Here we define terms that arise in the Curriculum on phase equilibria.

## Glossary

Chemical Potential, : The driving force for chemical reactions. Just as temperature is the driving force for heat flow, in which heat will flow from a body at high temperature to a body at low temperature until equilibrium is achieved when the two bodies have equal temperature, and pressure is the driving force for work to be done, in which a body at high pressure will expand against a body at low pressure until equilibrium is achieved when the pressures are equal, hence also chemical potential is the driving force for mass transfer, and a component will move from a body where its chemical potential is high to a body where its chemical potential is low until equilibrium is achieved when the chemical potential of each component is uniform (among phases within a system and also between the system and the environment if the system is an open system). The chemical potential of a component is defined as the derivative of the (extensive) Gibbs free energy with respect to the mass (or number of moles, either definition is good as long as we are consistent) of the component in the phase, at constant temperature, pressure, and masses of all the other components. That is: .
Defined this way, the chemical potential is also the partial specific gibbs free energy and hence it obeys the relation ,
where there are N components in the system, xi is the mass fraction of component i in the bulk composition and is the specific (i.e. per unit mass) Gibbs free energy of the system. If we restrict the system for the moment to a particular phase, the above equation is also true for the specific Gibbs free energy of the phase, component mass fractions in the phase, and chemical potentials of components in the phase.

Chemical Thermodynamics: For systems with a large enough number of atoms to treat statistically, thermodynamics can tell us the state towards which the system will evolve given enough time. The question of how long it might take to get there is the subject of a related field of science, kinetics. The behavior of a small number of discrete objects is also a separate field, mechanics, although the ideas of statistical mechanics show how the macroscopic behavior of large systems captured by thermodynamics emerges from the collective behavior of many individual objects. Although thermodynamics is often taught in physical terms, applied to the efficiency of steam engines and such, it also predicts the behavior of reactions among elements and compounds, the subject of chemistry. The ideas of chemical thermodynamics are useful also in geology, materials science, and other fields where one needs to be able to predict how chemical systems will behave, particularly at high temperature where kinetic rates are fast.

Clausius-Clapeyron Equation: Gives the slope of a univariant reaction in P-T space. If the reaction is in equilibrium then the change in Gibbs free energy between the products and reactants G = 0. Hence - SdT + VdP = 0 where S is the entropy of reaction and V is the volume change of the reaction. Hence

dP/dT = S/ V.
Always remember, and never forget, that the Clausius-Clapeyron equation applies only to univariant reactions.

Coincidence: A situation where two phases of equal composition coexist in a multicomponent system. This is a univariant assemblage in a system of any number of components, even though the number of phases is smaller than expected for a variance of one because of the additional restriction(s) of equal composition. In ternary and higher systems there are related sorts of degenerate univariant equilibria such as a collinearity where three phases plot along a line in composition space.

Component: A chemical formula that can be used to express the compositional range available to a system (when used to express the (generally more restricted) compositional range available to a phase, it is called a phase component). It is important to think of components as mere mathematical constructions, as basis vectors for expressing composition, and not to confuse them either with phases or species. Hence components with negative numbers of atoms in them are allowed (e.g., an exchange component like FeMg-1 is often useful in describing solid solutions), and compositional bases in which physically real systems contain a negative amount of some component are also allowed (e.g., if the components are FeO and Fe2O3 then the composition Fe = 3 FeO - 2 Fe2O3 has mole fraction XFeO = -2/(3-2) = -2). Although there is often some freedom in choosing components to describe a system, we characterize systems by the minimum number of components needed to express the compositional range available. Thus a pure system, in which all available phases have identical composition, has one component. A system of two components is called binary; although the absolute quantities of the two components are independent, when composition is expressed as any intensive parameter (such as mass fraction or mole fraction) it requires only one parameter to describe compositions is such a system. Our components only need to span the possible compositions of phases actually considered. Thus we can use SiO2 as a component if we are never going to deal with such high temperatures and energies that Si and O will act independently and form phases with compositions other than SiO2.

Composition: An intensive parameter defining the relative proportion of each component in a system or phase. Our symbol for composition is X. When applied to the whole system we use the term bulk composition; when applied to a particular phase we might say phase composition. In a one-component system all possible phases and the system as a whole have identical composition; no compositional parameters must be specified to determine the state of the system. In a binary system such as that represented by the Applet, although there are two components there is only one independent compositional variable because the fractions of the two components in the system or in a phase must sum to unity.

Critical Point: The boundary between pressure/temperature conditions where a phase has an unstable region and pressure/temperature conditions where the phase is continuously stable. The critical point is the termination of a spinodal and a solvus (see exsolution). The conditions of criticality reduce the variance by two; hence in a one-component system a critical point is invariant whereas in a binary system criticality is univariant and there is a family of critical points along a critical line.

Degenerate: Describes an equilibrium assemblage whose variance is lower than expected from the number of phases and components because some other restriction is active in the phase rule. Examples include coincidences, singular points, collinear phases, and equilibria in which some or all phases are restricted to a bounding subsystem.

Energy: Generally, the capacity to cause change, and the basic quantity studied by thermodynamics. Energy comes in many forms (including kinetic energy, potential energy, chemical energy), and can be interchanged among these forms and transferred from one place to another (as heat or work) but not created or destroyed. This is the essence of the First Law of Thermodynamics: conservation of energy and equivalence among different forms of energy. In relativistic systems or when nuclear reactions are involved, energy and mass can be interchangeable (as in E=mc2), but in classical thermodynamics each is conserved separately. We use the term internal energy and the symbol E (some books use U) for the sum of kinetic and potential energy in a system.

Enthalpy: A quantity with units of energy symbolized H and defined by H = E + PV, where E is internal energy, P is pressure, and V is volume. Enthalpy is useful because at constant pressure it measures the quantity of heat that flows into or out of a system, hence differences in enthalpy can be measured directly by calorimetry. The difference in enthalpy between a compound and the elements of which it is composed is called the enthalpy of formation.

Entropy: In macroscropic thermodynamic, entropy is simply defined as a state variable whose changes in value are defined by the Second Law and whose absolute value for some matierals can be fixed according to the Third Law. However, statistical mechanics provides more insight into the nature of entropy. It is a measure of the "disorder" of a system, by which is meant the number of available configurations or microscopic states that are consistent with a given macroscopic or average state. This relation, S = k ln , is inscribed on Boltzmann's tombstone.

Equilibrium: A challenging yet essential term to define in thermodymamics. Generally refers to a state of a system that does not change with time; in particular all the state variables remain constant. The difficulty is that some authors require an equilibrium state to be unchanging forever if not perturbed, whereas others require only that it remain unchanged during the period of observation. The latter definition is more practical, since any real situation imposes a limit on the time-scale of interest, but it includes within the definition of equilibrium states that are in fact spontaneously changing, but too slowly to be observed. See Stable for more discussion.

Eutectic: A univariant reaction involving a liquid and c solid phases (where c is the number of components in the system) where in composition space the liquid plots inside the polyhedron whose vertices are the solid compositions. Hence in a binary system (c=2), a eutectic is an equilibrium between two solids and a liquid which plots in between them. Therefore it is a reaction of the form A + B = liquid. See peritectic.

Exact Differential: A required property for a function to be a state variable. Much of thernodynamics is expressed in the language of multivariable calculus; if you are not comfortable with total derivatives and partial derivatives you might want to review some calculus first. If u(x,y) is a function of two variables, then it is an exact function and its total derivative is an exact differential if the total differential of u may be written and the second partial derivatives obey the reciprocal relation .

Exsolution: When a homogeneous phase breaks up into two instances of the same phase with different composition, as in immiscible liquid solutions or perthitic feldspars. Modeling of exsolution and related phenomena such as critical points requires a better model for the free energy of phases than the simple one implemented by the demonstration applet.

Extensive Parameter: A property of a system that scales linearly with the volume of the system or the total quantity of system present. Examples include mass, enthalpy, volume, etc. To be distinguished from intensive quantities, which are independent of the size of the system, like temperature, pressure, mole fraction, and specific volume. The ratio of two extensive quantities is an intensive quantity.

First Law of Thermodynamics: An empirical statement developed by James Joule and the other founders of thermodynamics that defines the essential properties of energy: that it is a conserved quantity or state variable and that all its forms are equivalent. For a closed system it is usually stated ,
where q is heat entering the system and w is work done on the system.

Gibbs free energy: A quantity with units of energy symbolized G and defined by G = E + PV - TS, where E is internal energy, P is pressure, V is volume, T is absolute temperature, and S is entropy. If you check the definition of enthalpy, H, you will note that G = H - TS. Gibbs free energy is useful because it is the quantity that reaches a minimum when equilibrium is achieved at prescribed temperature and pressure. The direction of decreasing G tells us which way a system will evolve as it tries to reach equilibrium. G is an extensive quantity, but we will frequently prefer its intensive equivalent, specific Gibbs free energy, , which is Gibbs free energy divided by mass. See also chemical potential.

Heat: That which is transferred from a hot body to a cold body. It is important not to think of heat as a form of energy but as a transfer of energy from one place to another. See the First Law. By convention, heat entering the system from the environment is positive, heat leaving the system to the environment is negative.

Intensive Parameter: A property of a system that is independent of the size of the system or the total quantity of system present. Examples include temperature, pressure, mole fraction, and specific volume. To be distinguished from extensive quantities, which scale linearly with the mass of the system, like mass, enthalpy, volume, etc. The ratio of two extensive quantities is an intensive quantity. We are usually concerned mostly with intensive quantities in chemical thermodynamics; the total amount of system present is not usually considered an interesting property of the state.

Invariant: Having a variance of zero. An invariant assemblage can exist only at a unique point in pressure-temperature-composition space, and projects to a point in the P-T projection of a system. A non-degenerate invariant point involves c+2 phases, where c in the number of components in the system (so 4 phases in a binary system).

Mass: In physics, the property of matter that is acted upon by gravity (as in Newton's Universal Law of Gravitation) and that leads to inertia (as in Newton's Laws of Motion) (for some reason these masses are the same). In chemical thermodynamics (neglecting radioactivity and high energy physics), mass is a conserved quantity, as are the masses (or numbers) of each individual kind of atom.

Peritectic: A univariant reaction involving a liquid and c solid phases (where c is the number of components in the system) where in composition space the liquid plots outside the polyhedron whose vertices are the solid compositions. Hence in a binary system (c=2), a peritectic is an equilibrium between two solids and a liquid which does not plot in between them. Therefore it is a reaction of the form A = B + liquid. See eutectic.

Phase: Strictly, a homogenous and mechanically separable portion of a system. Practically speaking, phases are the tangible states of matter such as solids, liquids, and vapors. Some phases have crystal structure (solids, including minerals), others are amorphous (liquids, vapors, glasses). Some only form with a definite chemical composition (pure phases), others can take on a range of compositions (solutions). It is critical to avoid confusing phases with components, even though they are often given the same name or symbol: e.g., quartz is a phase, whereas SiO2 is a component, but the two might be interchanged in casual usage (as in "the quartz component" or "a silica phase"). The same component can often form many pure phases (H2O can form steam, liquid water, and about ten different crystalline ice structures) and a given phase can often vary widely in composition within a multidimensional component space (e.g., olivine, in which the M crystal sites can be occupied by Mg, Fe, Ni, Mn, Co, Ca, etc.). It is the intent of studies in phase equilibria to identify and predict what phase or phases will be present in a system, the propotions of the phases, and the composition of each phase.

Phase Rule: A fundamental relation stating how much one needs to know to define the state of a thermodynamic equilibrium system. Usually given as f = c + 2 - - other, where f is the variance of the assemblage, c is the number of components, is the number of coexisting phases, and "other" stands for extra constraints like phases of equal composition or critical conditions. The phase rule amounts to setting the number of constraints (imposed or given by the conditions of homogeneous and heterogeneous equilbrium) equal to the number of variables so that a unique solution (i.e., an equilibrium state) is determined. See Section 5 of the tutorial.

Pressure: Force per unit area acting on the boundaries of a system and needed to confine it to a given volume. In equilibrium, then, also the force per unit area exerted by the system on its boundaries. Gases at finite temperature always exert pressure on any container because of the kinetic motions of their molecules. Solids and liquids will exert pressure on their containers or boundaries except at some particular volume where the attractive and repulsive forces between their consituent atoms happen to be perfectly balanced. Pressure is best defined in gases and liquids lacking shear strength, because the equilibrium state of stress in these materials must be hydrostatic, i.e. the same in all directions. Solids can be more complicated because they can support shear stress, in which case pressure is not a complete characterization of the mechanical confining conditions.

Reversible: A process that moves through a sequence of equilibrium states with no finite departure from equilibrium. To be contrasted with spontaneous processes, which happen by themselves because the system is not in equilibrium (remember that the definition of equilibrium is that nothing macroscopic changes!). Of course, reversible processes are an unachievable idealization, but they still occupy a critical place in thermodynamic thinking. We can think of a reversible process as the change of state driven by externally imposed variations in the intensive parameters, in the limit of infinitely slow change. In practical terms, if we can characterize the time for a system to relax to equilbrium after a perturbation, then we can achieve nearly reversible changes by driving the system at a time scale much longer than the relaxation time. The idea of a reversible change is embodied in the second law of thermodynamics, since it is used to define the sense in which entropy is a state variable.

Schreinemakers rules: A set of rules that determine the arrangement of stable and metastable univariant equilibria in P-T space where they intersect at an invariant point, and the relationship of this arrangement to the positions of the participating phases in composition space. The convexity of free energy surfaces leads to strict topological constraints which are conveniently expressed by a few simple rules. For a full discussion, I recommend E-An Zen's monograph, USGS Bulletin 1225, 1966.

Second law of thermodymamics: Like the first law, an empirical (but never yet disproven) statement embodied in a definition. Defines the quantity entropy, declares it to be a state variable, and establishes the fundamental assymmetry of time by requiring that any spontaneous process cause entropy to increase (and, by extension, any process at all to cause the entropy of a system large enough to encompass the cause of the process, up to and including the entire universe, to increase). In mathematical terms, the second law is stated ,
where S is entropy, q is the quantity of heat transferred during the process over which the differential of entropy is taken, and T is absolute (thermodynamic) temperature. Why does this statement of the second law mean what we just said it means? Well, just as the observation that neither heat nor work are state variables but there sum is leads to the first law and the definition of internal energy, we have here the observation that heat is not a state variable but for reversible processes the ratio of heat transferred to absolute temperature is independent of path and so defines a state variable. Or rather, that a temperature scale can be defined so that this statement is true. The inequality applies to irreversible, i.e. spontaneous, processes; for reversible processes dS = qrev/T. What about the universal interpretations? Evidently if heat flows out of a system, then q < 0 so dS < 0, and the entropy of the system decreased, right? Yes, but the heat has to go somewhere, such as into the environment, which receives heat -q > 0. And here is the key: in the definition of temperature we noted that heat always flows from high temperature to low temperature. Hence if heat flowed from our system at T to the environment at Tenv if follows that T > Tenv. Therefore q/T < q/Tenv. Therefore dSenv > -dS. That is, the same quantity of heat transferred from system to environment caused the entropy of the environment to increase more than the entropy of the system decreased, because the environment was at lower temperature. The second law forbids the opposite process, of tranferring heat from a cold body to a hot one.

Singular point: A type of degenerate> invariant point. In a binary system three phases are present rather than the four expected for a variance of zero, but two of the phases have the same composition, which elimninates a degree of freedom. A singular point forms at the intersection of a coincidence line with a unvariant reaction. Topology requires that this intersection be a tangency rather than a crossing.

Species: A microscopic atomic or molecular unit that actually has or is hypothesized to have some (at least transient) physical reality as a distinct entity in some phase. Species is meant to be distinguished from component, which is just a mathematical expression for composition, although it may look like a species or possible species. Species is also distinguished from phase, which is a macroscopic concept.

Spinodal: The curve bounding compositions of a phase that are unstable with respect to exsolution from those that are metastable. In a binary system the spinodal is defined by the second derivative of the intensive Gibbs energy with respect to composition being zero. In higher order systems it is a point where the Hessian of the intensive Gibbs energy is singular.

Stable: Closely related to "at equilibrium." No macroscopic, measurable parameter of a state that is stable will change during the time of observation, and after any perturbation from a stable state the system will return to the original state. A state that is time-stationary and stable with respect to small perturbations, but unstable with respect to some large perturbation that would allow the system to reach a more stable state is called metastable. There are two slightly subjective aspects to the definition of stable equilibrium: it is always possible that a system that appears stable on one timescale of observation might be seen to evolve if we wait longer, and it is always possible that some perturbation larger or different in kind from those previously considered could be found that would cause the system to tunnel through to a new, more stable state. Hence our definition of stable is a working definition only. Most equilibria are dynamic : at the microscropic level we do not require that nothing is happening at equilibrium, only that any ongoing process is balanced by a reverse process occuring at the same rate so that the macroscopic observables do not change.

State Variable: A quantity that describes the state of a system and is independent of the path taken to reach a state, so its path integral along any closed path that leads back to the original state is zero. State variables have no "memory" of the history of a system, since they are fixed by its current state. Whenever enough state variables are fixed to define the state (see the Gibbs phase rule), all other state variables are determined. Examples of state variables include pressure, temperature, volume, internal energy, entropy, and Gibbs free energy. Heat and Work are not state variables: they depend on path (see the First Law). It is a mathematical requirement that state variables have Exact differentials.

System: The region under consideration, as distinguished from the from the rest of the universe (the environment). Systems may be separated from environments by boundaries that prevent the transfer of mass (a closed system), of heat (an adiabatic system), or of any energy (an isolated system). Systems that exchange mass with the environment are open systems. Sometimes the word system is also used to refer to all possible compositions defined by a particular set of components (for example, the MgO-SiO2 system).

Temperature: Operationally, a measure of the tendency of a body or system to give up or take in heat from its surroundings. Heat always flows from high temperature to low temperature. Two bodies in equilibrium must have the same temperature (this is sometimes called the zeroth law of thermodymamics). This qualitative definition can be put on an absolute scale in a few ways, based either on the Second Law, the ideal gas law, or statistical mechanics. In each case there exists an absolute zero (0 Kelvin or -273.15 °C) where there are no vibrational degrees of freedom and the Third Law applies. Microscopically, of course, temperature is associated with kinetic energy of atoms, and quantum mechanically with occupancy of excited quantum states.

Third law of thermodynamics: Establishes an absolute scale for entropy (unlike energy, enthalpy, and Gibbs free energy, which can only be defined relative to a reference state). The Third Law states that the entropy of a perfect crystalline solid at the absolute zero of temperature is zero. In the context of statistical mechanics, this is to say that if only one configuration is available because all positions are fixed and all vibrations are in their ground state, then the entropy is zero.

Univariant: Having a variance of one. A univariant assemblage can exist along a one-dimensional array (line or curve) in pressure-temperature-composition space, and projects to a line or curve in the P-T projection of a system.

Variance: The number of degrees of freedom or unconstrained variables in a thermodynamic equilibrium assemblage, or the dimensionality of a region in pressure-temperature-composition space where a given phase assemblage can exist. See the Gibbs phase rule.

Volume: The quantity of space occupied by a system, in units of length3. Volume is an extensive quantity, but it can be converted into the intensive equivalents molar volume by dividing by the number of moles or specific volume by dividing by mass. The specific volume is the reciprocal of density.

Work: The transfer of energy into or out of a system by a force acting over some distance. In chemical thermodynamics, the force is most often pressure (force per unit area) acting to change the volume of the system. The quantity of work done is -PdV (the sign convention is that work done on the system, which would decrease the volume, is positive). Work can also be done by gravitational and magnetic fields.

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