1. Thermodynamic Models of Magmas
Introduction to Kinetics
Lecture 13

2. Silicate Magmas
Basic structural unit of silicates (solid & liquid) is the silica tetrahedron
These are variously joined by shared, or bridging, oxygens, to form various structures in solids and liquids.
Basic difference between solids and liquids is lack of long-range structure in the latter.
Liquids structure can be studied by quenching them to glass.

3. Liquid Structures
Bridging oxygens and joining tetrahedra results in polymerization of the melt, changing its properties.
Al3+, Ti, and Fe3+ can promote polymerization and, along with Si, are called network-forming ions.
Other ions, Ca2+, Mg2+, Fe2+, Na+, K+, and H+ tend to break up this structure and are called network modifiers.

4. Modeling Silicate Liquids
Silicate liquids are complex solutions of many components.
Solids crystallizing from them are generally solutions themselves.
Generally these solutions cannot be treated as ideal.
Crystallization (or melting) occurs over a wide range of T ( ˚C).
Problems are:
Decide on the components
Determine the nature of the model
Ghiorso et al. adopt a regular solution model for their MELTS model.
Determine the interaction parameters from experimental data.
The resulting program then iteratively computes free energy of the liquid plus free energy of all possible precipitating solids and calculates the equilibrium assemblage based on the principles that
the stable assemblage is the one with the lowest free energy.
The chemical potentials of components in coexisting phases are equal.

5. MELTS Model Free energy of the liquid solution is:
For network modifiers, Ghiroso et al. chose silicate components such as CaSiO3, Mg2SiO4, Na2SiO3, KAlSiO4, etc. because mole fractions of individual oxides tend to be small numbers, reducing influence of interaction parameters.
Network formers generally just the oxides (e.g., Al2O3).
Eleven components, plus water treated separately.
Free energy of the liquid solution is:
Activity coefficients calculated as:

6. Components & Interaction Parameters

7. Will Plagioclase Precipitate?
For anorthite, reaction of interest is:
CaSiO3(l) + Al2O3(l) + SiO2(l) ⇋ CaAl2Si2O8(plag)
But plag is usually a solid solution, so:
x[½NaSiO3 + ½Al2O3(l) + 2½SiO2(l)] + y[CaSiO3(l) + Al2O3(l) + SiO2(l)] ⇋ [yCaAl2Si2O8–xNaAlSi3O8](plag)
What thermodynamic condition must be met for plagioclase to precipitate?
∆Gr is negative.
The negative of ∆Gr is often referred to (particularly in MELTS lingo) as the affinity of reaction, Ar.
MELTS calculates affinities for all possible reactions.
If a plagioclase crystal has precipitated from a magma under equilibrium conditions, what can we say about the component?
Chemical potentials of components must be equal in both solid and liquid.

8. MELTS Iterations Otherwise increment T, P Calculate
∆G for all possible reactions
Precipitate any phases where ∆Gr is negative
Recalculate liquid comp and Calculate ∆G for all possible reactions
Precipitate any phases where ∆Gr is negative;
Recalculate liquid and all solid compositions
Otherwise increment T, P

9. pMELTS
Predicted and actual SiO2 concentrations in experimental melts of peridotite as a function of melt percent.
Predicted and actual pyroxene compositions in lavas.

10. Introduction to Kinetics
Lecture 14

11. Reading in Chapter 5
Read sections 5.1 through (p.160 to p. 199) and section 5.7 (p ).
We will probably skip the intervening sections – or cover them briefly.
Book errata:

12. Kinetics
Whereas thermodynamics concerns itself with equilibrium and the distribution of components between species and phases at equilibrium, kinetics concerns itself with the pathway to equilibrium, including the rates and mechanisms of reaction.
Rates depend on temperature and at the surface of the Earth reaction rates are often so slow that equilibrium is never achieved. This can also be true at higher temperature - and we have mentioned one example (the spinodal).
The microscopic perspective becomes somewhat more important in kinetics than it was in thermodynamics.

13. Overall & Elementary Reactions
The reaction:
CaAl2Si2O8 + 3H2O + CO2 = CaCO3 + 2Al(OH)3 + 2SiO2
describes a key process at the surface of the Earth, namely weathering igneous minerals (plagioclase) to form common sedimentary ones (calcite, gibbsite, and quartz). But does this overall reaction describe what actually happens?
NO.
In thermodynamics ,we might not care, but in kinetics, we do. The first step in understanding reaction pathways and reaction mechanics is to breakdown overall reactions such as this into the elementary reactions.
An elementary reaction is one that involves only one step a describes what occurs on the microscopic level.

14. Reaction Mechanisms
We can begin to breakdown the overall reaction. Some steps are:
CO2(g) + H2O = CO2(aq) + H2O
CO2(aq) + H2O = H2CO3
H2CO3 = H+ + HCO3–
Producing acidity necessary for weathering.
Next step is likely absorption of H+ to the surface:
CaAl2Si2O8 + 2H+ = H2CaAl2Si2O82+
Followed by replacement of the Ca by H:
H2CaAl2Si2O82+ = H2Al2Si2O8 + Ca2+
etc.

15. Defining Reaction Rates
For a reaction such as:
Ca2+ + Mg2+ + 2CO32– = CaMg(CO3)2
We define the rate of reaction as the rate of production of the products, or equivalently, the rate of consumption of the reactants divided by the stoichiometric coefficient:
Equivalently:
The equation tell us nothing about what the reaction rate is, we are just defining what it means.
We’ll shortly see that rates generally do depend on concentrations or reactants and products, so don’t be confused.

16. Reaction Rates & Concentration
Consider the gas phase reaction:
N0 + O2 = NO + O0
First thing that must happen is we must bring the reactants together.
We can imagine a reference frame in which the N atom sweeps out a volume V = v × t × πr2 (velocity times time times area).
Whether a reaction will occur in that time will depend on whether or not the center of an oxygen molecule is present within that volume.
Number of collisions (per N)will be:
Overall collision rate will be:

17. Bottom Line:
For an elementary reaction, we expect the rate of reaction to depend on the concentration of reactants

18. Dependence on Temperature
Just because two people meet on a date, doesn’t mean they will tie the knot.
Kinda depends on how ‘hot’ the date was!
Similarly, just because two atoms or molecules collide, doesn’t mean they will react.
Depends on whether the collision is energetic enough to overcome coulomb repulsion and the electron orbits can reorganize.
An energy barrier, EB, must be overcome.
That means it depends on temperature.
Hence the date analogy!

19. Temperature and Barrier Energy
Since energy levels are closely spaced, we can integrate, so the probability of a molecule having E ≥ EB is:
Our reaction rate is now:
Maxwell-Boltzmann Law gives ave. velocity in a gas as:
where µ is reduced mass of gas:
µ = mNmO2/(mN + mO2)
We suppose that a reaction will proceed if the N atom has at least certain energy, E ≥ EB.
What function tells us how energy is distributed among molecules?
Boltzmann Distribution Function.

20. Arrhenius Relation Our equation now is: Let:
A describes the frequency of opportunity for reaction and is called the frequency factor.
We can express the temperature dependence of the reaction rate as:
This is known as the Arrhenius relation and describes the dependence of reaction rates on temperature.

21. The Rate Constant Arrhenius Relation k is known as the rate constant.
So many K’s!
We’ll use upper case roman K for the equilibrium constant
Lower case roman k for Boltzmann’s constant
Lower case italic k for the rate constant.
We can now write the rate of our N+O2 reaction as:
R = knN nO2

22. Reaction Rates and Temperature
The Arrhenius Relation tells us that reaction rates depend exponentially on temperature (fits everyday experience).
This is reason high-T rocks survive at the surface of the Earth out of equilibrium.
In the gas phase reaction, A depended on square root of T - much weaker than the exponential factor.
Other kinds of reactions show difference dependence of A on T.
In many cases we can view A as a constant independent of T.
T = 300K
EB = 15kJ
Dependence of the rate constant on T and barrier energy