Study of interactions among work, energy, and heat. Calorimetry
1. 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
2. Energy States
Unstable: falling or rolling
Stable: at rest in lowest
energy state
Metastable: in low-energy
perch
Figure 5.1. Stability states. Winter (2001) An Introduction to
Igneous and Metamorphic Petrology. Prentice Hall.
3. Gibbs Free Energy
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)
4. Thermodynamics
a Phase: a mechanically separable portion of a system
Mineral
Liquid
Vapor
a Reaction: some change in the nature or types of phases
in a system
reactions are written in the form:
reactants = products
5. Thermodynamics
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
6. Thermodynamics
For a phase we 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
Arbitrary reference state and assign an equally arbitrary
value of H to it:
Choose 298.15 K and 0.1 MPa (lab conditions)
...and assign H = 0 for pure elements (in their natural
state - gas, liquid, solid) at that reference
7. Thermodynamics
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
8. Thermodynamics
For other temperatures and pressures we can use the
equation:
dG = VdP – SdT (ignoring DX for now)
where V = volume and S = entropy (both molar)
We can use this equation to calculate G for any phase at
any T and P by integrating
z
z
G G VdP SdT
T P T P
T
T
P
P
2 1 1
1
2
1
2
2
- = -
9. Thermodynamics
If V and S are constants, our equation reduces to:
GT2 P2
- GT1 P1
= V(P2 - P1) - S (T2 - T1)
which ain’t bad!
10. Thermodynamics
In 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
Low quartz 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
Agreement is quite good
(< 2% for change of 500o and 500 MPa or 17 km)
11. Thermodynamics
Summary thus far:
G is a measure of relative chemical stability for a phase
We can determine G for any phase by measuring H and S for
the reaction creating the phase from the elements
We can then determine G at any T and P mathematically
Most accurate if know how V and S vary with P and T
• dV/dP is the coefficient of isothermal compressibility
• dS/dT is the heat capacity (Cp)
Use?
If we know G for various phases, we can determine which is
most stable
Why is melt more stable than solids at high T?
Is diamond or graphite stable at 150 km depth?
What will be the effect of increased P on melting?
12. Does the liquid or
solid have the larger
volume?
High pressure favors
low volume, so which
phase should be stable
at high P?
Does liquid or solid have a
higher entropy?
High temperature favors
randomness, so which
phase should be stable at
higher T?
We can thus predict that the slope
of solid-liquid equilibrium should
be positive and that increased
pressure raises the melting point.
Figure 5.2. Schematic P-T phase diagram of a melting reaction.
Winter (2001) An Introduction to Igneous and Metamorphic
Petrology. Prentice Hall.
13. Does the liquid or solid
have the lowest G at
point A?
What about at point B?
The phase assemblage with the lowest G under a specific set of
conditions is the most stable
Figure 5-2. Schematic P-T phase diagram of a melting reaction.
Winter (2001) An Introduction to Igneous and Metamorphic
Petrology. Prentice Hall.
14. Free Energy vs. Temperature
dG = VdP - SdT at constant pressure: dG/dT = -S
Because S must be (+) G for a phase decreases as T
increases
Would the slope for the
liquid be steeper or
shallower than that for
the solid?
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.
15. Free Energy vs. Temperature
Slope of GLiq > Gsol since
Ssolid < Sliquid
A: Solid more stable than
liquid (low T)
B: Liquid more stable than
solid (high T)
Slope dP/dT = -S
Slope S < Slope L
Equilibrium at Teq
GLiq = GSol
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.
16. Now consider a reaction, we can then use the equation:
dDG = DVdP - DSdT (again ignoring DX)
For a reaction of melting (like ice water)
DV is the volume change involved in the reaction (Vwater - Vice)
similarly DS and DG are the entropy and free energy changes
dDG is then the change in DG as T and P are varied
DG is (+) for S L at point A (GS < GL)
DG is (-) for S L at point B (GS > GL)
DG = 0 for S L at point x (GS = GL)
DG for any reaction = 0 at equilibrium
17. 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
X
Y
dP
dT
DS
=
DV
18. Figures I don’t use in class
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.
19. Figures I don’t use in class
Figure 5.5. Piston-and-cylinder apparatus to compress a gas. Winter
(2001) An Introduction to Igneous and Metamorphic Petrology. Prentice
Hall.
Editor's Notes
a system:
Some portion of the universe that you wish to study
a glass of water
plagioclase
the mantle
Changes in a system are associated with the transfer of energy
lift an object: stored chemical potential
drop an object: potential kinetic
pump up a bicycle tire: chemical mechanical heat (friction + compression)
add an acid to a base: chemical heat
The side of the reaction with lower G will be more stable
How do we go about determining this for a reaction?
First we must be able to determine G for the phases in the reaction at any P and T
To integrate properly we must know how V and S vary with P and T (hence the calculus), but we shall simplify the math and assume V and S are constant
This simplifies our math considerably (but may lead to some errors)
We can check the validity of our assumptions by comparing to experiments or more rigorous calculations (performed by computer)
The P and T corrections can be done separately in any order, since we are concerned only with the initial and final states, not the path the system followed to get from one to the other
All of these questions depend on reactions comparing 2 or more phases
Consider the phase diagram (as we anticipate igneous petology)
We can apply thermodynamics qualitatively to assess these diagrams
From dG = VdP - SdT we can deduce that at constant pressure: dG = -SdT
And since S must be (+) G for a phase decreases as T increases