Part 4. Introductory Phase Equilibria and Thermodynamics

Metamorphic Reactions

Each experiment is conducted by mixing oxides or minerals in a sealed capsule and then subjecting the capsule to elevated pressure and temperature for periods of time from minutes to months. The products of such experiments look like this back-scattered electron image:

And yield phase diagrams that appear similar to this:

Metamorphic reactions are recognized in the field as isograds, which are mapped surfaces defined by the (dis)appearance of one or more minerals. The Kangmar Dome in Tibet shows a sequence of four prograde isograds surrounding a structurally lower orthogneiss:

Metamorphic Phase Equilibria

The figure above shows the mechanical analogy for H₂O at -5°C and + 5°C and 1 atm. Left: at -5°C, solid H₂O has the lowest possible energy state. Right: at +5°C, liquid H₂O has the lowest possible energy state. When solid H₂O is actually present at +5°C, the difference between the free energy of solid H₂O and liquid H₂O is available to drive the reaction to form the stable solid H₂O phase, and the reaction will go to completion if kinetically possible.

Gibbs Free Energy Change and Reactions
To understand how the energy that is available to drive reactions changes as a function of pressure and temperature within the Earth, we write the (Gibbs) free energy change of a reaction, _rG, as:

• _rG = _rH + P_rV - T_rS

• volume: the volume of a phase is simply the amount of space taken up by a particular mass or number of moles of that phase. For example, the volume of quartz is 22.7 cm³/mol.
  coordination number: number of anion atoms bonded to single cation atom
  For example, quartz ^IVSi, stishovite ^VISi
  

  You know intuitively that denser phases are stable at higher P. We can see this in terms of the Gibbs free energy by looking at the _rV term:

  • _rG P_rV
  • _rV = V_products - V_reactants
  Thus, if the products are smaller than the reactants, _rV< 0.
  For example, 1 atm V of graphite = 5.3 cm³;
  1 atm V of diamond = 3.4 cm³;
  thus, at any given P, diamond has a lower volume-related energy than graphite;
  with increasing P, therefore, diamond has a lower free energy than graphite and is the polymorph favored by increasing P

• entropy: the degree of randomness of atom arrangement
  • S = (C_P/T) dT
  More-disordered phases are stable at higher temperature. The entropic contribution to the free energy change of reaction is:
  • _rG -T_rS
  For example, the entropy of graphite at 1 atm = 5.7 J/mol K;
  the entropy of diamond at 1 atm = 2.4 J/mol K;
  thus, at any given T, graphite has a lower (more negative) entropic energy than diamond; with increasing T, therefore, graphite has a lower free energy than diamond and is the polymorph favored by increasing T.
  From this you can guess that higher entropy phases always lie on the high-temperature side of reactions.

  
  

• enthalpy: the change in heat of a reaction; equivalent to the heat capacity integrated over the temperature range of interest;
  • H = C_PdT
  • _rG _rH
  By definition, the heat capacity of water at 15°C is 1 cal K^-1 g^-1 or 18 cal K^-1 mol^-1 (i.e., the heat required to heat 1 gram of water from 14.5 to 15.5°C is 1 calorie). Generally, only old farts use calories--joules are more in vogue today (1 cal = 4.184 cal).
  The figure below shows how the heat capacities of various minerals change in response to temperature (at 1 atm). The enthalpy change for a particular temperature change is just the area under the curve.
  

phase: a identifiable solid (e.g., plagioclase), liquid (e.g., silicate melt), or vapor (e.g., H₂O) in the system
component: a specific chemical constituent of a phase that is useful for understanding thermodynamics or phase equilibria (e.g., for plagioclase, one could use the components ab or an--even though neither is present as a separate phase).
degrees of freedom: the number of variables (typically P, T, or composition) that can change without changing the number of phases that are stable
phase rule: the utility of the phase rule is to be able to

• specify the number of phases P that exist for given components C at a specific pressure and temperature
  
• know how many variables such as pressure and temperature may vary for a given set of components and phases

• F + P = C + 2