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Gibbs' Phase Rule
Minerals are the monitors of the physical and chemical conditions under which they formed. The
occurrences of minerals, their parageneses (stable associations), types of reactions, and compositional
variation (e.g. zoned minerals) all provide important information about geologic history and processes.
Of particular importance to geologists are:
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Estimates of pressure and temperature (geothermobarometry)
Estimates of other physico-chemical conditions such as acidity (pH) and oxidation state (eH)
Partial pressures of gases (e.g. fugacities of H2O, CO2 , etc.)
Partitioning of major and trace elements between phases (e.g. minerals, melts and/or fluids) to
characterize and quantify petrogenetic processes; and
Use of minerals in geochronology and thermochronology
Gibbs' Phase Rule provides the theoretical foundation, based in thermodynamics, for characterizing the
chemical state of a (geologic) system, and predicting the equilibrium relations of the phases (minerals,
melts, liquids, vapors) present as a function of physical conditions such as pressure and temperature.
Gibbs' Phase Rule also allows us to construct phase diagrams to represent and interpret phase equilibria
in heterogeneous geologic systems. In the simplest understanding of phase diagrams, stable phase
(mineral) assemblages are represented as "fields" (see colored areas on the figure to the right) in "P-T
space", and the boundaries between stable phase assemblages are defined by lines (or curves) that
represent reactions between the phase assemblages. The reaction curves actually represent the
condition (or the locus of points in P-T space) where ΔGrxn =0; for more information on this point see
Gibbs Free Energy. A solid understanding of Gibbs' Phase Rule is required to successfully master the
applications of heterogeneous phase equilibria presented in this module.
NAME OF THE FACULTY: Shaik.Mehboob Subhani
SUBJECT: Engg. Chemistry
LORDS INSTITUTE OF ENGINEERING & TECHNOLOGY
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Definitions
Gibbs Phase Rule is expressed by the simple formulation:
P + F = C + 2, where
P is the number of phases in the system
A phase is any physically separable material in the system. Every unique mineral is a phase
(including polymorphs); igneous melts, liquids (aqueous solutions), and vapor are also
considered unique phases. It is possible to have two or more phases in the same state of matter
(e.g. solid mineral assemblages, immiscible silicate and sulfide melts, immiscible liquids such as
water and hydrocarbons, etc.) Phases may either be pure compounds or mixtures such as solid
or aqueous solutions--but they must "behave" as a coherent substance with fixed chemical and
physical properties.
C is the minimum number of chemical components required to constitute all the phases in the system
For historical reasons, geologists normally define components in terms of the simple oxides (e.g.
SiO2, Al2O3, CaO, etc.). If two possible components always occur in the same proportions in
multiple phases in a system, these can be combined into a single component (remember, we are
always trying to define the minimum number of components required to make all the phases in
the system). Consider the reaction:
CaMg(CO3)2 + 2 SiO2 = CaMgSi2O6 + 2 CO2
Dolomite + 2 Quartz = Diopside + 2 Carbon Dioxide
Normally we would pick the 4 components: CaO, MgO, SiO2, and CO2 based on the simple
oxides. However, because Ca and Mg are in a 1:1 ratio in both dolomite and diopside (and not
present in quartz or carbon dioxide), we can consider this a ternary system with components:
CaMgO2, SiO2, and CO2
In some geologic systems it is convenient to define the components in terms of end-member
compositions (e.g. binary systems such as carbonates, CaCO3 - MgCO3; alkali feldspars, NaAlSi3O8
- KAlSi3O8, etc.).
In some cases, a given mineral assemblage may be represented by a sub-set of the whole
system if fewer components are needed to define the compositions of the observed mineral
assemblage--this is known as a degenerate system. A good example of this can be seen on a
ternary (3 component) chemographic projection if the phases of interest plot in a co-linear
fashion (i.e. requiring only two components to define the phase compositions in what is
otherwise a three component system).
NAME OF THE FACULTY: Shaik.Mehboob Subhani
SUBJECT: Engg. Chemistry
LORDS INSTITUTE OF ENGINEERING & TECHNOLOGY
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A more complete discussion of components can be found in the section on Plotting Mineral
Compositions and Chemographic Projections.
F is the number of degrees of freedom in the system (also referred to as the variance of the system).
For geologic applications, this generally refers to the number of variables (e.g. pressure and
temperature) that can be independently changed without altering the state of the system (i.e.
the number of phases and their compositions are constant). Three common types of equilibria
are possible:
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Invariant equilibria, in which neither P or T can be changed; on a phase diagram, this is
represented as a singular invariant point
Univariant equilibria, in which either P or T can be changed independently, but to
maintain the state of the system, there must be a corresponding change in the other
variable; on a phase diagram this is referred to as a univariant curve
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Divariant equilibria, in which both P and T are free to change independently without
changing the state of the system (but bounded by the conditions defined by the
univariant equilibria).
The integer in the Gibbs Phase Rule
is related to the number of intensive parameters (i.e. those that are independent of mass; such
as pressure and temperature) that are being considered. Note that many phase diagrams
consider only one intensive parameter (e.g. T-Xfluid, where T is intensive, and the mole fraction of
fluid is related to the amount of mass of the fluid components); in this case, the Gibbs Phase
Rule would be: P + F = C + 1.
It is generally the case that as the number of components increase, the variance of the system must also
increase. Conversely, as the number of phases increase in the system, the variance of the system must
correspondingly decrease. A corollary to Gibbs' Phase Rule is Goldschmidt's Mineralogical Phase Rule:
for a given rock in equilibrium at a fixed P and T, the number of phases is less than or equal to the
number of components. Geologic systems are really rather simple! This means that we have a
reasonable expectation that we can successfully interpret the conditions of formation of rocks in their
natural settings, and to model their behavior using the thermodynamic databases and programs
presented at the end of this module.
Some Simple One-Component Examples
NAME OF THE FACULTY: Shaik.Mehboob Subhani
SUBJECT: Engg. Chemistry
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LORDS INSTITUTE OF ENGINEERING & TECHNOLOGY
Let's consider the simple 1 component system for H2O:
Phase diagram for the one component system H2O. This diagram is not to scale; a scale version looks like
the thumbnail in the white box.
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The system is entirely composed of H2O, so there is only one component present.
The phases present represent three states of matter: liquid (water), solid (ice), and vapor
(steam). All have distinct physical properties (e.g. density, structure--or lack of, etc.) and
chemical properties (e.g. ΔGformation, molar volume etc.) so they must be considered distinct
phases.
Note that there is only one point on this diagram where all three phases coexist in equilibrium-this "triple point" is also referred to as an invariant point; because P and T are uniquely
specified, there are zero degrees of freedom.
Each of the curves represents a chemical reaction that describes a phase transformation: solid
to liquid (melt/crystallization), liquid to vapor (boiling/condensation), solid to vapor
(sublimation/deposition). There are three univariant curves around the invariant point; it is
always the case that for a C-component system, there will always be C+2 univariant curves
radiating around an invariant point. This relationship is further explained in the unit on the
Method of Schreinemakers. There is only one degree of freedom along each of the univariant
curves: you can independently change either T or P, but to maintain two coexisting phases along
the curve the second variable must change by a corresponding fixed amount.
There are three distinct areas where only ice, liquid, or vapor exit. These are divariant fields. T
and P are both free to change within these fields and you will still have only one phase (a bit
hotter or colder, or compressed or expanded, but nonetheless the same phase).
The end of the "boiling curve", separating the liquid to vapor transition, is called the "critical
point". This is a particularly interesting part of the phase diagram because beyond this region
the physico-chemical properties of water and steam converge to the point where they are
identical. Thus, beyond the critical point, we refer to this single phase as a "supercritical fluid".
Now, let's consider a simple 1 component system that describes the mineral phases in the
aluminosilicate system:
NAME OF THE FACULTY: Shaik.Mehboob Subhani
SUBJECT: Engg. Chemistry
LORDS INSTITUTE OF ENGINEERING & TECHNOLOGY
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Phase diagram for the one component system Al2SiO5.
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The entire system is defined by one component: Al2SiO5 (i.e. all the phases can be completely
made of this one component)
There are three solid phases shown in this diagram: the polymorphs of Al2SiO5 andalusite,
kyanite and sillimanite.
There is only one unique place on this diagram where all three phases can coexist in equilibrium-the invariant point at 3.8 Kb and 500oC; at this point there are zero degrees of freedom.
There are three univariant reactions on this diagram, each representing the phase transitions:
andalusite = sillimanite, andalusite = kyanite, and kyanite = sillimanite. In each of these
reactions, either pressure or temperature can be changed independently, but for the state of
the system to remain the same (i.e. two solid phases coexisting in equilibrium), the other
variable must change by a fixed amount to maintain the assemblage on the univariant curve -so there is one degree of freedom. In a later section, we will see that the univariant curves
represent the condition where ΔGrxn = 0 (i.e. the intersection of the "free energy surface" with
the Pressure-Temperature plane represented by the phase diagram).
There are three divariant fields in which only a single mineral phase is stable. Within these fields
pressure and temperature may be changed independently without changing the state of the
system--thus there are two degrees of freedom in the divariant fields.
These principles extend to the more complicated binary, ternary and multi-component systems
presented in the next section(s) of this module on Phase Diagrams and Pseudosections.
Assumptions
There are a number of important assumptions that must be considered when applying Gibbs' Phase
Rule:
NAME OF THE FACULTY: Shaik.Mehboob Subhani
SUBJECT: Engg. Chemistry
LORDS INSTITUTE OF ENGINEERING & TECHNOLOGY
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The relationships described by the Gibbs' Phase Rule apply the concepts of equilibrium
thermodynamics. It is assumed that geologic systems have enough time to naturally equilibrate.
It is also assumed that geologic systems are "closed" systems--that is the mass of the system
remains essentially constant, and the system is free to consume or liberate energy to do work.
For geologic systems, heat and work are considered to be the major contributors to the system's
energetics--that is, "P-V work" is the dominant source of energy consumed/liberated by the
system. We do not consider the effects of external fields (e.g. electrical, magnetic, gravitational),
and we do not consider surface energy or boundary effects.
No kinetic effects are considered. This may present problems with an equilibrium
thermodynamics approach if metastable phases are present in the system. In natural settings it
is common for a given mineral to exist under physical conditions that exceed its predicted
stability limits. A good example is the persistence of kyanite into the sillimanite stability field.
This particular reaction a) has a very small ΔGrxn (and thus, a small "driving force" to make the
reaction go), and b) the polymorphic reaction is of the "reconstructive" kind, and requires the
breaking and reforming of strong covalent bonds. The presence of metastable phases may
complicate matters in trying to determine the equilibrium conditions for stable mineral
assemblages, but they are essential indicators of changing physical conditions that allow us to
interpret Pressure-Temeperature-time Paths.
In some cases, surface reactions (e.g. sorption, catalysis) may operate in geological environment
and contribute to the overall energetics of the system. Surface energies are not included in the
Gibbs Phase Rule formulation.
Gibbs' Phase Rule cannot be applied indiscriminately. It is important to obtain the best possible
characterization of the identity, composition and structural state of the phases present. For
example, complex materials such as mixed layer silicates may cause problems, or minerals that
exhibit a range of structural states (e.g. the alkali feldspars). An overview of analytical
techniques that are used to characterize minerals, melts and fluids can be found at the
companion website on Geochemical Instrumentation and Analysis
Recognizing these assumptions and limitations, the Gibbs Phase Rule still provides a powerful
way to analyze and interpret geological systems. Although geologic systems are dynamic and
often in a state of disequilibrium, the equilibrium approach demonstrates what the state of the
system would be given sufficient time and energy to achieve that equilibrium. It also provides
information on the phase changes or reactions that would be expected in a system that is
thrown out of equilibrium due to changes in physical conditions. Thus a knowledge of the
sequence of possible reactions provides important information about "pathways" in geologic
systems that allow interpretations of tectonic environments, petrogenetic processes, and the
evolution of geologic systems.
Derivation of the Phase Rule
It is important to recognize that the simple formulation of Gibbs' Phase Rule is derivative from
fundamental thermodynamic principles. The Gibbs- Duhem equation establishes the relationship
between the intensive parameters temperature (T) and pressure (P) and the chemical potential of
all components (μi) in the system:
dG = Vdp - Sdt + ΣNidμi
This means that there are
NAME OF THE FACULTY: Shaik.Mehboob Subhani
SUBJECT: Engg. Chemistry
LORDS INSTITUTE OF ENGINEERING & TECHNOLOGY
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C + 2 independent variables that describe the system: P, T and one each for the chemical
potential for all components, and
P independent equations (of the Gibbs-Duhem form) that describe the energetics of the system-one equation for each phase.
In mathematical terms, the variance (F) is determined by the difference between (C+2) variables and (P)
equations. Thus,
F = C + 2 - P or as originally written, P + F = C + 2
A more extensive derivation of the phase rule can be found at the website of W. R. Salzman,
Department of Chemistry, University of Arizona or this PowerPoint presentation on the
derivation of the Phase Rule (PowerPoint 337kB Sep18 07) by Marty Rutstein, SUNY New
Paltz.
Phase Diagrams (and Pseudosections) for Petrologists
Compiled by Dexter Perkins (University of North Dakota) and John Brady (Smith College)
Equilibrium phase relationships between calcite, quartz, wollastonite and CO2. Image courtesy of Darrell
Henry.
Standard 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 H2O). Other diagrams (such as the one shown on the right), depict the
NAME OF THE FACULTY: Shaik.Mehboob Subhani
SUBJECT: Engg. Chemistry
LORDS INSTITUTE OF ENGINEERING & TECHNOLOGY
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relationships between a number of compounds having different compositions. Still others show
how compositions of phases change under different conditions.
Pseudosections are a type of phase diagram that shows the fields of stability of different
equilibrium mineral assemblages for a single bulk-rock composition. Standard phase diagrams
may include many reactions but, depending on its composition, a particular rock may only
experience a few (or none) of them. Pseudosections only include those reactions that affect a
particular composition. Fields on a pseudosection are labeled (specifying the equilibrium mineral
assemblage) with the reaction lines unlabeled (although the specific reaction can be deduced).
Phase Diagram Archive
This is a diverse collection of phase diagrams, intended to be used by instructors when teaching
petrology or mineralogy classes.
Most of the thumbnail images are jpg files. However, we provide links to pdf files of all diagrams. They
can be viewed, printed, or edited with Adobe Illustrator or Corel Draw. An easy way to do this is to open
the files using Acrobat or its equivalent and then save them as eps files or in some other format that your
graphics program can import.
Besides single page diagrams, we also provide animations for some of the more useful systems. The
animations are multi-page pdf files.
Binary Igneous Diagrams
NAME OF THE FACULTY: Shaik.Mehboob Subhani
SUBJECT: Engg. Chemistry
LORDS INSTITUTE OF ENGINEERING & TECHNOLOGY
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These temperature-composition (TX) diagrams show melting relationships, and some subsolidus
relationships, involving two chemical components. Some involve incongruent melting; most do
not. Some of these diagrams involve eutectic and peritectic relationships.
Ternary Igneous Diagrams
Ternary igneous phase diagrams are triangular diagrams that show melting relationships
involving three chemical components. The diagrams may involve eutectic, peritectic and cotectic
relationships.
Metamorphic P-T Phase Diagrams
NAME OF THE FACULTY: Shaik.Mehboob Subhani
SUBJECT: Engg. Chemistry
LORDS INSTITUTE OF ENGINEERING & TECHNOLOGY
Page |
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Natural metamorphic systems may involve many chemical components. To simplify analysis and
interpretation, petrologists may study model systems containing only a few components. Some of
the diagrams available here involve only a few components and a few phases. Others are more
complex.
Phase diagrams that show many reactions that take place in a single chemical system are termed
"petrogentic grids." Petrognetic grids generally involve more components and more reactions
that simple model metamorphic systems. The idea is to include enough components and phases
so that the grid may be used to interpret natural metamorphic mineral assemblages.
Metamorphic T-X Phase Diagrams
For some rock compositions, metamorphic assemblages vary greatly depending on the
composition of the fluid present. Metamorphic fluids are generally dominated by H2O and CO2,
and the ratio H2O:CO2 can control mineral stability. So, phase equilibria are plotted on T-X
diagrams instead of P-T diagrams. (X refers to the mole fraction of 2 or H2O in the metamorphic
fluid that is present.)
NAME OF THE FACULTY: Shaik.Mehboob Subhani
SUBJECT: Engg. Chemistry
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Other Phase Diagrams
The above examples are of phase diagrams commonly used by geologists. Phase diagrams are
also useful in other fields, including chemistry and engineering.
Example diagrams:
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H2O (solid-liquid-vapor relationships)
NaCl-KCl (eutectic and solvus involving alkali halides)
CH3CH2OH-H2O ("ethanol")-H2O (diagram similar to plagioclase and olivine melting)
C12H22O11 ("sucrose")-H2O (binary eutectic)
NAME OF THE FACULTY: Shaik.Mehboob Subhani
SUBJECT: Engg. Chemistry