G EOL 2312 I GNEOUS AND M ETAMORPHIC P ETROLOGY Lecture 4 Introduction to Thermodynamics Jan. 27, 2016

T HERMODYNAMICS IS THE STUDY OF THE RELATIONSHIPS BETWEEN HEAT, WORK, AND ENERGY SYSTEM- Some portion of the universe that we wish to study SURROUNDINGS - The adjacent part of the universe outside the system Changes in a system are associated with the transfer of energy from one form to another Energy of a system can be lost or gained from its surroundings, but collectively energy is conserved. Types of Energy include: PotentialKineticChemicalMechanical Thermal Gravitational

S TATES OF E NERGY N ATURAL SYSTEMS TEND TOWARD STATES OF MINIMUM ENERGY Stable – at minimum energy state Unstable – energy state in flux (disequilibrium) Metastable – temporary energy state that is not lowest, but requires energy to push it to lower energy state GOOD THING FOR GEOLOGY! Winter (2001), fig. 5-1

G IBBS F REE E NERGY MEASURE OF THE ENERGY CONTENT OF A CHEMICAL SYSTEM All chemical systems tend naturally toward states of minimum Gibbs free energy (G) G = H - TS Where: G = Gibbs Free Energy H = Enthalpy (heat content) T = Temperature in Kelvins (= o C + 273) S = Entropy (randomness) Basically, Gibbs free energy parameter allows us to predict the equilibrium phases of a chemical system under particular conditions of pressure (P), temperature (T), and composition (X)

E QUILIBRIUM OF A C HEMICAL R EACTION Phase - a mechanically separable portion of a system (e.g., Mineral, Liquid, Vapor) Reaction - some change in the nature or types of phases in a system. Written in the form: Reactants Products e.g. 2A + B + C = 3D + 2E To know whether the products or reactants will be favored (under particular conditions of T, P, and X, we need to know the Gibbs free energy of the product phases and the reaction phases at those conditions  G =  (n  G) products -  (n  G) reactants = 3G D + 2G E - 2G A - G B - G C If  G is positive, the reactants are favored; if negative, the products are more stable

G IBBS F REE E NERGY OF A P HASE AT ITS R EFERENCE S TATE It is not possible to measure the absolute chemical energy of a phase. We can measure changes in the energy state of a phase as conditions (T,P,X) change. Therefore, we must define a reference state against which we compare other states. The most common reference state is to consider the stable form of pure elements at “room conditions” (T=25 o C (298 o K) and P = 1 atm (0.1 MPa)) as having G=0 joules. Because G and H are extensive variables (i.e. dependent on the volume of material present), we express the G of any phase as based on a quantity of 1 mole (called the molar Gibbs free energy).

M OLAR G IBBS F REE E NERGY OF F ORMATION With a calorimeter, we can then determine the enthalpy (H- heat content) for the reaction: Si (metal) + O 2 (gas) = SiO 2  H = -910,648 J/mol Since the Enthalpy of Si metal and O 2 is 0 at the reference state, the value for  H of this reaction measures is the molar enthalpy of formation of quartz at 298 K, 0.1MPa. Entropy (S) has a more universal reference state: entropy of every substance = 0 at 0 o K, so we use that (and adjust for temperature) Then we can use G = H - TS to determine molar Gibbs free energy of formation of quartz at it reference state  G o f = -856,288 J/mol

D ETERMINING THE G OF A P HASE AT ANOTHER T EMPERATURE AND P RESSURE The differential equation for this is: dG = VdP – SdT Assuming V and S do not change much in a solid over the T and P of interest, this can be reduced to an algebraic expression: G T2 P2 - G T1 P1 = V(P 2 - P 1 ) - S (T 2 - T 1 ) and G 298, 0.1 = -856,288 J/mol to calculate G for quartz at several temperatures and pressures Low quartzEq. 1SUPCRT P (MPa)T (C)G (J) eq. 1G(J)V (cm3)S (J/K) , , , , , , , , V/dP – isothermal compressibility S = Cp/T, Cp – heat capacity (heat required to raise 1 mole of substance 1°C)

G IBBS F REE E NERGY FOR A R EACTION S OLID L IQUID Here, X is constant (one comp) so we just have to consider affects of T and P on G dG = VdP – SdT We can portray the equilibrium states of this reaction with a phase diagram What does this say about the  G of the reaction at Points A, X, and B? High temperature favors randomness, so which phase should be stable at higher T? High pressure favors low volume, so which phase should be stable at high P? Let’s look at the effects of P and T on G individually

T EMPERATURE E FFECT ON F REE E NERGY dG = VdP - SdT at constant pressure: dG/dT = -S Because S must be (+) G for a phase decreases as T increases Would the slope for the liquid be steeper or shallower than that for the solid?

T EMPERATURE E FFECT ON F REE E NERGY Slope of G Liq > G sol since S solid < S liquid A: Solid more stable than liquid (low T) B: Liquid more stable than solid (high T) Slope  P/  T = -S Slope S < Slope L Equilibrium at T eq G Liq = G sol = crystallization/melting temperature

P RESSURE E FFECT ON F REE E NERGY dG = VdP - SdT at constant temperature: dG/dP = V Note that Slopes are + Why is slope greater for liquid?

P HASE DIAGRAM PORTRAYS THE L OWEST F REE E NERGY S URFACES P ROJECTED ON TO T-P SPACE From Philpotts (1990), Fig. 8-2

MELTS D ETERMINES P HASE E QUILIBRIUM BASED ON T HERMODYNAMIC M EASUREMENTS

PELE – M ELT FOR PC USE A. B OUDREAU 2002