Lecture 16 – Metamorphic Rocks 3

Last time we talk about mineral reactions that could be used as geothermometers and as geobarometers to constrain the pressure and temperaure of metamorphism. Garnets are chemically slow to react so that, once formed, the composition of garnet can be “locked in”, even though the composition of later-formed garnet may be very different. Thus natural garnets are commonly strongly zoned in Mg, Fe, Ca, and Mn, as shown by the image to the right (a garnet from n. Idaho). This is a false-color image showing the concentration of Ca in the mineral. Red represents high Ca, blue-green show intermediate values. Look closely at the blue zones, which suggest at least two, and probably three, stages of garnet growth.

We can then use these chemical zones to determine the P-T path over which that garnet grew.

Next step is to determine over what time period this growth happened. The figure below shows the radiometric age of peak metamorphism in n. Idaho related to the Idaho batholith. ,Note the rapid uplift of this region between 90 and 50 Ma, during which time the decompression was nearly isothermal:

This sort of uplift (decompression) path requires that the rocks be heated while they were being uplifted. The source of heat in this area can only be the Idaho batholith, thus batholith emplacement and metamorphism were intimately related. Additionally, the high temperatures of metamorphism are sufficient to cause dehydration melting, thus providing an explanation of the source of the batholith. Once formed, the batholith rose rapidly (since it was hot and buoyant), carrying the rocks along with it.

The thermodynamics of reactions

The question was raised about equilibrium, and how we know it has been achieved. One of the most important indications of equilibrium is that the mineral assemblage is stable, particularly as determined by Gibbs’ phase rule (p + f = C + 2) which tells us the number of phases that are allowed to exist at equilibrium for a given number of components, when the only intensive variables that can change are pressure and temperature.

F = C system is divariant (common)

F < C occurs when systems exhibit solid solution

F> C either P or T is fixed, OR we didn’t pick the right number of components (a common problem!), OR the system is not at equilibrium (e.g., partially completed retrograde reactions)

Last time we looked at the simple one-component system Al2SiO5. Now let’s look at a more complicated 3-component system Al2O3 – SiO2 – H2O (Fig. 7.13 in your book). The reactions shown are univariant (degree of freedom = 1), and thus involve four phases:

Al₂Si₂O₅(OH)₄ + 2SiO₂ = Al₂Si₄O₁₀(OH)₂ + H₂O

kaolinite + 2quartz = pyrophyllite + water (vapor)

This is a low temperature dehydration reaction. NOTE: this reaction does NOT limit the stability field of quartz IF kaolinite is not present!

Al₂Si₄O₁₀(OH)₂ = Al2SiO5 + 3SiO₂ + H₂O

pyrophyllite = kyanite + 3quartz + water (vapor)

Pyrophyllite is stable over only about 100˚C, at which point it dehydrates to form the anhydrous phases kyanite and quartz. In fact, in this system the only phases that are stable over 450˚C are corundum, quartz, and an aluminosilicate.

We can also use chemographic diagrams to look at phase relations graphically…. [EX]

Let’s say that the rocks in our area exist of the following mineral assemblages:

x-xy-x2z

xy-xyz-x2z

xy-xyz-y

xyz-z-x2z

y-z-xyz

Minerals that coexist are connected by tie lines..any specific assemblage of minerals represents a bulk rock composition…

The Gibbs free energy of a mineral is a way of expressing the relative stability of that mineral in a specific P,T,X space. The Gibbs free energy of a reaction is the difference between the Gibbs free energy of the products and of the reactants.

Consider the equilibrium between calcite and its polymorph, aragonite (Fig. 7.16). the reaction relating the two is simply CaCO₃ (cc) = CaCO₃ (ar)

The Gibbs energy of the reaction ∆G = G_aragonite - G_calcite

If G_aragonite < G_calcite, then ∆G is negative and aragonite is stable. If G_aragonite > G_calcite then ∆G is positive and calcite is stable. Calcite and aragonite can coexist only when ∆G = 0; the P-T conditions at which this is true are defined by the phase boundary.

How do we determine ∆G? There are thermodynamic tables that give the Gibbs free energy of formation (∆G˚_f) for most minerals, given in J/mol (or Kcal/mol). This number refers to the amount of energy released (in joules or Kcal) when pure elements react to produce one mole of the mineral in question. Because ∆G˚_f is negative (exothermic, that is, energy is released), calcite is more stable than the individual elements. We can use the formation energy to determine the energy of reaction:

∆G_rxn = ∆G_f(aragonite) - ∆G_f(calcite)

So far we haven’t talked about the effect of P,T. Gibbs free energy varies with P,T (which is why phase boundaries vary with P,T). The variations depend on internal energy (E, also called enthalpy), molar volume (V), and molar entropy (S) as

G = E + PV – TS

Similarly, we can write this equation in terms of reactions:

∆G_rxn = ∆E_rxn + P∆V_rxn – T∆S_rxn

where in each case, the change of the parameter with reaction is the difference between the products and reactants. Let’s think about what these equations mean. If P∆V is large (that is, high pressure and/or large molar volumes) then ∆G will be large and the mineral will be unstable. Therefore, at high pressures, samples with low molar volume (high densities) will be most stable. Similarly, at high T, high entropy samples are most stable (also decreases G).

If a reaction is at equilibrium, then

∆G_rxn = 0 = ∆E_rxn + P∆V_rxn – T∆S_rxn

From this equation we can derive an important equation known as the Clausius-Clapeyron equation, used to calculate the slope of a reaction on a P-T diagram:

slope = dP/dT = ∆S_rxn/∆V_rxn

For most solid-solid reactions, S and V vary little so phase boundaries are straight lines. In contrast, dehydration and decarbonation reactions plot as curves because the volumes and entropies of fluids like H2O and CO2 vary greatly with P and T, leading to great variations in ∆V and ∆S.

In theory, we could use this approach to predict phase diagrams for systems of any composition. In practice, things get complicated, particularly for minerals that form solid solutions. Additionally, we don’t have good thermodynamic data for all mineral species.

The Rock Cycle

Over the past several weeks, we have seen that Earth material can start as igneous rock, then be weathered, eroded, transported and deposited as sediment. This sediment can then be lithified and converted to a sedimentary rock which may be subsequently metamorphosed. If metamorphism is sufficiently intense, the rock can melt and start the cycle all over again. This sequence is known as the rock cycle (or recycle!).