**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

,where there are N components in the system, x

**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

**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
Fe_{2}O_{3} then the composition Fe = 3 FeO - 2 Fe_{2}O_{3} has
mole fraction X_{FeO} = -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 SiO_{2} 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 SiO_{2}.

**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=mc^{2}), 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

.

**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 SiO_{2} 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 (H_{2}O 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 = q

**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-SiO_{2} 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
length^{3}. 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.

Send suggestions, whines, and flames to Paul Asimow.