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