Thermodynamics. a system: Some portion of the universe that you wish to study. The surroundings: The adjacent part of the universe outside the system. Changes in a system are associated with the transfer of energy. Natural systems tend toward states of minimum energy. Energy States.
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Some portion of the universe that you wish to study
The adjacent part of the universe outside the system
Changesin a system are associated with the transfer of energy
Natural systems tend toward states of minimum energy
Figure 5.1. Stability states. Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
Gibbs free energy is a measure of chemical energy
All chemical systems tend naturally toward states of minimum Gibbs free energy
G = H  TS
Where:
G = Gibbs Free Energy
H = Enthalpy (heat content)
T = Temperature in Kelvins
S = Entropy (can think of as randomness)
aPhase: a mechanically separable portion of a system
a Reaction: some change in the nature or types of phases in a system
reactions are written in the form:
reactants = products
The change in some property, such as G for a reaction of the type:
2 A + 3 B = C + 4 D
DG = S (n G)products  S(n G)reactants
= GC + 4GD  2GA  3GB
For a phasewe can determine V, T, P, etc., but not G or H
We can only determine changes in G or H as we change some other parameters of the system
Example: measure DH for a reaction by calorimetry  the heat given off or absorbed as a reaction proceeds
Arbitraryreference stateand assign an equally arbitrary value of H to it:
Choose298.15 K and 0.1 MPa(lab conditions)
...and assignH = 0 for pure elements(in their natural state  gas, liquid, solid) at that reference
In our calorimeter we can then determine DH for the reaction:
Si (metal) + O2 (gas) = SiO2 DH = 910,648 J/mol
= molar enthalpy of formation of quartz (at 298, 0.1)
It serves quite well for a standard value of H for the phase
Entropy has a more universal reference state: entropy of every substance = 0 at 0K, so we use that (and adjust for temperature)
Then we can use G = H  TS to determine G of quartz
= 856,288 J/mol
We can use this equation to calculate G for any phase at any T and P by integrating
z
z
P
T
2
2

=

G
G
VdP
SdT
T
P
T
P
2
2
1
1
P
T
1
1
ThermodynamicsFor other temperatures and pressures we can use the equation:
dG = VdP – SdT(ignoring DX for now)
where V = volume and S = entropy (both molar)
If V and S are constants, our equation reduces to:
GT2 P2  GT1 P1 = V(P2  P1)  S (T2  T1)
which ain’t bad!
Low quartz T and P by integrating
Eq. 1
SUPCRT
P (MPa)
T (C)
G (J) eq. 1
G(J)
V (cm3)
S (J/K)
0.1
25
856,288
856,648
22.69
41.36
500
25
844,946
845,362
22.44
40.73
0.1
500
875,982
890,601
23.26
96.99
500
500
864,640
879,014
23.07
96.36
ThermodynamicsIn Worked Example 1 we used
GT2 P2  GT1 P1 = V(P2  P1)  S (T2  T1)
and G298, 0.1 = 856,288 J/mol to calculate G for quartz at several temperatures and pressures
Agreement is quite good
(< 2% for change of 500o and 500 MPa or 17 km)
Summary thus far:
Use?
If we know G for various phases, we can determine which is most stable
Does the liquid or solid have the larger volume? T and P by integrating
High pressure favors low volume, so which phase should be stable at high P?
Does liquid or solid have a higher entropy?
Figure 5.2.Schematic PT phase diagram of a melting reaction. Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
High temperature favors randomness, so which phase should be stable at higher T?
We can thus predict that the slope of solidliquid equilibrium should be positive and that increased pressure raises the melting point.
Does the liquid or solid have the lowest G at point T and P by integratingA?
What about at pointB?
Figure 52. Schematic PT phase diagram of a melting reaction. Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
The phase assemblage with the lowest G under a specific set of conditions is the most stable
dG = VdP  SdT at constant pressure:dG/dT = S
Because S must be (+)G for a phase decreases as T increases
Figure 5.3. Relationship between Gibbs free energy and temperature for a solid at constant pressure. Teq is the equilibrium temperature. Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
Would the slope for the liquid be steeper or shallower than that for the solid?
Slope of GLiq > Gsol since Ssolid < Sliquid
A:Solid more stable than liquid (low T)
B:Liquid more stable than solid (high T)
Equilibriumat Teq
Figure 5.3. Relationship between Gibbs free energy and temperature for the solid and liquid forms of a substance at constant pressure. Teq is the equilibrium temperature. Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
Now consider a T and P by integratingreaction, we can then use the equation:
dDG = DVdP  DSdT (again ignoring DX)
For a reaction of melting (like ice water)
dDG is then the change in DG as T and P are varied
DG for any reaction = 0 at equilibrium
dP T and P by integrating
DS
=
DV
dT
Y
X
Pick any two points on the equilibrium curve
DG = ? at each
Therefore dDG from point X to point Y = 0  0 = 0
dDG = 0 = DVdP  DSdT
Figure 5.4. Relationship between Gibbs free energy and pressure for the solid and liquid forms of a substance at constant temperature. Peq is the equilibrium pressure. Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
Figure 5.5. Pistonandcylinder apparatus to compress a gas. Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.