Phase Equilibrium

Reporters:Cruz, JericoCaringal, AngeloCastillanes, JobertDion, CanlasCandidato, VinceTañedo, Cassidy

ABSTRACT

The recent interest in the determination of the phase diagrams for many systems involving water as one component has been in those systems which are of greatest importance to clay mineralogy. The importance of both published and more recent work on synthesis of the clay is demonstrated in the ability to prepare chemically pure, mineralogically homogeneous kaolinites, montmorillonites, micas, and chlorites. The stability and identification of the clay minerals are discussed in relation to some of the major problems of clay mineralogy.

INTRODUCTIONStudies of phase equilibria are highly relevant to many areas of geosciences because in most cases, mineral systems have the time to reach thermodynamic equilibrium and lack the energy input needed to sustain disequlibrium. With increasing pressure, most crystal structures that are stable at low pressure (e.g. 1 bar) are replaced by denser structures. This has important implications for Earth's interior as well as for the synthesis of novel materials for industrial applications. A significant area of mineral physics research is therefore dedicated towards discovering new phases that may be present either in the deep earth or within other planets and characterizing their physical and chemical properties. Equilibrium relations between phases at high pressure are governed by the same thermodynamic principles that operate at low pressure. 

A given mineral crystallizes as a stable phase only within the restricted ranges of pressure and temperature. Subjecting the mineral to conditions that fall outside its stability will cause another mineral that is table under those conditions to crystallize in its place.

Testing mineral stability: laboratory experiment looking at the T, P, water vapour pressure,etc)

Note: Phase diagrams

I. Introduction

Concepts that are particularly important to understand are: Gibbs Free energy, enthalpy, entropy, chemical activity, Clausius Clapeyron equation, endothermic and exothermic reactions and the phase rule.

II. Description of Terms:

1) Phase – part or parts of a system occupying a specific volume and having a uniform physical and chemical characteristics which distinguish it from all other parts of the system.2) System – a collection of geological phasesa) Open system – one that is free to exchange both matter and energy with its surroundingsb) Closed system - one that is sealed with respect to the transfer of matter, but that can still exchange energy with the surroundings.c) Isolated system - one that is incapable of exchanging both mass and energy with its surroundings.

II. Description of Terms:3) Component – basic chemical constituents of a system, of which the various phases are composed; comprises the minimum number of chemical (atomic and molecular) species required to specify completely the compositions of all the phases present.

III. EquilibirumA. Thermal equilibirum

All parts of the system have the same temperature; there is no net transfer of heat.

B. Chemical equilibriumDistribution of components among the phases of a system has become constant, showing no net change with time; the flux of atoms across the crystal boundary is the same in both directions.

Note: disequilibrium

IV. The Gibbs Phase Rule

Q: How many phases can be in equilibrium with each other at any one time?

(William Gibbs-1870- pioneer of modern thermodynamics)

Phase Rule- expresses the number of phases that can coexist in mutual equilibrium (Ф) in terms of the number of components (C) in the system and variance (F).

Variance = number of degrees of freedom

Ф + F = C + 2

Degrees of freedom of a system is the number of variables (i.e., temperature, pressure and concentration of the components, which must be arbitrarily fixed in order that the condition of the system may be completely defined.

Phase CharacteristicA phase does not have to be BOTH physically and chemically distinct

Example:1.Water and ice2.Oil and water

Equilibrium Assemblages• At equilibrium, the mineralogy (and the composition of each mineral) is determined by T, P, and X • “Mineral paragenesis” refers to such an equilibrium mineral assemblage • Relict minerals or later alteration products are thereby excluded from consideration unless specifically stated

Driving Forces and Gibbs Free Energy

EnthalpyEnthalpy is a measure of the total energy of a thermodynamic system. It includes theinternal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing itsenvironment and establishing its volume and pressure.Enthalpy is athermodynamic potential. It is a state function and anextensive quantity. The unit of measurement in theInternational System of Units (SI) for enthalpy is the joule, but other historical, conventional units are still in use, such as the small and the large calorie.

ENTROPY

a measure of the unavailable energy in a closed thermodynamic system that is also usually considered to be a measure of the system's disorder, that is a property of the system's state, and that varies directly with any reversible change in heat in the system and inversely with the temperature of the system;

CHEMICAL ACTIVITYThe active mass or chemical activity of any chemical system is simply the chemical activity of its components, which are the reactants or the products of a chemical reaction. The rate of the chemical reaction CaCO3 --> CaO + CO2 is then proportional to the activity of CaCO3, while the rate of the reverse reaction, CaO + CO2 --> CaCO3 is proportional both to the activity of CaO and to the activity of CO2. Likewise, the driving force of the reaction of CaCO3 is directly proportional to the activity of CaCO3 while the driving force of the reverse reaction is directly proportional both to the activity of CaO and to the activity of CO2. 

Clausius Clapeyron equation

The relationship between the temperature of a liquid and its vapor pressure is not a straight line. The vapor pressure of water, for example, increases significantly more rapidly than the temperature of the system. This behavior can be explained with the Clausius-Clapeyron equation.

The Clausius-Clapeyron Relation

Along the phase boundary:P

T

P

T

phaseboundary

Since S1 - S2 = L/T (L is the latent heat), we arrive at the Clausius-Clapeyron Relation :

For the slope of the boundary we have:

TPGTPG ,, 21

TPGTPG ,, 21

TPVTPV

TPSTPSdTdP

,,,,

21

21

TVTTL

dTdP

dPVdTSdPVdTS 2211

ENDOTHERMIC AND EXOTHERMIC REACTIONSexothermic: 1. release heat, therefore the room temperature will increase 2. ΔH = negative (kJ mol^-1) 3. bond will be formed, and energy will be release in form of heat, the molecules that are formed had a lower kinetic energy but very stabil 

endothermic: 1. absorb heat, therefore the room temperature will decrease 2. ΔH = positive (kJ mol^-1) 3. bonds will broke down, because the energy of reactants are less than the energy of products in the chemical reaction

PHASE RULE

The Phase Rule describes the possible number of degrees of freedom in a (closed) system at equilibrium, in terms of the number of separate phases and the number of chemical constituents in the system. It was deduced from thermodynamic principles by J. W. Gibbs in the 1870s.

PHASE RULE

Phase diagrams in P-T space (Reaction or equilibrium between compounds)

NaAlSi2O6 + SiO2 = NaAlSi3O8Jadeite (px) silica pyroxene

Note: reaction or equilibrium boundary

Gibbs’ Phase Rule

The Phase Rule- expresses the number of phases that can coexist in mutual equilibrium

F = C - P+ 2 F – is the degree of freedomP number of phasesC – minimum number of components.Note: the 2 represent 2 intensive parameters particularly temperature and pressure.

High Pressure Phase Equilibrium Studies

There are a number of different types of high pressure studies related to phase equilibria. There are true phase equilibrium studies that seek to characterize the range of pressure, temperature or chemical activity (e.g. for O2) over which a given phase (or phase assemblage) is stable.Synthesis studies seek to discover and produce new phases. However, they are distinct from phase equilibrium studies. There are a number of circumstances that can lead to minerals forming outside their thermodynamic stability field. Another major area of interest is phase transformation mechanisms and kinetics. 

Reversals: the difference between synthesis and phase equilibrium studies

One of the most important features of a good phase boundary determination is the reversal. A reversal is a pair of experiments where the reactants and products in one experiment become the products and reactants in another experiment.

Reversals: the difference between synthesis and phase equilibrium studies

The position of the phase boundary can be said to be somewhere between these bracketing experiments. In-situ experiments (e.g. synchrotron x-ray diffraction) are particularly efficient for conducting phase equilibrium studies because the transformation from one phase assemblage to the other can be run first one way and then back the other at a number of pressure and temperatures during a single experiment.

Since phase boundaries are lines (or curves if volatiles are present) at minimum two reversal are required to fully describe a phase boundary. Reversals are necessary for two reasons:

1)Due to kinetics, phases can persist in some cases indefinitely outside of their stability field. Merely witnessing a phase at a given P and T, especially at lower temperatures, does not guarantee that it is the stable phase.

2) As mentioned above, phases can be synthesized outside of their stability field. Some of the reasons that this can occur are related to a phenomena referred to as Ostwald's step rule – which says that the first phase to from in a reaction is not always the one with the lowest free energy but the phase with a free energy closest to the free energy of the reactants. In order to start with a sample that is highly homogeneous, experimentalists may choose a gel or glass; the free energy of these reactants can be quite large. 

Reactants ground in a mortar and pestle can be highly strained (e.g. Kingma et al., 1993) which can also substantially raise their free energy. So for example, coesite can be produced from highly strained quartz at P and T conditions in the quartz stability field (Green, 1972). For these reasons if one wants to measure the position of a phase boundary or confirm that a given phase in fact has a stability field, reversals are required.

Discovery and Synthesis of New Phases

The simplest way to discover a new phase is to put reactants of interest together and bring them to some new pressure temperature condition and either observe them with x-rays at temperature and pressure or quench them and use x-ray diffraction to identify the phases present and search for new phases. However, there are more sophisticated strategies for searching P,T chemistry spaces for new phases.

Discovery and Synthesis of New PhasesHowever, there are more sophisticated strategies for searching P,T chemistry spaces for new phases. One strategy that has been around for a long time is to use one's knowledge of phase relations in one chemical system to make intelligent guesses about phase relations in another similar chemical system. For example, the observation that Mg2GeO4 formed in both an olivine and a denser spinel structured polymorph, led researchers in the 1930's to suggest that (Mg, Fe)2SiO4 olivine might also have a spinel structured polymorph which could account for the increase in seismic velocity at ~400 km depth. Many compounds that share the same stoichiometry have phase diagrams that share the same topology. 

Phase Equilibria Relevant to the Deep EarthPhase diagrams illustrate the stability of phases in Pressure,Temperature, chemical activity space. Many igneous and metamorphic systems contain up to 11 significant chemical components. This means that a full depiction of phase equilibria in these systems requires 13 dimension. Petrologists have devised a host of sections, pseudo sections and projections to create illustrative 2D diagrams from these multi-dimensional spaces. Mineral physics has mostly confined itself to working in relatively simple chemical systems (e.g. MgO-SiO2) that serve as a proxy for the chemically more complex Earth. Another strategy is to look at changes with pressure and temperature in a single bulk composition (e.g. MORB or pyrolite). A selection of relevant phase diagrams is given below.

Phases: Kyanite, Sillimanite and AndalusiteComponent: Al2SiO5

Pressure and Temperature Phase Diagram (One component)

Analyze at point X, Y and Z.

Analyze point F

Binary Diagrams

Composition is now variable

Condensed phase rule

P + F = C + 1Note: pressure is no longer a variable: only Temperature and composition matter.

Crystallization in systems with no solid solution •no compounds/solid

solutions in solid state•only single phase liquid at high temperatures•partial melting at intermediate temperatures

Liquidus Curve- specifies the maximum temperature at which crystals can co-exist with the melt in thermodynamic equilibrium

Tie Line- links together the compositions of two phases which can coexist stably

Analyze point x1 and E

Liquidus Curve

System with a complete Solid SolutionExample would be plagioclase

m, cools down, encounters liquidus at a where plag b will begin to crystallize

Because b is more calcic, it will deplete the melt with anorthite and thereby enrich it with albite

The changing melt composition causes a corresponding evolution in the equilibrium composition of the plagioclase crystals.

LEVER RULE

The lever rule is a tool used to determine weight percentages of each phase of a binary equilibrium phase diagram. It is used to determine the percent weight of liquid and solid phases for a given binary composition and temperature that is between the liquidus and solidus.

Consider a cooling alloy at the composition and temperature marked on the diagram. As shown on the phase diagram, the alloy is, at the given temperature, a mixture of alpha and liquid phases - but what are their exact compositions at this temperature?

An isothermal (constant temperature) line through the alloy's position on the phase diagram when it is in a two phase field, intersecting the two adjacent solubility curves, is called a tie line (yes, that's the horizontal yellow line on the diagram).The ends of the tie lines show the compositions of the two phases that exist in equilibrium with each other at this temperature. From the diagram we know that alpha and liquid phases will exist. The tie line shows that the alpha phase is 5.2%B and the liquid phase is 34.5%B at this temperature.

Remember, though, that the overall composition of the sample is unchanged - we are only discovering the compositions of the constituent phases within the sample.

For a cooling alloy at composition Co and temperature Tx , tie lines may be used to answer questions such as:what phases are present ?•what are their compositions ?•if the temperature is reduced to Ty, how do the compositions of the two phases vary ?

The answer to "what phases are present ?" is easy. Composition Co and temperature Tx meet in the beta + liquid phase field, so these are the two phases present.

•To answer "what are their compositions ?" we must draw a horizontal tie line from the point to the nearest phase diagram boundaries. The tie line shows us that the compositions are:

•Liquid: X wt% B•Beta: Y wt% B

To answer the last question "if the temperature is reduced to Ty, how do the compositions of the two phases vary?"consider the new tie-line, shown in yellow on the diagram.The compositions of liquid and beta phase have both decreased in wt%B to:

Liquid: X' wt% B Beta: Y' wt% B

Thus, both the liquid and the beta phases are getting richer in A as the sample is cooled.

Now that we know the compositions of the two phases, we need to find how much of each phase exists at the given temperature. The ratio of the two phases present can be found by using the lever rule.At first sight the lever rule can appear confusing. It is really invoking the conservation of mass, and can be proved mathematically, as shown below the diagram.Essentially, we start off with an overall composition of our alloy - Co. From the tie-line we know that the two phases at a given temperature have two different compositions, but overall the amounts of these two compositions must add up to the alloy's overall composition, Co.This is the basis for the lever rule. Using the lever rule itself is very simple, we'll show you with a diagram.....

Basically, the proportions of the phases present are given by the relative lengths of the tie line. So, the proportions of alpha and liquid present on the diagram (showing a portion of the whole phase diagram) are: 

X Y X+Y and X+Y

Simple, isn't it ?But... which equation corresponds to which phase ?

Now, consider the same alloy as it crosses the liquidus line. It seems reasonable to assume that, at this point, the alloy will be nearly all liquid. Looking at the diagram it can be seen that Y1 is very small here and so must be the proportion of alpha present. Similarly X1 is relatively large and so it corresponds to the amount of liquid.

So, the left side of the tie line gives the proportion of the liquid phase (the phase to the right), and the right side of the tie line gives the proportion of the alpha phase (the phase on the left).

Remember: you use the length of the line which is furthest from the phase in which you are interested.

Distances along the tie line can be found very simply by using a ruler on an accurate phase diagram or, more correctly, by using data from the composition axis (the x-axis).For example, on the diagram shown, the percentage of alpha present can be calculated from the three pieces of composition data given:

Fraction of alpha = (34.5 - 23.7) / (34.5 - 5.2) = 0.3686

Thus, percentage of alpha = 0.3686 x 100 = 36.86%

and, as the alpha and the liquid make up 100% of the alloy's composition:

Percentage of liquid = 100 - 36.86 = 63.14%

Magmatic Processes

Some useful terms:Primary magma - magma originating in (mantle) source directly from melting.Primitive magma - magma that undergo minimal differentiation.Parental magma - least differentiated magma in a series leading to evolved rocks.

Magmatic Differentiation:the process whereby, magma originally homogeneous splits up into contrasted parts, which may form separate bodies of rocks or may remain within the boundaries of single unitary mass.

It is usually favored by two factors:a.Rate of cooling.b.Settling of early crystallized heavy minerals.

Fractional CrystallizationSeparation of crystals from liquidGravitative settling or flotation play a significant role

ASSIMILATIONIt is the method of creating different daughter magmas from a parent is by having the latter react with its wall rocks assimilation is accompanied by crystallization, it is likely that both fractional crystallization and assimilation will take place simultaneously.

Combined Process (AFC = Assimilation + Fractional Crystallization)

HOT (1200°C) basaltic magma

Blocks of continental crust fall into basaltic magma and dissolve.

basaltic magma + assimilated blocks = andesitic magma

Heat transfer from hot, basaltic magma melts wall rock

magma composition progressively changes as crystal are physically “removed” from the magma

Incongruent Melting● Incongruent melting occurs when a substance does not melt

uniformly and decomposes into another substance. For example, potassium feldspar (KAlSi3O8) decomposes to leucite (KAlSi2O6) when it melts. The decomposition is not complete, however. Most of the feldspar does melt, a portion of it decomposes to leucite and some quartz (SiO2) is left over, since the chemical formulas of potassium feldspar and leucite differ by SiO2. Another mineral that melts incongruently is enstatite (MgSiO3), which decomposes to forsterite (Mg2SiO4). Enstatite does melt congruently between pressures of 2.5 and 5.5 kilobars.

Congruent Melting● Congruent melting occurs during melting of a

compound when the composition of the liquid that forms is the same as the composition of the solid. It can be contrasted with incongruent melting. This generally happens in two- component systems.

Congruent melting VS Incongruent melting

● In congruent melting of rocks, the minerals simply melt and join the liquid. In incongruent melting, a new solid minerals precipitates from the melt.

● For example of a gneiss containing quartz, biotite and sillimanite is melted, then the result is a liquid plus the new solid mineral cordierite (magnesium iron aluminum cyclosilicate)