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 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).

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 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

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 + PVEnthalpy = Internal Energy + PV

Upon differentiation and combing with our earlier definitionfor internal energy:

dH = dEi + PdV + VdPdEi = dq - PdV

dH = dq + VdP

Enthalpy, Melting, and HeatFor isobaric (constant pressure) systems, dP = 0 and then thechange 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

Cp = (dH/dT)p

Heat capacity is defined by the amount of heat that may be absorbedas a result of temperture change at constant pressure:

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.

S = Sfinal - Sinitial > 0

Change in Entropy

Ssteam > Sliquid water > Sice

Relative Entropy Example:

Third Law Entropies:All crystals become increasingly orderedas absolute zero isapproached (0K =-273.15°C) and at0K all atoms are fixedin space so that entropyis zero.

ISOLATED SYSTEM

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 = -SdTdGT = VdP

Melting Relations for Selected Minerals

dGc = dGl

VcdP - ScdT = VldP - SldT

(Vc - Vl)dP = (Sc - Sl)dT

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dPdT

=(Sc − Sl)(Vc − Vl)

=ΔSΔV

Clapeyron Equation

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

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XA ≡nA

n∑=

nA

(nA + nB + nC +L ),

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 componentA in phase i as

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XAi

For a simple binary system, XA + XB = 1

Partial Molar Volumes & MixingTemperature Dependenceof 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 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, µ, for the partial molar Gibbs free energy.

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μA ≡∂GA

∂XA

⎛ ⎝ ⎜

⎞ ⎠ ⎟P ,T ,XB ,XC ,K

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

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dG = VdP − SdT + μ idX ii

∑

μ idX ii

∑ = μ AdXA + μB dXB + μC dXC +L + μn dXn

MASTER EQUATION

This equation demonstrates that changes in Gibbs free energy aredependent on:

• changes in the chemical potential, µ, 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

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.

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μH 2Omelt = μ H2O

gas = μ H 2Obiotite = L = μ H 2O

n

μCaOmelt = μCaO

gas = μCaObiotite =L = μCaO

n

If

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μH 2Omelt > μ H2O

gasthen system will have to adjust the mass(concentration) to make them equal:

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μH 2Omelt = μ H2O

gas

Gibbs Free Energy of Mixing

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).

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μA < GA° ;μB < GB

°

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μAi = GA

° + RT lnaAi

aAi = γ A

i ⋅XAi

For ideal solutions (remember dG of mixing is linear),

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γAi →1

such that the activity is equal to the mole fraction.

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μAi = GA

° + RT ln XAi

P, T, X Stability of Crystals

Equilibrium 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

Fugacity Defined

For gaseous phases at fixed temperature: dGT = VdP

- Assume Ideal Gas Law

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PV = nRT;n = 1

V =RT

P

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dGT = VdP =RTP

⎛ ⎝

⎞ ⎠dP = RT ln dP

PA = 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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μA = GA° + RT ln fA

Equilibrium Constants

Mg2SiO4 + SiO2 = 2MgSiO3

olivine melt opx

G =

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2μMgSiO3

opx − μMg2SiO4

ol − μSiO2

melt = 0

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μSiO2

melt = GSiO2

° glass + RT lnaSiO2

melt

μMg2SiO4

ol = GMg2SiO4

° ol + RT lnaMg2SiO4

ol

μMgSiO3

opx = GMgSiO3

° opx + RT lnaMgSiO3

opx

Equilibrium Constants, con’t.

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2GMgSiO3

° opx − GMg2SiO4

° ol − GSiO2

° glass =−RT ln(aMgSiO3

opx )2

(aMg2SiO4

ol ⋅aSiO2

melt )

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K eq =(aMgSiO3

opx )2

(aMg2SiO4

ol ⋅aSiO2

melt )

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GF° = −RT lnK eq

where dG°F is referred to as the change in standard state

Gibbs free energy of formation, which may be obtainedfrom tabulated information

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/2.303RIntercept = b = log A

All processes that are thermally activated havesimilar form!

Gibbs Free Energy - Temperature Relations

State A is stable for T > Te

because GA < GB

Metastability for polymorphs A & B

Undercoolingallows metastabilityof phase A over B

State B is stable for T < Te

because GB < GA

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