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. Entropy can also be viewed as a measure of disorder.
ΔS = 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 ) "S = = dT (Vc ! Vl ) "V
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 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. • Need to define a standard state (i.e. reference) from which to measure variations in thermodynamic properties. The simplest and most common one is a pure phase at STP.
Partial Molar Volumes & Mixing Temperature Dependence of Partial Molar Volumes
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 & µA ! % ( $ "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 + " µi dX i i
" µ dX i
i
= µ A dX A + µB dX B + µC dX C + ! + µn 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 • 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 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
µ
melt H 2O
µ
melt H 2O
=µ
µ
melt CaO
=µ
>µ
gas H2 O
gas H2 O
biotite H 2O
=µ
=!=µ
gas CaO
biotite CaO
=µ
=! = µ
n H 2O
n CaO
then system will have to adjust the mass melt (concentration) to make them equal: µ H 2O
= µ H2 O gas
Activity - Composition Relations The activity of any diluted 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
µA < G ; µB < G
° B
µAi = GA° + RT ln aAi
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
µ = G + RT ln X i A
i A
i A
! "1 i A
Gibbs Free Energy of Mixing
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, 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. ° A
µA = G + RT ln f A
Equilibrium Constants Mg2SiO4 + SiO2 = 2MgSiO3 olivine melt opx At Equilibrium
ΔG = 2µ
µ
melt SiO 2
µ
ol Mg 2 SiO 4
µ
opx MgSiO3
!µ
opx MgSiO3
=G
° glass SiO 2
ol Mg 2 SiO 4
+ RT ln a
° ol Mg 2 SiO 4
=G =G
!µ
° opx MgSiO3
=0
melt SiO 2
melt SiO2
+ RT ln a
ol Mg 2 SiO4
+ RT ln a
opx MgSiO3
Equilibrium Constants, con’t. ° opx MgSiO3
2G
° ol Mg 2 SiO 4
° glass SiO 2
!G
!G
opx 2 !RT ln(aMgSiO ) 3 = ol melt (aMg 2 SiO 4 "aSiO 2 )
° F
!G = " RT lnK 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, and
K eq =
opx 2 (aMgSiO ) 3
ol melt (aMg ! a ) SiO SiO 2 4 2
