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Learning Objectives and Fundamental QuestionsWhat 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 matteror
energy cross systemboundaries. No work can bedone on the
system.Open System: Free exchangeacross system boundaries.Closed
System: Energy can beexchanged but matter cannot.Adiabatic System:
Special casewhere no heat can be exchangedbut work can be done on
thesystem (e.g. PV work).
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Thermodynamic State PropertiesExtensive: 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 ProcessesIrreversible: 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 ThermodynamicsThe increase in internal energy as a
result ofheat absorbed is diminished by the amount ofwork 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 EnthalpyWe can define a new state variable (one
where the path to its current state does not affect its value)
called enthalpy:H = Ei + PVEnthalpy = Internal Energy + PVUpon
differentiation and combing with our earlier definitionfor internal
energy:dH = dEi + PdV + VdPdEi = dq - PdVdH = dq + VdP
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Enthalpy, Melting, and HeatFor isobaric (constant pressure)
systems, dP = 0 and then thechange in enthalpy is equal to the
change in heat:dHp = dqpThree 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 changeCp = (dH/dT)pHeat
capacity is defined by the amount of heat that may be absorbedas a
result of temperture change at constant pressure:
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Enthalpy of Melting
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Second Law of ThermodynamicsOne 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
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Change in EntropySsteam > Sliquid water > SiceRelative
Entropy Example:Third Law Entropies:All crystals become
increasingly orderedas absolute zero isapproached (0K =-273.15C)
and at0K all atoms are fixedin space so that entropyis
zero.ISOLATED SYSTEM
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Gibbs Free Energy DefinedG = Ei + PV - TSdG = dEi + PdV + VdP -
TdS - SdT dw = PdV and dq = TdSdG = 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. LiquiddGp = -SdTdGT = VdP
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Melting Relations for Selected MineralsdGc = dGlVcdP - ScdT =
VldP - SldT (Vc - Vl)dP = (Sc - Sl)dTClapeyron Equation
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Thermodynamics of SolutionsPhases: 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 Fractionswhere 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 componentA in phase i
asFor a simple binary system, XA + XB = 1
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Partial Molar Volumes & MixingTemperature Dependenceof
Partial Molar Volumes
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Partial Molar QuantitiesDefined 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 EnergyAs 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, , 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 ThermodynamicsMASTER EQUATIONThis equation demonstrates
that changes in Gibbs free energy aredependent on: changes in the
chemical potential, , through theconcentration 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 PotentialChemical 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 RelationsThe 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 CrystalsEquilibrium stability surface where
Gl=Gc is defined by three variables:
1) Temperature2) Pressure3) Bulk Composition
Changes in any of thesevariables can move thesystem from the
liquid to crystal stability field
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Fugacity DefinedFor gaseous phases at fixed temperature: dGT =
VdP
- Assume Ideal Gas LawPA = XAPtotal and the fugacity coefficient
is defined as fA/PA, whichIs analogous to the activity coefficient.
As the gas componentBecomes more ideal, fA goes to unity.
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Equilibrium ConstantsMg2SiO4 + SiO2 = 2MgSiO3 olivine melt opxDG
=
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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 obtainedfrom tabulated information
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Silica Activity, Buffers, and SaturationMg2SiO4 + SiO2 = 2MgSiO3
olivine melt opxNeAlSiO4 + SiO2 = NaAlSi3O8nepheline melt
albite
- Oxygen Buffers
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Arrhenius Equation and Activation EnergyKinetic Rate = A exp
-Ea/RTlog D = log A - Ea/2.303RT y = b + m xSlope = dy/dx =
-Ea/2.303RIntercept = b = log AAll processes that are thermally
activated havesimilar form!
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Gibbs Free Energy - Temperature RelationsState A is stable for T
> Tebecause GA < GBMetastability for polymorphs A &
BUndercoolingallows metastabilityof phase A over BState B is stable
for T < Tebecause GB < GAIrreversible PathSYSTEM STATE
CHANGES YIELD REACTION OVERSTEPPING
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Silica Polymorph Free Energy Relations and Reaction
ProgressOstwalds 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