Second Law of Thermodynamics

• One statement defining the second law is that aspontaneous natural processes tend to even outthe energy gradients in a isolated system.

• Can be quantified based on the entropy of thesystem, S, such that S is at a maximum whenenergy is most uniform. Entropy can also beviewed as a measure of disorder.

ΔS = 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

�

dP

dT=(S

c! S

l)

(Vc! V

l)

="S

"V

Clapeyron Equation

Thermodynamics of Solutions

• Phases: Part of a system that is chemically andphysically homogeneous, bounded by a distinctinterface with other phases and physically separablefrom other phases.

• Components: Smallest number of chemical entitiesnecessary to describe the composition of everyphase in the system.

• Solutions: Homogeneous mixture of two or morechemical components in which their concentrationsmay be freely varied within certain limits.

Mole Fractions

�

XA!

nA

n"=

nA

(nA

+ nB

+ 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 componentA in phase i as

�

XA

i

For a simple binary system, XA + XB = 1

Partial Molar Quantities• Defined because most solutions DO NOT mix ideally, but

rather deviate from simple linear mixing as a result of atomicinteractions of dissimilar ions or molecules within a phase.

• Partial molar quantities are defined by the “true” mixingrelations of a particular thermodynamic variable and can becalculated graphically by extrapolating the tangent at the molefraction of interest back to the end-member composition.

• Need to define a standard state (i.e. reference) from which tomeasure variations in thermodynamic properties. Thesimplest and most common one is a pure phase at STP.

Partial Molar Volumes & MixingTemperature Dependenceof Partial Molar Volumes

Partial Molar Gibbs Free EnergyAs noted earlier, the change in Gibbs free energy function determines thedirection in which a reaction will proceed toward equilibrium. Because ofits importance and frequent use, we designate a special label called thechemical potential, µ, for the partial molar Gibbs free energy.

�

µA!

"GA

"XA

#

$ %

&

' ( P ,T ,X

B,X

C,…

We must define a reference state from which to calculate differences inchemical potential. The reference state is referred to as the standard stateand 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

�

dG = VdP ! SdT + µidX

i

i

"

µidX

i

i

" = µAdX

A+ µ

BdX

B+ µ

CdX

C+! + µ

ndX

n

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• changes 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 overallpotential is lowest.

• At equilibrium, the chemical potentials for any specificcomponent in ALL phases must be equal. This means thatthe system will change spontaneously to adjust by the Law ofMass Action to cause this state to be obtained.

�

µH2O

melt = µH2O

gas = µH2O

biotite =! = µH2O

n

µCaO

melt= µCaO

gas= µCaO

biotite=! = µCaO

n

If

�

µH 2O

melt> µH2O

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

�

µH 2O

melt= µH2O

gas

Activity - Composition RelationsThe activity of any diluted component is always less than thecorresponding Gibbs free energy of the pure phase, where theactivity is equal to unity by definition (remember the choiceof standard state).

�

µA

< GA

°;µ

B< G

B

°

�

µA

i = GA

° + RT lnaA

i

aA

i= !

A

i"X

A

i

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

�

!A

i

"1such that the activity is equal to the mole fraction.

�

µA

i= G

A

°+ RT ln X

A

i

Gibbs Free Energy of Mixing

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

Fugacity DefinedFor gaseous phases at fixed temperature: dGT = VdP

- Assume Ideal Gas Law

�

PV = nRT;n = 1

V =RT

P

�

dGT

= VdP =RT

P

!

"

#

$ dP = RT lndP

PA = XAPtotal and the fugacity coefficient, γA, is defined as fA/PA, which is analogous to the activity coefficient. As the gas component becomes more ideal, γA goes to unity and fA = PA.

�

µA = GA

°+ RT ln fA

Equilibrium Constants

Mg2SiO4 + SiO2 = 2MgSiO3 olivine melt opx

ΔG =

�

2µMgSiO3

opx! µMg2SiO4

ol! µSiO2

melt= 0

�

µSiO2

melt = GSiO2

° glass + RT lnaSiO2

melt

µMg2SiO4

ol= GMg2SiO4

° ol+ RT lnaMg2SiO4

ol

µMgSiO3

opx = GMgSiO3

° opx + RT lnaMgSiO3

opx

At Equilibrium

Equilibrium Constants, con’t.

�

2GMgSiO3

° opx!GMg2SiO4

° ol!GSiO2

° glass=!RT ln(aMgSiO3

opx )2

(aMg2SiO4

ol "aSiO2

melt )

�

Keq =(aMgSiO3

opx )2

(aMg2SiO4

ol !aSiO2

melt )

�

!GF

°= "RT lnKeq

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

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