Part 1. Overview of Metamorphism and Tectonics
Part 2. Introduction to Metamorphism
Part 3. Physical Processes of Metamorphism
Part 4. Introductory Phase Equilibria and Thermodynamics
Part 5. Ultramafic Rocks
Part 6. Mafic Rocks
Part 7. Pelitic Rocks
Part 8. Diffusion
Part 9. Thermobarometry
Part 10. Kinetics
Part 11. Interaction Between Metamorphism and Deformation
Part 12. Metamorphism and Geochronology
Part 13. Metamorphism and Tectonics I
Part 14. Metamorphism and Tectonics II
Thermodynamics Notes

Part 9. Thermobarometry

Read pages 53-70 of Vernon and Clarke or Chapter 16 of I&M Petrology by Best or Chapter 27 of Igneous and Metamorphic Petrology by John Winter or Chapter 19 of Igneous and Metamorphic Petrology by Philpotts

Qualitative thermobarometry can be done simply by inspection of rocks and minerals in the field and in thin section, based on reaction textures and mineral assemblages. These garnets show thin rims of plagioclase, likely related to decompression:

Examples of P-T Determinations

Let's look at some examples of quantitative determinations of metamorphic pressures and temperatures.
Pressures and temperatures of equilibration in lower crustal xenoliths from Tibet were constrained using compositions of garnet, pyroxene, and plagioclase:

Pressure-Temperature Paths

One of the important goals of metamorphic petrologists is to reconstruct the P-T paths experienced by rocks. In general, this goal is attained by textural analysis combined with thermobarometry.

peak temperature: the maximum T experienced by a rock
peak pressure: the maximum P experienced by a rock
metamorphic field gradient: a spatially varying sequence of peak pressures and temperatures.

Quantitative Thermobarometry

More sophisticated, quantitative determination of metamorphic pressures and temperatures involves four steps:

  1. careful assessment of reaction textures
  2. identification of zoning
  3. measurement of mineral compositions
  4. application of thermodynamics.

  • Mineral compositions can be measured with an electron microprobe to high accuracy; they can be measured to lower detection limits with an ICP mass spectrometer.


    1. Assessment of reaction textures. The purpose of this step is to identify which minerals are early, which are late, and which are part of a stable assemblage. Early minerals are likely to be inclusions or broken, late minerals may be in cracks or strain shadows, and minerals that are in textural equilibrium should not be separated by reaction zones. Back-scattered electron imaging is often a convenient and powerful way to study textures:

      This back-scattered electron image shows zoning in titanite from a Norwegian gneiss:

    2. Identification of zoning. Compositional zoning in minerals can be done by tuning the electron microprobe spectrometers--or focusing the ICP mass spectrometer detectors--to specific elements and then making element maps:

      An electron microprobe works by measuring element-characteristic x-rays emitted by a sample being bombarded with electrons on a 2-micron spot:

      Zoning can be quantified over an entire grain with good coverage of spot analyses:

      Such zoning can quantified using line scanning:

    3. Application of thermodynamics. Experiments have been used to determine the PT positions of many different kinds of endmember reactions.
      For example, experiments have shown that the distribution of Fe and Mg between garnet and biotite is a function of inverse temperature:

      thermometry: The assessment of metamorphic temperatures.
      Quantitative thermometry typically is based on the exchange of Fe and Mg between pairs of minerals. For example, the exchange of Fe and Mg between garnet and biotite can be written as a reaction among the Fe and Mg endmembers of these minerals:

      • Fe3Al2Si3O12 + KMg3AlSi3O10(OH)2 = Mg3Al2Si3O12 + KFe3AlSi3O10(OH)2

      barometry: The assessment of metamorphic pressures.
      Qualitative barometry is based on mineral assemblages-such as the coexistence of kyanite + staurolite.
      Quantitative barometry typically is based on net transfer reactions
      For example, garnet - aluminumsilicate - silica - plagioclase (GASP)

      • 3CaAl2Si2O8 = Ca3Al2Si3O12 + 2Al2SiO5 + SiO2
      • 3 anorthite = grossular + 2 kyanite + quartz

    Quantitative Geothermometry and Geobarometry

    Quantitative thermometry and barometry are based on thermodynamics.

    Clapeyron Relation

    There is a useful relation between the slope of a reaction in PT space (i.e., dP/dT) and the entropy and volume changes of the reaction that follows from At equilibrium G = 0, such that or So, the P-T slope of a reaction is equal to the ratio of the entropy change to the volume change. Alternatively, along the equilibrium curve, the changes in pressure times the volume change are equal to changes in temperature times the entropy change. This is the Clapeyron Equation.

    Popular thermometers include garnet-biotite (GARB), garnet-clinopyroxene, garnet-hornblende, and clinopyroxene-orthopyroxene; all of these are based on the exchange of Fe and Mg, and are excellent thermometers because rV is small, such that

    is large (i.e., the reactions have steep slopes and are little influenced by pressure). In contrast, as we will see, reactions with gentle slopes are the best barometers.

    Calculating the PT Position of a Reaction

    Because rG = 0 at equilibrium, we can write the following approximation (ignoring fluids and variable heat capacities):
    • 0 = rH1,Tref - TrS1,Tref + rVsP
    and thus we can calculate the pressure of a reaction at different temperatures by
    • P = rH1,Tref - TrS1,Tref / -rVs
    and we can calculate the temperature of a reaction at different pressures by
    • T = Tref + rH1,Tref + rVsP / rS1,Tref
    Let's do this for the albite = jadeite + quartz reaction at T = 400 K and T = 1000 K:
    • P = (15,860 - 5147 * 1000) / -1.7342 = 20.6 kbar
    Assuming that dP/dT is constant (incorrect in detail), the reaction looks like this

    This is fine and dandy for determining P and T based on the appearance and disappearance of minerals, but how do we determine P and T based on changes in mineral composition?

    Equilibrium Constant

    For the reaction we write the equilibrium constant: Moreover, or At equilibrium, where rG°= 0, ln K = 0 and K = 1. This enables us to write a very important relationship:

    Let's see what K looks like for jadeite + quartz = albite at 800 K and 20 kbar:

    If we do this for all of PT space, we can contour PT space in terms of lnK:

    Solid Solutions

    Almost no phases are pure, but typically are mixtures of components. For example, olivine varies from pure forsterite Mg2SiO4 to pure fayalite Fe2SiO4, and can have any composition in between--it is a solid solution. We need a way to calculate the thermodynamic properties of such solutions.
    As a measure of convenience, we use mole fraction to describe the compositions of phases that are solid solutions. For example, a mix of 1 part forsterite and 3 parts fayalite yields an olivine with 25 mol% forsterite and 75 mol% fayalite, which can be written as (Mg0.25Fe0.75)2SiO4 or fo25fa75, etc. Mole fractions are denoted as Xi.

    Activity Models (Activity-Composition Relations)

    Garnets are solid solutions of
    component abbrev. formula
    pyrope prp Mg3Al2Si3O12
    almandine alm Fe3Al2Si3O12
    grossular grs Ca3Al2Si3O12
    spessartine sps Mn3Al2Si3O12
    andradite and Ca3Fe23+Si3O12

    The simplest type of useful activity model is the ionic model, wherein we assume that mixing occurs on crystallographic sites. For a Mg-Fe-Ca-Mn garnet with mixing on one site, which we can idealize as (A,B,C,D)Al2Si3O12, the activities are
    • aprp = Mg3XMg3
    • aalm = Fe3XFe3
    • agrs = Ca3XCa3
    • asps = Mn3XMn3

    In general, for ideal mixing in a mineral with a single crystallographic site that can contain ions,
    • ai = Xj
    where a, the activity of component i, is the mole fraction of element j raised to the power.

    Exchange Reactions

    Many thermometers are based on exchange reactions, which are reactions that exchange elements but preserve reactant and product phases. For example:
    • Fe3Al2Si3O12
    + KMg3AlSi3O10(OH)2 = Mg3Al2Si3O12 + KFe3AlSi3O10(OH)2
    • almandine
    + phlogopite = pyrope + annite

    We can reduce this reaction to a simple exchange vector:
    • (FeMg)gar+1 = (FeMg)bio-1

    Let's write the equilibrium constant for the GARB exchange reaction

    • K = (aprpaann)/(aalmaphl)
    • rG = -RT ln (aprpaann)/(aalmaphl)

    This equation implies that the activities of the Fe and Mg components of biotite and garnet are a function of Gibbs free energy change and thus are functions of pressure and temperature.
    If we assume ideal behavior ( = 1) in garnet and biotite and assume that there is mixing on only 1 site
    • aalm = Xalm3 = [Fe/(Fe + Mg + Ca + Mn)]3
    • aprp = Xprp3 = [Mg/(Fe + Mg)]3
    • aann = Xann3 = [Fe/(Fe + Mg)]3
    • aphl = Xphl3 = [Mg/(Fe + Mg)]3
    Thus the equilibrium constant is
    • K = (XMggar XFebio)/(XFegar XMgbio)
    • or K = (Xprp3 Xann3)/(Xalm3 Xphl3)
    Long before most of you were playground bullies (1978) a couple of deities named John Ferry and Frank Spear measured experimentally the distribution of Fe and Mg between biotite and garnet at 2 kbar and found the following relationship:

    If you compare their empirical equation
    • ln K = -2109 / T + 0.782
    this immediately reminds you of
    • ln K = - (rG° / RT) = -(rH / RT) - (PrV / RT) + (rS / R)
    Molar volume measurements show that for this exchange reaction rV = 0.238 J/bar, thus
    • rH = 52.11 kJ/mol
    The full equation is then

    • 52,110 - 19.51*T(K) + 0.238*P(bar) + RT ln K = 0
    To plot the K lines in PT space

    Net-Transfer Reactions

    Net-transfer reactions are those that cause phases to appear or disappear. Geobarometers are often based on net-transfer reactions because rV is large and relatively insensitive to temperature. A popular one is GASP:
    • 3CaAl2Si2O8
    = Ca3Al2Si3O12 + 2Al2SiO5 + SiO2
    • anorthite
    = grossular + kyanite + quartz

    which describes the high-pressure breakdown of anorthite.

    For this reaction
    • rG = -RT ln [(aqtzaky2agrs) / aan3] = -RT ln agrs / aan3
    (the activities of quartz and kyanite = 1 because they are pure phases). A best fit through the experimental data for this reaction by Andrea Koziol and Bob Newton yields
    • P(bar) = 22.80 T(K) - 7317
    for rV = -6.608 J/bar. Again, if we use
    • ln K = -(rH / RT) - (PrV / RT) + (rS / R)
    and set ln K = 0 to calculate values at equilibrium, we can rewrite the above as
    • (PrV / R) = -(rH / R) + (TrS / R)
    • P = TrS / rV - rH / rV
    if TrS / rV = 22.8 then rS = -150.66 J/mol K
    if rH / rV = 7317 then rH = -48.357 kJ/mol
    So, we can write the whole shmear as

    • 0 = -48,357 + 150.66 T(K) -6.608 P (bar) + RT ln K
    Contours of ln K on a PT diagram for GASP look like this:

    A Complete Example

    Let's suppose you analyze a group of minerals that are in equilibrium and find the following compositions:
    • garnet: Ca0.42Mg0.51Fe2.04Mn0.03Al2Si3O12
    • biotite: KMg1.62Fe1.38AlSi3O10(OH)2
    • plagioclase:Na0.64Ca0.36Al1.36Si2.64O8
    • kyanite
    • quartz
    Let's determine the equilibrium P and T.
    1. Determine mole fractions
      • Xgrs = 0.14
      • Xprp = 0.17
      • Xalm = 0.68
      • Xann = 0.46
      • Xphl = 0.54
      • Xan = 0.36
    2. Determine activities, assuming ideal behavior
      • agrs = 0.143 = 0.0027
      • aprp = 0.173 = 0.0049
      • aalm = 0.683 = 0.31
      • aann = 0.463 = 0.097
      • aphl = 0.543 = 0.16
      • aan = 0.36
    3. Calculate equilibrium constants:
      • KGASP = agrs / aan3 = 0.0025 / 0.363 = 0.054
      • KGARB = (aprpaann) / (aalmaphl) = (0.0049 * 0.097) / (0.31 * 0.16) = 0.0096
    4. Calculate P and T:
      • GASP: P (bar) = ( -48,357 + 150.66 * T(K) + RT ln K ) / 6.608
      • GARB: P (bar) = ( 52,110 - 19.51 * T(K) + RT ln K ) / -0.238

        Choose T = 873 K:

      • GASP P = 1.7 kbar
      • GARB P = -7.7 kbar

        Choose T = 1073 K:

      • GASP P = 13.2 kbar
      • GARB P = 40.6 kbar

        These two lines intersect at ~955K and 10.9 kbar.

    Part 1. Overview of Metamorphism and Tectonics
    Part 2. Introduction to Metamorphism
    Part 3. Physical Processes of Metamorphism
    Part 4. Introductory Phase Equilibria and Thermodynamics
    Part 5. Ultramafic Rocks
    Part 6. Mafic Rocks
    Part 7. Pelitic Rocks
    Part 8. Diffusion
    Part 9. Thermobarometry
    Part 10. Kinetics
    Part 11. Interaction Between Metamorphism and Deformation
    Part 12. Metamorphism and Geochronology
    Part 13. Metamorphism and Tectonics I
    Part 14. Metamorphism and Tectonics II
    Thermodynamics Notes