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Studiengang Geowissenschaften M.Sc.
Wintersemester 2004/05
Thermodynamics and Phase
Transitions in Minerals
Victor Vinograd & Andrew Putnis
Basic thermodynamic concepts
One of the central themes in Mineralogy is the study
of mineral behaviour - the response of a mineral to
changing physical and chemical conditions.
Equilibration always tends to reduce the free
energy to a minimum.
The internal energy U, of a mineral is the potential
energy stored in the interatomic bonding + the
kinetic energy of the atomic vibrations.
Adding more heat increases the kinetic energy and
hence the temperature and the internal energy.
If the crystal is allowed to expand (and hence do
some work on its surroundings) the total change in
the internal energy is:
dU = dQ - PdV
Change in
heat content
Work done
in
expansion
It is also convenient to define another energy function
called the enthalpy H
as
H = U + PV
Heat Capacity
A fundamental property of a material is its heat
capacity C which describes the amount of heat
dQ, required to change the temperature of one
mole of the material by
C = dQ/dT
The heat capacity can be defined at constant
volume (or at constant pressure) :
From
dU = dQ - PdV
dU/dT = dQ/dT - P dV/dT
At constant volume dV/dT = 0
and
(dU/dT)V = (dQ/dT)V = CV
The heat capacity can also be defined at constant
pressure :
From
H = U + PV
dH = dU + PdV + VdP
Since
dU = dQ - PdV
dH = dQ + VdP
dH/dT = dQ/dT + VdP/dT
At constant pressure
and
dP/dT = 0
(dH/dT)P = (dQ/dT)P = CP
In experiments it is easier to consider constant pressure
rather than constant volume, and so enthalpy changes
are easier to measure than internal energy changes.
Enthalpy is analogous to internal energy at constant
pressure i.e. heat input is equal to the enthalpy change if
the pressure is constant.
where α is the thermal
CP - CV = TVα2/β
expansion coefficient and β is the compressibility.
CP - CV is a very small quantity and becomes
significant only at high T.
Variation in heat capacity CP with temperature.
Classical 3NkT
Heat Capacity, CP
T2
T1
0
1000
Temperature (K)
The enthalpy change between T1 and T2 is given by
H = HT1 +
T2
∫C
P
dT
P
dT
T1
The enthalpy at T1
H = H0 +
T1
∫C
0
H0 includes the enthalpy due to potential energy of the crystal at 0K as well as
the zero point vibrational energy
The heat capacity of pyrope
3mR = 498.8 J/mol K
Heat capacity (J/mol K)
500
Pyrope
Mg3Al2Si3O12
375
250
m=3+2+3+12=20
125
0
0
200
400
600
Temperature, K
800
1000
Enthalpy changes in mineral transformations and reactions
Meaning of standard states : 298K and 1 atm P
Enthalpies are defined as the enthalpy of formation from the
elements (enthalpies of elements assigned to be zero at the standard
state). For ternary oxides enthalpies of formation are often listed from
component oxides.
e.g.
Mg + 1/2O2
⇒ MgO
at 1 atmosphere and 298K
∆Ho = -601.5 kJ/mol.
Si
+
O2
⇒ SiO2 (low quartz)
∆Ho = -910.7 kJ/mol.
2Mg + Si + 2O2 ⇒ Μg2SiO4
∆Ho = -2170.39 kJ/mol.
(all exothermic).
Exercise : Using Hess’s Law calculate the enthalpy of formation of olivine
from the oxides at 1 atm and 298K
(-56.69 kJ/mol)
Enthalpy changes during a polymorphic transformation
If ∆H is positive heat is absorbed i.e. endothermic
If ∆H is negative heat is given out i.e. exothermic
For example: tridymite to quartz
Si + O2 ⇒ SiO2 (tridymite)
∆Hfo = -907.5 kJ/mol
Si + O2 ⇒ SiO2 (low quartz)
∆Hfo = -910.7 kJ/mol
SiO2 (tridymite) ⇒ Si + O2 ⇒ SiO2 (low quartz)
∆ H = + 907.5 - 910.7 = -3.2 kJ/mol
i.e. exothermic
Calcium carbonate CaCO3 : calcite and aragonite
The standard state enthalpy of calcite is -1207.37 kJ/mol
The standard state enthalpy of aragonite is -1207.74 kJ/mol
Therefore the transformation from aragonite to calcite involves an
enthalpy increase of 0.37 kJ/mole i.e. endothermic
But calcite is more stable than aragonite at 25oC and 1 atm. pressure,
showing that a reduction in enthalpy is not a criterion for an increase
in stability
The concept of Entropy
When a mineral changes from one structure to another it exchanges
heat with its surroundings. The entropy is defined as the quantity
which measures the change in the state of order associated with this
process. The overall entropy change is the sum of the entropy change
in the mineral (i.e. the system under consideration) and the entropy
change in the surroundings, i.e.
dS = dSsystem + dSsurr.
For a reversible reaction , i.e. one which passes through a continuous
sequence of equilibrium states, dS = 0, but for any natural reaction
proceeding towards equilibrium, dS > 0, according to the Second Law
of Thermodynamics.
Entropy
The entropy change is defined by
dS > dQ/T
where dQ is the amount of heat exchanged by the system at
temperature T. In a system free to exchange heat with its
surroundings the change in entropy of the system is related to the
enthalpy change by the relation
dS > dH/T
noting that at constant pressure dQ = dH.
Thus the criterion for a mineral transformation or reaction to
proceed is that
dH - TdS < 0.
If dH - TdS = 0 no further change is possible, i.e. the system is at
equilibrium. If
dH - TdS is > 0 the reaction will not proceed.
Entropy and the direction of change in a reaction
The quantity (dH - TdS) therefore can be used to define a criterion for
the direction of change in a mineral reaction and a definition of
equilibrium. This quantity is known as the change in the Gibbs free
energy, dG of the system. Thus
dG = dH - TdS
or
G = H -TS
Therefore the change in free energy ∆G must always be negative
in a reaction. Equilibrium is defined as the state with the minimum
Gibbs free energy.
Configurational entropy and disorder
In the statistical definition, according to Boltzmann, the entropy of a
system in a given state is related to the probability of the existence
of that state. The "state" in this context refers to a particular
distribution of atoms and their vibrational energy levels.
Mathematically this is expressed as
S = k ln ω
where ω is the probability that a given state will exist and k is
Boltzmann's constant (1.38 x 10-23 JK-1)
The probability is related to the state of disorder or randomness in
the structure which may be expressed statistically by the number of
different ways in which atoms can arrange themselves in that state.
Entropy - an example
Consider a distribution of atoms of A and B on a simple cubic
lattice which contains a total of N atomic sites.
Ordered
Disordered
To determine the entropy of a completely random distribution of A
and B atoms we need to calculate the number of ways in which the
atoms can be arranged
If the atomic fraction of A atoms is xA and of B atoms is xB, there are
xAN atoms of A and xBN atoms of B to be distributed over N sites. The
number of such arrangements, ω, is determined from statistics as
ω = N! / (xAN)! (xBN)!
The entropy associated with this disorder is therefore
S = k ln ω = k ln [N! / (xAN)! (xBN)!]
When the number of sites N is very large, as is the case in a mole of a
mineral, we can simplify this expression by using Stirling's
approximation:
ln N! = N ln N - N
Thus
S = -Nk(xA lnxA+ xB ln xB)
For a mole of sites N is Avogadro's number (6.02 x 1023 mol-1) and
Nk = R, the gas constant (8.31 J mol-1K-1).
Hence
S = -R(xA lnxA+ xB ln xB)
In a complex mineral structure there may be more than one site
per formula unit over which disorder can occur and the more
general form of the above expression is
S = -nR(xA lnxA+ xB ln xB)
where n is the number of sites on which mixing occurs
Entropy S
This expression is known as the entropy of mixing.
Since xA and xB are both fractions, S is always positive as shown
below, which gives the general form of the curve for the entropy
of mixing ∆S as a function of the atomic fraction xB of B atoms.
0.0
0.2
0.4
0.6
0.8
Composition X B
1.0
Example 2
Entropy of mixing Fe and Mg over the
M1 and M2 sites in olivine (Mg,Fe)2SiO4
S = -nR(xA lnxA+ xB ln xB)
For a composition 50 mole% forsterite (Mg2SiO4) and 50 mole%
fayalite (Fe2SiO4) :
S = -nR(xA lnxA+ xB ln xB)
(Note: n=2 for olivine
because there are 2 sites for mixing in each formula unit)
= -2R(0.5 ln 0.5 + 0.5 ln 0.5)
= 11.52 J mol-1 K-1
For a composition 25 mole% forsterite (Mg2SiO4) and 75 mole%
fayalite (Fe2SiO4) :
S = -nR(xA lnxA+ xB ln xB)
= -2R(0.25 ln 0.25 + 0.75 ln 0.75)
= 9.34 J mol-1 K-1
Vibrational entropy
- the entropy associated with lattice vibrations
Energy of lattice vibrations is quantised - each
quantum of vibrational energy is a phonon.
Increasing the amplitude of atomic vibrations
increases the number of phonons. The phonon
spectrum defines the number of phonons in each
frequency range : the phonon density of states.
Vibrational entropy arises from the number of ways
of distributing phonons over the vibrational energy
levels in a crystal.
Vibrational entropy S is related to the heat capacity Cp :
For a reversible process dS = dQ/T and since Cp
=dQ/dT dS/dT=Cp/T
and
S=
∫
CP
dT
T
S = S0 +
T1
∫
0
T1
0
CP
dT
T
Cp
dT
T
Cp / T
0
T1
Temperature (K)
1000
The Gibbs free energy, G, and equilibrium
Example: the transformation of aragonite to calcite:
aragonite ⇒ calcite
∆Ho = +370 Joules at 25oC and 1 atm.
The entropy of aragonite and calcite under the same conditions is 88
J mol-1 K-1 and 91.7 J mol-1 K-1 respectively. Thus for the
transformation
aragonite ⇒ calcite
∆So = +3.7 J mol-1 K-1 at 25oC and 1 atm.
Therefore ∆G for the aragonite ⇒ calcite transformation equals
∆H - T∆S = +370 - (298 x 3.7) = -732.6 Joules.
Calcite thus has the lower free energy and is the stable polymorph of
calcium carbonate at 25oC and 1 atm. pressure.
Variation of enthalpy and free energy as a function of T and P
H
TS
G
G, H
G
T (at constant P)
(a)
P (at constant T)
(b)
We can derive the relations between G, T and P from the basic definitions
of the thermodynamic functions as follows:
G = H - TS
Substituting H = U + PV into this equation gives
G = U + PV - TS
Differentiating gives
dG = dU + PdV + VdP - TdS - SdT.
From the first law, dU = dQ - PdV, and for a reversible process (which is
always in equilibrium) dQ = TdS.
On substitution
dG = VdP - SdT
at equilibrium
Therefore at constant pressure:
(∂G/∂T)p = -S
and at constant temperature:
(∂G/∂P)T = V
The figure in the previous slide shows the way in which the free energy of a
mineral structure changes when it maintains equilibrium with a changing T
and P.
For a polymorphic transformation:
Hβ
Hα
∆H
G, H
Gα
Gβ
Tc
T
The figure shows the free energy and enthalpy
curves for two phases α and β as a function of T
Above Tc the β phase is more stable than the α phase (it has a
lower free energy). The enthalpy change at Tc is called the latent
heat of transformation (∆H )
If we consider both temperature and pressure stability fields for two
polymorphs α and β, the free energy curves become surfaces in G–T–P
space and their intersection defines the equilibrium between α and β.
G
α
β
B
A
(a)
T
The slope of this intersection line when projected on the P–T plane, i.e.
dP/dT is given by the Clapeyron relation which arises from the
equilibrium relation ∆G = 0. At equilibrium,
∆VdP = ∆SdT
and hence
dP/dT = ∆S/∆V
B
G
P
α
β
dP = ∆S
dT
α
stable
B
A
(a)
A
T
(b)
∆V
β
stable
T
Reversible and irreversible processes. Metastability
G
G
Gα
Tc
Reversible at Tc
Gα
∆T
Gβ
(a)
G
Gβ
Gβ
T
(b)
Tc
Large undercooling
Gα
∆T
T
(c)
Tc
T
Smaller superheating
Case (a) is never observed in practice because at Tc (equilibrium) the
free energy of both phases is the same. Undercooling (case b) is always
required to produce a free energy ‘drive’ for the transition from α to β.
During heating the overheating is generally less.
Reversible and irreversible processes. Metastability
Gγ
G
Gβ
T2
Tc
Gα
T
As phase β is cooled from high T it should transform to phase α
below Tc. If the transition is kinetically impeded, then phase α
could persist to temperature T2. In that case a transition to the
metastable phase γ can occur. This is the case in the case of
high tridymite (β) transforming to low tridymite (γ). Phase α would
be high quartz.
First- and second-order phase transitions
At the equilibrium temperature (or pressure), the free energies
of the two polymorphs are equal, and there is no discontinuity in
the free energy G on passing from one structure to another.
In first-order phase transitions the first derivatives of the free
energy ∂G/∂T and ∂G/∂P are discontinuous. Since ∂G/∂T = -S
and ∂G/∂P = V, first order phase transitions are characterized by
discontinuous changes in entropy and volume at the critical
temperature.
β
α
G
H,S
α
β
Tc
T
In second-order phase transitions the first derivatives of the free
energy are continuous, but the second derivatives ∂2G/∂T2 and
∂2G/∂P2 are discontinuous. The enthalpy change is continuous and
so there is no latent heat associated with second order transitions.
Since
∂2G/∂T2 = - ∂S/∂T = - Cp/T
and
∂2G/∂P2 = - Vβ ;
∂2G/∂T∂P = Vα
the discontinuities occur in the specific heat capacity Cp , the
compressibility β and the thermal expansion α.
β
α
G
H,S
β
Tc
α
Tc
T
Although the thermodynamic classification of a phase
transition cannot be simply related to a transformation
mechanism, we can say that:
First order phase transitions are generally
reconstructive transitions
e.g. quartz ⇔ tridymite ⇔ cristobalite
Second order phase transitions are generally
displacive transitions
e.g. high quartz ⇔ low quartz
1/--pages
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