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Thermodynamics
Dexter Perkins (University of North Dakota), Andrea Koziol (University of Dayton), John Brady (Smith College)
Basic Definition of Thermodynamics
The Oxford English Dictionary says: "Thermodynamics: the theory of the relations between heat and mechanical energy, and of the conversion of either into the other." In simpler terms, we can think of thermodynamics as the science that tells us which minerals or mineral assemblages will be stable under different conditions.
In practical terms, thermodynamics not only allows us to predict what minerals will form at different conditions ( forward modeling ), but also allows us to use mineral assemblages and mineral compositions to determine the conditions at which a rock formed ( thermobarometry ). The calculations are often complex and are best carried out using thermodynamic modeling programs or programs specifically designed for thermobarometry.
The Basis for Thermodynamic
Calculations
All phases, whether mineralogical or not, have an associated Gibbs Free Energy of Formation value abbreviated ∆G (^) f. The ∆G (^) f value describes the amount of energy that is released or consumed when a phase is created from other phases.
We can calculate the Gibbs Free energy of any reaction (∆G (^) rxn ) by summing the energies of the right-hand side of the reaction and subtracting the energies of the left hand side. If the calculation reveals that ∆G (^) rxn < 0, the reaction proceeds to the right. If the ∆G (^) rxn
0, the reaction proceeds to the left.
Consider the reaction albite = jadeite + quartz (Figure 1). Under normal earth surface conditions, the Gibbs Energy of this reaction is greater than zero. Consequently albite is stable and the assemblage jadeite + quartz is unstable.
The ∆G (^) f of a mineral varies with changes in pressure (P), temperature (T) and mineral composition (X). Consequently, the ∆G (^) rxn for any reaction will vary with P, T and X,
Figure 1. Phase diagram showing the reaction albite = jadeite + quartz. Along the reaction line, the Gibbs Energy of Reaction , ∆Grxn = 0. Above the line ∆Grxn < 0; below the line ∆Grxn > 0.
being positive in some portions of P-T-X space and negative in others. The result is that we can plot reactions on phase diagrams, such as the one in Figure 1. Along any reaction line, such as the one separating the albite and jadeite + quartz fields shown in Figure 1 ∆G (^) rxn = 0.
Doing thermodynamic calculations requires reliable thermodynamic data. Additionally, although the calculations can be done by hand or with a calculator, they are complicated and time consuming.
Gibbs Free Energies of a Mineral
Gibbs Free Energy values have been tabulated for many phases. Gibbs Energy values are, most often today, given in units of joules/mole or (less commonly) calories/mole.
Consider, for example, enstatite (MgSiO 3 ). The Gibbs Free Energy of Formation for enstatite from pure elements (Mg, Si and O) = ∆G (^) f (enstatite, elements) is about -1,460.9 KJ/mole at room temperature and pressure. The Gibbs Free Energy of Formation for enstatite from oxides (MgO and SiO 2 ) = ∆G (^) f(enstatite, oxides) is about -35.4 KJ/mole at room temperature and pressure.
The ∆G (^) f values given above for enstatite are both negative. This means that enstatite is more stable than, and will form from, the separate elements or separate oxides at room temperature and pressure (although reaction rate is extremely slow). Some of the energy produced will be given off as heat; some will contribute to entropy.
Gibbs Free Energies are Relative Values
Gibbs free energies are relative values, not absolute values. They allow us to compare energies of different phases but individual values by themselves have no significance. Because Gibbs Energy values are relative, we can arbitrarily assume some values in order to calculate others. So, by convention the ∆G (^) f for any pure element is assumed to be as zero.
The Gibbs Free Energy of Reaction
We can calculate the Gibbs Free energy of any reaction by summing the energies of the right-hand side of the reaction and subtracting the energies of the left hand side. For example, we can write a reaction describing the formation of enstatite from separate elements:
Mg + Si + 3O = MgSiO 3 (rxn 1)
Phases and Reactions
The Gibbs Free Energy of any phase varies with pressure and temperature. The fundamental relationship is:
G = E + PV - TS (Eqn 2)
or
G = H - TS (Eqn 3)
In the above expressions, P and T refer to pressure and temperature. E, V, H and S refer to the internal energy, volume, enthalpy and entropy of the phase. It follows that:
H = E + PV (eqn 4)
Similarly, for any reaction, the Gibbs Free Energy of reaction varies with pressure and temperature:
∆G (^) rxn = ∆Erxn + P∆Vrxn -T∆Srxn (Eqn 5)
or
∆G (^) rxn = ∆Hrxn -T∆Srxn (Eqn 6)
As above, P and T refer to pressure and temperature ∆G (^) rxn is the Gibbs energy of reaction, ∆Erxn is the internal energy of reaction, ∆Vrxn is the volume of reaction, ∆Hrxn is the enthalpy of reaction and ∆Srxn is the entropy or reaction.
Look closely at Equation (5). The right hand side contains three terms. The first is the change in internal energy -- a constant depending on the phases involved. The second is a PV term -- it equates Gibbs Energy with volume and pressure. More voluminous phases have greater Gibbs Free Energy. (Recall that the energy of an ideal gas = PV = nRT.) The third term involves entropy (S). Entropy is a measure of disorder. Some phases can absorb energy simply by becoming disordered. Temperature may not increase, volume may remain the same, but energy disappears.
Chemical systems seek to minimize energy and, consequently, phases of greater Gibbs Free Energy are unstable with regard to phases with lower Gibbs Free Energy. So, at high temperature, phases with high entropy are very stable. This is because the TS term in Equation (5) has a negative sign. Similarly, at high pressure, phases with high volume are unstable. The PV term has a positive sign. (Although your intuition may not work well when considering entropy, it should seem reasonable that low volume, very dense, phases are more stable at high pressure than phases of less density.)
Intensive and Extensive Variables, and Units
P and T are termed intensive variables. G, E, H, V and S are extensive variables. The difference is that intensive variables (P and T) do NOT depend on the size of the system or the amount of material present. G, E, H, V and S do depend on system size (e.g., the larger the system, the larger the volume).
Pressure is typically given in units of pascals (Pa), Gigapascals (GPa), bars (bar) or Kilobars (Kbar). G, E, and H are typically given in units of J/mole. V is in cm^3 /mole, and entropy in J/deg-mole. (Calories may be substituted for Joules, 1 cal = 4.186 J). Note: The PV term in the above expressions is not in the same units as the other terms. A necessary conversion factor is 1 J = 10 cc-bar.
What is the significance of the different thermodynamic variables?
The Gibbs Free Energy (∆G (^) rxn ) tells us whether a reaction will take place. ∆G (^) rxn is the Gibbs Free Energy of the right hand side of a reaction, minus the Gibbs Free Energy of the left hand side. If ∆G (^) rxn < 0, the reaction will proceed to the right; if it is > 0, the reaction will proceed to the left.
The Enthalpy of Reaction (∆Hrxn ) tells us how much heat will flow in or out of the system. If ∆Hrxn < 0, the reaction is exothermic -- it releases heat. For example, combustion of carbon based compounds (C + O 2 = CO 2 ) gives off a lot of heat. If ∆Hrxn > 0, the reaction is endothermic -- it consumes heat. Melting ice [H 2 O(ice) = H 2 O (water)] is endothermic and, consequently, cools our gin and tonics in the summer.
The entropy of a reaction (∆Srxn ) tells us whether the products or reactants are more disordered. For example the reaction of liquid water to steam (boiling) has a large associated entropy. The steam molecules are more dispersed, are less well bonded together, and have greater kinetic energy.
The volume of a reactions (∆Vrxn ) tells us whether the products or the reactants have greater volume. The reaction of graphite to diamond, both made entirely of carbon, proceeds at high pressure because diamond is more dense (has smaller volume) than graphite.
Internally Consistent Thermodynamic Data Bases
Thermodynamic data are obtained by calorimetry (enthalpy and entropy values), X-ray diffraction (volumes) or derived on the basis of experimental studies, each with associated uncertainties. Combining thermodynamic values from different sources
Perplex is a thermodynamic calculation package suitable for rapidly creating phase diagrams of all types, creating pseudosections (phase diagrams that include only those reactions experienced by a particular bulk composition). Find out more about Perplex, and how to get a copy, at: http://serc.carleton.edu/dev/research_education/equilibria/Perplex.html
Phase Diagrams: The Results of
Thermodynamic Calculations
Phase diagrams are graphical representations of the equilibrium relationships between minerals (or others phases). These relationships are governed by the laws of thermodynamics. Standard phase diagrams show how phases or phase assemblages change as a function of temperature, pressure, phase composition, or combinations of these variables.
Some phase diagrams (those for 1-component diagrams) depict relationships involving multiple phases having the same composition (for example, the relationships between the vapor, liquid and solid forms of H 2 O). Other diagrams (such as the one shown on the right), depict the relationships between a number of compounds having different compositions. Still others show how compositions of phases change under different conditions.
Thermobarometry
Thermobarometry refers to the quantitative determination of the temperature and pressure at which a metamorphic or igneous rock reached chemical equilibrium. Many programs exist to facilitate such calculations, but most are only applicable to specific kinds of rocks and mineral assemblages.
The Math behind the Calculations
The basis for calculating a reaction curve (a curve/line on a phase diagram) is the understanding that the Gibbs Free Energy of a reaction (?Grxn) can be calculated for any P-T-X conditions, provided a starting point and the requisite thermodynamic data
Figure 3. Phase diagram showing reactions in the system Al 2 O 3 -SiO 2 -H 2 O
are available. The fundamental relationship is:
∆G, ∆V, and ∆S refer to the Gibbs Energy, volume and entropy of reaction. The degree superscript reminds us that these values are for pure phases. The P and T subscripts remind us that the values change with pressure and temperature.
The starting Gibbs Free Energy value, ∆G (^) P1,T1 , may be directly obtained from calorimetric studies. Alternatively, and more commonly, experimental studies may provide a reference point at which a particular reaction is in equilibrium (∆G (^) P,T = 0). In either case, the equation above is solved to find the location of the reaction curve, points at which ∆G= 0. The result is a curve (often nearly a straight line) in P-T-X space that separates fields of stability for different minerals or mineral assemblages.
For reactions involving pure, end member minerals, the above equation suffices. However, many minerals are solid solutions, so the effective activity (reactivity) of mineral components is diluted. The basic thermodynamic equation is modified to take this into account by adding a term involving the equilibrium constant (K):
The equilibrium constant for a reaction (K) is the product of the activities of the reaction products, divided by the product of the activities of the reactants:
Putting It All Together: An Example
For an example, let's combine the above equations, and consider the reaction:
3 anorthite = grossular + 2 kyanite + quartz
For this reaction, the conditions for thermodynamic equilibrium are:
The Slope of a Reaction Curve
Sometimes, petrologists have a starting point, perhaps from experimental studies, and only need to know the slope of a reaction in order to plot it on a phase diagram. Differentiation of the equations above reveals that the slope may be calculated:
dP/dT = ∆SP,T/∆VP,T
This relationship is called the Clausius-Clapeyron equation.
Books on Fundamental Thermodynamics
Cemic, L. (2005) Thermodynamics in Mineral Sciences: An Introduction. Springer. 386 p.
Fraser, D. (1977) Thermodynamics in Geology. NATO Science Series: C. Kluwer Academic Publishers. 424 p.
Nordstrom, D.K. (2006) Geochemical Thermodynamics. Blackburn Press. 504 p.
Kern, R. and Weisbrod, A. (1967) Thermodynamics for Geologists. Freeman, Cooper and Co. 3-4 p.
Spear, F.S. (1994) Metamorphic Phase Equilibria and Pressure-Temperature-Time Paths. Monograph of the Mineralogical Society of America. 799 p.
Wood, B.J. (1977) Elementary Thermodynamics for Geologists. Oxford University Press. 318 p.
TWQ References
Berman, R.G. (1988) Internally-consistent thermodynamic data for minerals in the system Na 2 O-K 2 O-CaO-MgO-FeO-Fe 2 O 3 -Al 2 O 3 -SiO 2 -TiO 2 -H 2 O-CO 2. Journal of Petrology, 29, 445-522.
Berman, R.G. (1991) Thermobarometry using multi-equilibrium calculations: a new technique, with petrological applications; in, Quantitative methods in petrology: an issue in honor of Hugh J. Greenwood; Eds. Gordon, T M; Martin, R F. Canadian Mineralogist v. 29, 833-855.
Berman, R.G. (2007) winTWQ (version 2.3): a software package for performing
internally-consistent thermobarometric calculations. Geological Survey of Canada, Open File 5462, (ed. 2.32) 2007, 41 pages.
Thermocalc References
Holland, TJB, & Powell, R, 1998. An internally-consistent thermodynamic dataset for phases of petrological interest. Journal of Metamorphic Geology 16, 309–344.
Holland, TJB, & Powell, R, 2003. Activity–composition relations for phases in petrological calculations: an asymmetric multicomponent formulation. Contributions to Mineralogy and Petrology 145, 492-501.
Powell, R, 1978. Equilibrium Thermodynamics in Petrology Harper and Row, 284 pp.
Powell, R, Guiraud, M, & White, RW, 2005. Truth and beauty in metamorphic mineral equilibria: conjugate variables and phase diagrams. Canadian Mineralogist, 43, 21–33.
Powell, R, & Holland, TJB, 1988 An internally consistent thermodynamic dataset with uncertainties and correlations: 3: application methods, worked examples and a computer program. Journal of Metamorphic Geology 6, 173-204.
Powell, R, & Holland, TJB, 1994. Optimal geothermometry and geobarometry. American Mineralogist 79, 120-133.
Powell, R, Holland, TJB, & Worley, B, 1998. Calculating phase diagrams involving solid solutions via non-linear equations, with examples using THERMOCALC Journal of Metamorphic Geology 16, 577–588.
Worley, B, & Powell, R, 1999. High–precision relative thermobarometry: theory and a worked example Journal of Metamorphic Geology 18, 91–102.
Melts References
Asimow PD, Ghiorso MS (1998) Algorithmic Modifications Extending MELTS to Calculate Subsolidus Phase Relations. American Mineralogist 83, 1127-
Ghiorso, Mark S., 1997, Thermodynamic models of igneous processes. Annual Reviews of Earth and Planetary Sciences, v. 25 p. 221-241.
Ghiorso, Mark S., Hirschmann, Marc M., Reiners, Peter W., and Kress, Victor C. III (2002) The pMELTS: An revision of MELTS aimed at improving calculation of phase relations and major element partitioning involved in partial melting of the mantle at
Connolly JAD, Kerrick DM (1987) An algorithm and computer program for calculating composition phase diagrams. CALPHAD 11:1-55.
