Learning Objectives and Fundamental Questions

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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?

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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

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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).
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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).

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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.

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Spontaneous Reaction Direction
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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
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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
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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
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Enthalpy of Melting
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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 gt 0
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Change in Entropy
Relative Entropy Example
Ssteam gt Sliquid water gt Sice
Third Law Entropies All crystals become
increasingly ordered as absolute zero
is approached (0K -273.15C) and at 0K all
atoms are fixed in space so that entropy is zero.
ISOLATED SYSTEM
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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
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Change in Gibbs Free Energy
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Gibbs Energy in Crystals vs. Liquid
dGp -SdT dGT VdP
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Melting Relations for Selected Minerals
dGc dGl
VcdP - ScdT VldP - SldT
(Vc - Vl)dP (Sc - Sl)dT
Clapeyron Equation
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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.

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Mole Fractions
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
For a simple binary system, XA XB 1
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Partial Molar Volumes Mixing
Temperature Dependence of Partial Molar Volumes
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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.

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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.
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 (25C 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.
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Chemical Thermodynamics
MASTER EQUATION
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
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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
then system will have to adjust the
mass (concentration) to make them equal
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Gibbs Free Energy of Mixing
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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).
For ideal solutions (remember dG of mixing is
linear),
such that the activity is equal to the mole
fraction.
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P, T, X Stability of Crystals
Equilibrium stability surface where GlGc 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
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Fugacity Defined
For gaseous phases at fixed temperature dGT
VdP
- Assume Ideal Gas Law
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.
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Equilibrium Constants
Mg2SiO4 SiO2 2MgSiO3 olivine melt
opx
DG
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Equilibrium Constants, cont.
where dGF is referred to as the change in
standard state Gibbs free energy of formation,
which may be obtained from tabulated information
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Silica Activity, Buffers, and Saturation
Mg2SiO4 SiO2 2MgSiO3 olivine melt
opx
NeAlSiO4 SiO2 NaAlSi3O8 nepheline melt
albite
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Oxygen Buffers
lt--- Calculated fO2 from Fe-Ti oxides
Fe2TiO4 Fe2O3 FeTiO3 Fe3O4
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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!
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Gibbs Free Energy - Temperature Relations
Metastability for polymorphs A B
State A is stable for T gt Te because GA lt GB
State B is stable for T lt Te because GB lt GA
Undercooling allows metastability of phase A over
B
Irreversible Path
SYSTEM STATE CHANGES YIELD REACTION OVERSTEPPING
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Silica Polymorph Free Energy Relations and
Reaction Progress
Ostwalds 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 -gt Qtz (favored) Glass -gt Cristobalite or
Tridymite