and Fundamental Questions
• What is thermodynamics and how are its
concepts used in petrology?
• How can heat and mass flux be predicted or
interpreted using thermodynamic models?
• How do we use phase diagrams to visualize
thermodynamic stability?
• How do kinetic effects affect our
interpretations from thermodynamic
models?
What is Thermodynamics?
• Thermodynamics: A set of of mathematical
models and concepts that allow us to describe the
way changes in the system state (temperature,
pressure, and composition) affect equilibrium.
– Can be used to predict how rock-forming systems will
respond to changes in state
– Invert observed chemical compositions of minerals and
melts to infer the pressure and temperature conditions
or origin
Thermodynamic Systems - Definitions
Isolated System: No matter
or energy cross system
boundaries. No work can be
done on the system.
Open System: Free exchange
across system boundaries.
Closed System: Energy can be
exchanged but matter cannot.
Adiabatic System: Special case
where no heat can be exchanged
but work can be done on the
system (e.g. PV work).
Thermodynamic State Properties
• Extensive: These variables or properties
depend on the amount of material present
(e.g. mass or volume).
• Intensive: These variables or properties DO
NOT depend on the amount of material (e.g.
density, pressure, and temperature).
Idealized Thermodynamic Processes
• Irreversible: Initial system state is unstable or
metastable and spontaneous change in the system
yields a system with a lower-energy final state.
• Reversible: Both initial and final states are stable
equilibrium states and the path between them is a
continuous sequence of equilibrium states. NOT
ACTUALLY REALIZED IN NATURE.
Spontaneous Reaction Direction
First Law of Thermodynamics
The increase in internal energy as a result of
heat absorbed is diminished by the amount of
work done on the surroundings:
dEi = dq - dw = dq - PdV
By convention, heat added to the system, dq,
is positive and work done by the system, dw,
on its surroundings is negative.
This is also called the Law of Conservation of Energy
Definition of Enthalpy
We can define a new state variable (one where the path to
its current state does not affect its value) called enthalpy:
H = Ei + PV
Enthalpy = Internal Energy + PV
Upon differentiation and combing with our earlier definition
for internal energy:
dH = dEi + PdV + VdP
dEi = dq - PdV
dH = dq + VdP
Enthalpy, Melting, and Heat
For isobaric (constant pressure) systems, dP = 0 and then the
change in enthalpy is equal to the change in heat:
dHp = dqp
Three possible changes in a system may occur:
1) Chemical reactions (heterogeneous)
2) Change in state (e.g. melting)
3) Change in T with no state change
Heat capacity is defined by the amount of heat that may be absorbed
as a result of temperture change at constant pressure:
Cp = (dH/dT)p
Enthalpy of Melting
Second Law of Thermodynamics
• One statement defining the second law is that a
spontaneous natural processes tend to even out
the energy gradients in a isolated system.
• Can be quantified based on the entropy of the
system, S, such that S is at a maximum when
energy is most uniform. Can also be viewed as a
measure of disorder.
DS = Sfinal - Sinitial > 0
Change in Entropy
Relative Entropy Example:
Ssteam > Sliquid water > Sice
ISOLATED SYSTEM
Third Law Entropies:
All crystals become
increasingly ordered
as absolute zero is
approached (0K =
-273.15°C) and at
0K all atoms are fixed
in space so that entropy
is zero.
Gibbs Free Energy Defined
G = Ei + PV - TS
dG = dEi + PdV + VdP - TdS - SdT
dw = PdV and dq = TdS
dG = VdP - SdT
(for pure phases)
At equilibrium: dGP,T = 0
Change in Gibbs Free Energy
Gibbs Energy in Crystals vs. Liquid
dGp = -SdT
dGT = VdP
Melting Relations for Selected Minerals
dGc = dGl
VcdP - ScdT = VldP - SldT
(Vc - Vl)dP = (Sc - Sl)dT
Clapeyron Equation
dP (Sc  Sl ) DS


dT (Vc  Vl ) DV
Thermodynamics of Solutions
• Phases: Part of a system that is chemically and
physically homogeneous, bounded by a distinct
interface with other phases and physically separable
from other phases.
• Components: Smallest number of chemical entities
necessary to describe the composition of every
phase in the system.
• Solutions: Homogeneous mixture of two or more
chemical components in which their concentrations
may be freely varied within certain limits.
Mole Fractions
nA
nA
XA 

 n (nA  n B  nC 
)
,
where XA is called the “mole fraction” of
component A in some phase.
If the same component is used in more than one phase,
Then we can define the mole fraction of component
A in phase i as X Ai
For a simple binary system, XA + XB = 1
Partial Molar Volumes & Mixing
Temperature Dependence
of Partial Molar Volumes
Partial Molar Quantities
• Defined because most solutions DO NOT mix
ideally, but rather deviate from simple linear
mixing as a result of atomic interactions of
dissimilar ions or molecules within a phase.
• Partial molar quantities are defined by the “true”
mixing relations of a particular thermodynamic
variable and can be calculated graphically by
extrapolating the tangent at the mole fraction of
interest back to the end-member composition.
Partial Molar Gibbs Free Energy
As noted earlier, the change in Gibbs free energy function determines the
direction in which a reaction will proceed toward equilibrium. Because of
its importance and frequent use, we designate a special label called the
chemical potential, µ, for the partial molar Gibbs free energy.
GA 
A  

X A P,T ,X
B ,X C
,
We must define a reference state from which to calculate differences in
chemical potential. The reference state is referred to as the standard state
and can be arbitrarily selected to be the most convenient for calculation.
The standard state is often assumed to be pure phases at standard atmospheric
temperature and pressure (25°C and 1 bar). Thermodynamic data are tabulated
for most phases of petrological interest and are designated with the superscript °,
for example, G°, to avoid confusion.
Chemical Thermodynamics
MASTER EQUATION
dG  VdP  SdT   i dX i
i
  dX
i
i
  A dX A  B dX B  C dX C 
 n dX n
i
This equation demonstrates that changes in Gibbs free energy are
dependent on:
• changes in the chemical potential, µ, through the
concentration of the components expressed as mole
fractions of the various phases in the system
• changes in molar volume of the system through dP
• chnages in molar entropy of the system through dT
Equilibrium and the Chemical Potential
• Chemical potential is analogous to gravitational or electrical
potentials: the most stable state is the one where the overall
potential is lowest.
• At equilibrium the chemical potentials for any specific
component in ALL phases must be equal. This means that the
system will change spontaneously to adjust by the Law of
Mass Action to cause this state to be obtained.
If

melt
H 2O

melt
H 2O


melt
CaO


gas
H2 O
gas
H2 O
biotite
H 2O



gas
CaO
biotite
CaO



n
H 2O
n
CaO
then system will have to adjust the massmelt
(concentration) to make them equal:  H 2O
  gas
H2 O
Gibbs Free Energy of Mixing
Activity - Composition Relations
The activity of any component is always less than the
corresponding Gibbs free energy of the pure phase, where the
activity is equal to unity by definition (remember the choice
of standard state).

A
A  G ; B  G

B
  G  RT ln a

A
i
A
a
i
A
i
A
 X
i
A
For ideal solutions (remember dG of mixing is linear),
such that the activity is equal to the mole fraction.

A
  G  RT ln X
i
A
i
A
i
A
 1
i
A
P, T, X Stability of Crystals
Equilibrium stability
surface where Gl=Gc
is defined by three
variables:
1) Temperature
2) Pressure
3) Bulk Composition
Changes in any of these
variables can move the
system from the liquid
to crystal stability field
Fugacity Defined
For gaseous phases at fixed temperature: dGT = VdP
- Assume Ideal Gas Law
PV  nRT;n  1
RT
V
P
RT 
dGT  VdP   dP  RT ln dP
P
PA = XAPtotal and the fugacity coefficient is defined as fA/PA, which
Is analogous to the activity coefficient. As the gas component
Becomes more ideal, fA goes to unity.

A
A  G  RT ln f A
Equilibrium Constants
Mg2SiO4 + SiO2 = 2MgSiO3
olivine
melt
opx
DG = 2

opx
MgSiO3
ol
Mg 2 SiO4

0
mel t
SiO2
mel t
 glass
t
SiO
 GSiO
 RT ln a mel
SiO
2

ol
Mg 2 SiO4

opx
MgSiO3
2
2
 ol
Mg 2 SiO4
G
G
 opx
MgSiO3
 RT ln a
ol
Mg 2 SiO4
 RT ln a
opx
MgSiO3
Equilibrium Constants, con’t.
 opx
MgSi O3
2G
 ol
Mg 2 SiO4
 glass
SiO2
G
G

RT ln( a
)
(a
)
ol
Mg 2 SiO4
opx
2
MgSi O3
melt
SiO2
a

F
DG   RT ln K eq
where dG°F is referred to as the change in standard state
Gibbs free energy of formation, which may be obtained
from tabulated information
K eq 
opx
2
(aMgSi
)
O3
ol
melt
(aMg

a
)
SiO
SiO
2
4
2
Silica Activity, Buffers, and Saturation
Mg2SiO4 + SiO2 = 2MgSiO3
olivine melt
opx
NeAlSiO4 + SiO2 = NaAlSi3O8
nepheline melt
albite
Oxygen Buffers
<---
Calculated fO2 from Fe-Ti oxides
Fe2TiO4 + Fe2O3 = FeTiO3 + Fe3O4
Arrhenius Equation and Activation Energy
Kinetic Rate = A exp -Ea/RT
log D = log A - Ea/2.303RT
y = b
+ m•x
Slope = dy/dx = -Ea/2.303R
Intercept = b = log A
All processes that are
thermally activated have
similar form!
Gibbs Free Energy - Temperature Relations
Metastability for polymorphs A & B
State A is stable for T > Te
because GA < GB
State B is stable for T < Te
because GB < GA
Undercooling
allows metastability
of phase A over B
Irreversible Path
SYSTEM STATE CHANGES YIELD REACTION OVERSTEPPING
Silica Polymorph Free Energy Relations
and Reaction Progress
Ostwald’s Step Rule: In a change of state the kinetically most
favored phase may form at an intermediate step rather than
the most thermodynamically favored (lowest G) phase!
Glass -> Qtz (favored)
Glass -> Cristobalite or
Tridymite