,This curriculum teaches the elementary principles of phase equilibria, a branch of chemical thermodynamics. This knowledge is useful to anyone dealing with chemical systems that may involve more than one phase, especially when there is also more than one component. This includes geologists, since rocks are generally multiphase systems (more than one mineral coexists), especially in igneous, metamorphic, and hydrothermal systems where fluid phases and high temperatures contribute to the achievement of thermodynamic equilibrium. It also includes all branches of materials engineering, including metallurgy, ceramics, composites, and chemical engineering; if a phase change (freezing, boiling, precipitation, recrystallization, etc.) occurs then the principles of phase equilibria are needed to understand it.
The approach of this curriculum emphasizes the use of geometric constructions that allow a visual interpretation of the rules of phase stability. The basic idea is to understand how we predict what phases will be stable in a system when we prescribe the temperature, pressure, and composition, and how those stability relations will change as the prescribed parameters are varied. If all the definitions, equations, and/or graphical representations ever start to make your head spin, come back and read this paragraph again to remind yourself what we are doing here.
In order to derive the equations underlying the graphical constructions and understand their meaning, we must go all the way back to the foundations of thermodynamics and build up some definitions and concepts.
The First and Second laws of thermodynamics allow us to define criteria for whether or not a chemical system is at equilibrium. For a closed system, the first law can be written
where E is the internal energy of our system, q is heat entering the system, and w is work done on the system. See the notes on exact differentials to understand what is meant by dE. Considering only mechanical work associated with changes in the volume of the system, the work term may be written w = -PdV.
The Second Law is an equality if we consider only reversible processes but an inequality if we consider both reversible and spontaneous processes:
Solving for q and substituting the second law into the first, we obtain what is sometimes called the fundamental equation of thermodymamics,.
Though fundamental, this equation is not very useful: it tells us that at constant entropy (dS = 0) and volume (dV = 0), the internal energy will decrease in any spontaneous process and will reach a minimum at equilibrium. But it is very hard to imagine a natural process or construct an experiment in which entropy and volume are controlled. We therefore perform a change of variables by introducing the Gibbs free energy, G = E + PV - TS. Now the total differential of G is.
from which we get a much more useful criterion for equilibrium:,
This equation shows why Gibbs free energy is so important: at constant temperature (dT = 0) and pressure (dP = 0), conditions which are easy to impose in the laboratory or to imagine in nature, it becomes dG < 0, where the less than sign applies to irreversible or spontaneous processes (it is inherited from the second law) and the equals sign applies to reversible processes or equilibrium states. In other words, if we control P and T, then the direction of approach to equilibrium is always a decrease in Gibbs free energy, until equilibrium is achieved when Gibbs free energy reaches a minimum. If there are no lower values of G accessible to the system (a global minimum), the equilibrium is stable; if a perturbation could bump the system out of a local minimum in G and allow it to evolve down to a lower minimum, then we were at a metastable equilibrium..
Though we will here explore only the common situation of constant T and P, wherein equilibrium is defined by minima in G, it is worth keeping always in mind that there are other situations where the equilibrium criterion is different. At constant temperature and volume, equilibrium is found at the minimum in Helmholtz free energy F = E - TS. At constant pressure and entropy, equilibrium is found at the minimum in enthalpy H = E + PV.
For a phase, the Gibbs free energy is a function of P, T, and composition. Like all energies in thermodynamics, G is expressed not in absolute terms but relative to a standard state, usually the energy difference relative to the elements at 1 bar and 298.15 K. If the enthalpy of formation and third law entropy are measured, then we get G from G = H - TS. If we know G at one temperature and pressure, then since it is an exact differential, we can see from the expression for dG above that for reversible paths
which allows us to calculate G at any P and T from calorimetric and volumetric data.and
,
The intensive Gibbs free energy of a phase, which
we denote 
, is also, in general,
a function of composition.
All real phases have some range of variability in composition, though some are
always nearly pure. Thermodynamically, that is to say that their free energy may increase very fast
as one tries to add other components to such a phase. The extent to which a phase will vary in composition in equilibrium with other phases
is apparent from a graph of its intensive Gibbs free energy vs. composition; for a binary system with one intensive compositional
variable, X, this is a graph of (X).
For a binary phase to be stable, it is necessary that
the (X) curve be concave-up. You can prove this to yourself by considering what
happens if the curve is concave down: if a homogenous phase at composition X were to unmix at constant
temperature, pressure, and bulk composition into two phases of composition X+e and X-e, then the
Gibbs free energy would change from (X) to ((X+e) +
(X-e))/2. But the second derivative is defined as the limit as e goes to zero
of [((X+e) + (X-e)) - 2(X)]/e2,
so if the second derivative is less than zero than the unixing lowers the Gibbs free energy. Hence the
phase is unstable to decomposition by an infinitesimal perturbation where one region becomes slightly
richer in one component and another slightly poorer. The situation of concave-down free energy surfaces does in
fact arise; the boundary between a region where the free-energy surface is concave-up (stable) and concave-down (unstable)
is called the spinodal and leads to the phenomenon
of
exsolution. At any pressure, there is generally a maximum temperature where the spinodal terminates, which is called a
critical point. The family of critical points
at various pressures forms a critical line.
In ternary and higher-order systems, there is more than one independent intensive compositional variable and the
the simple criterion d2/dX2 > 0 is not sufficient to define stability. The criterion is
still that the surface be concave-up, but to define this idea
mathematically we must consider the matrix of second derivatives of , also called the
Hessian matrix. If this matrix is positive definite (has all positive eigenvalues and a positive determinant), the surface
is concave-up and the phase is stable at this point. If any of the eigenvalues are negative, the surface is either hyperbolic
(concave-down along some directions) or concave-down and hence unstable. On the boundary between these regions, the
spinodal, the determinant of the Hessian matrix is zero.
In multicomponent systems, where G depends on composition, we need to modify our equilibrium
criterion so we can examine both
the composition of the stable phases and how the stable phase assemblage
varies with bulk composition. This requires introducing a quantity called the
chemical potential, 
. Chemical
potential has the important property that the chemical potential of each component is the same
in all phases coexisting at equilibrium. It is easy to understand why: the definition of
is the change in G that results from adding an increment of component mass to a phase at constant
temperature, pressure, and masses of the other components. If
for some component is higher in one phase than in another, it follows that we can lower the overall
G of the system by moving a mass increment of that component from the high phase
to the low phase. Hence G was not at a minimum. But being at a minimum of G is
a criterion for equilibrium.
It can be shown that the intensive Gibbs free energy either of the system or of a phase is related to the sum of the mass fractions of the components times their partial specific Gibbs free energies or chemical potentials; for the binary case, there are two components (say, 1 and 2) and:
Since the two mass fractions sum to unity in the binary case, however, let us define X = x2 = 1 - x1 and simplify the above to.
which is the equation of a line in,
-X space with intercept
1 and slope 2-1.
In a plot of vs. X at constant T and P, this line is tangent to the
(X) curve. Do not misinterpret the above equation as an expression that tells
how varies with X: the chemical potentials of the components are themselves
functions of X. Now, if two or more binary phases are in equilibrium, the chemical potential of both
components must be the same in the coexisting phases (see previous paragraph), so their
(X) curves must share a common tangent line, whose intercept and slope are
given by the above equation. Furthermore, the compositions that coexist are given by the tangency
points where this "chemical potential line" touches the free energy-composition curves. This is the
basis of the visual construction that we use to find equilibria. This concept is easily extended to systems of more components, where it remains
true that the chemical potentials of all components must be equal among all phases at equilibrium. In the ternary system, the tangent plane
to a free-energy surface is defined by the chemical potentials and all coexisting phases must share a common tangent plane in
(X1, X2) space.
Consider the -X diagram in Figure 1. This diagram always shows the properties of two
phases of a binary system as a function of composition (X) at contant temperature and
pressure. In Figure 1 we show the individual (X) curves for two phases. If we imagine
fixing P, T and total X of the system, how do we find the
equilibrium state of a system in which these are the only two possible phases, and how do we see the
changes in that state as a function of X? By state of the system we mean what phases are present, how much
of each, and the composition of each phase.
Recall what we said in the previous paragraph about chemical
potential: it is a necessary and sufficient condition for equilibrium between the two phases that
the chemical potential of the each component be equal between the phases, and this is shown graphically
by both curves sharing a common tangent line, as shown in Figure 2. The intercept of the tangent line
at X = 0 (remember X is the mass fraction of component 2) is 1 in phase A
and in phase B, and the intercept of the tangent line at X = 1 is 2 in
phase A and in phase B. These two points are sufficient to define a line, and the line is tangent to
both (X) curves. But there is more information here: not just any composition of
phase A can be in equilibrium with phase B. Only phase A with the composition marked XA(B),
which is the point where the tangent line touches the A(X) curve, has
the correct chemical potential. Likewise only if phase B has the composition marked XB(A),
where the tangent line touches the B(X) curve, can it be in equilibrium
with A under these conditions.
There is still more information on these diagrams. We have shown that there exists an equilibrium
where A and B of particular phase composition will coexist. But how do we decide what the stable
phase assemblage will be at given bulk composition X? Well, we need to find the configuration with
minimum Gibbs free energy. There are two possibilities: either (1) the system will contain only one
homogeneous phase whose phase composition equals the bulk composition, or (2) the system will contain
a mechanical mixture of more than one phase which are all in equilibrium with each other and whose
compositions add up to the bulk composition. In case (1), the intensive Gibbs free energy of the system is
equal to the intensive Gibbs free energy of the single phase and we can read this directly off the
diagram from the (X) curve for the phase. In case (2), begin by imagining there
are two coexisting phases A and B, making up mass fractions fA and fB of the
system, respectively, and having composition XA(B) and XB(A). Then the bulk
composition of the system is X = fAXA(B) + fBXB(A), and
the intensive Gibbs free energy of the system is (X) =
fAA(XA(B)) +
fBB(XB(A)). Now look again at Figure 2: these
two equations define a line segment that connects the two points (XA(B),
A(XA(B))) [when fA=1 and fB=0] and
(XB(A), B(XB(A))) [when fA=0 and
fB=1]. But this line segment is exactly the same as the segment between the points
of tangency of the common tangent line defined by the chemical potentials of the coexisting phases.
Let's call this part of the tangent line an "interior tangent segment". When two phases coexist, we can
read the bulk (X) off the diagram by taking the where the
tangent line between their curves crosses the bulk composition of interest. The part of the common tangent
line exterior to the two tangency points is still useful for extrapolating to the chemical potentials
at X=0 and X=1, but it does not represent a physically achievable free energy for the system, since one
of the phase proportions would have to be negative to obtain a point along the line exterior to the tangency points.
Now we have what we need to read the stable phase assemblage off a binary -X
diagram. Recall that the criterion for equilibrium at prescribed P, T and X is that be a minimum;
graphically this means finding the lowest physically achievable state on the diagram for a given X
(the diagram already represents prescribed P and T). A physically achievable state is either the entire
system as a single phase or a mechanical mixture of phases that can coexist with one another. In the case
of a single phase, the phase composition equals the bulk composition and the value of
is given by the curve for that phase. So, on
our plot, at any X, the lowest available is represented either by the
(X) curve of a phase (these are colored red on the
diagram) or by the interior segment of a tangent line connecting
two such curves (this is colored green on the diagram). If a curve is lower than any interior tangent
segment, then we are in a one-phase
region in which the phase with the lowest (X) curve exists alone and the
phase composition equals the bulk composition. On the other hand, if at some X an interior tangent segment
is lower than any curve, then a mechanical mixture of the phase compositions at the two endpoints will
have lower free energy than any single phase with composition X. Hence we will be in a two-phase region with
the compositions of the two phases fixed at the endpoints of the interior tangent segment. The relative
proportions of the two phases follow from the lever rule,
In Figure 3, we have colored red the sections of the.
A(X) and
B(X) curves where A and B are stable alone in one-phase regions, and green
the interior tangent segment where A and B coexist. We have also labelled the sequence of stable
phase assemblages (A, A+B, B) across the bottom and divided the X-axis into the three regions.
This sequence, and the points where the phase assemblage changes across the X-axis,
contain the essential information derived from this diagram: once we have found the minimum Gibbs free
energy assemblages we usually do not care any further about . Therefore, since it is
easiest to make graphs in two dimensions, it is not an efficient use of graph paper (or computer
screen) to show a whole plot which only describes the system at a single pressure and temperature. Instead,
we make T-X (isobaric) or P-X (isothermal) diagrams that show a stack of the X-axis stability sequences
derived from -X diagrams at a sequence of temperatures and equal pressure (T-X) or
a sequence of pressures and equal temperature (P-X). The one-phase and two-phase intervals of these
segments link up with those at adjacent T or P to form regions, bounded by the compositions of phases
that coexist in two-phase regions. Note that with two phases whose (X) curves are
everywhere concave-up, there are only a few possible sequences:
either one (X) curve is lower than the other everywhere and one phase will be stable
alone at all X; or the curves cross once, in which case the sequence will be like (A,A+B,B); or the
curves cross twice, in which case the sequence will be (A,A+B,B,B+A,A).
Much of what we have discussed so far can be seen by exploring the old binary applet; just click on some
random points in the P-T projection at upper left. The applet will bring up a -X
diagram corresponding to the P-T point you clicked. There are four phases involved (or three in the
simplified version), but if you do not click on a line or intersection of
lines, you will see that no more than two phases ever share a common tangent, and the sequences of
stability across the X-axis all follow the rules we have just developed. If you wish to continue with the old binary applet, proceed to
Page 2 of the old tutorial. However, we recommend that instead you work through the series of
examples below that make use of the new applet. When you finish with binary systems, you can proceed to ternary systems.
At this point we need to step aside a minute and consider the apparently unrelated issues of how
many phases can coexist at equilibrium and how many variables need to be fixed to determine the state
of the system. We have shown already that in a binary system we can have one-phase regions where
the phase can have any composition at fixed P and T or two-phase
regions where both phase compositions are fixed. This suggests that actually the number of phases
and the number of free variables is related. This relationship is explicitly captured by the
Gibbs Phase Rule, which relates the
variance (f) of an assemblage to the number of
phases 
,
number of components c, and other restrictions imposed. The
phase rule is best understood by thinking
of the conditions of equilibrium as a set of simultaneous equations that nature must solve.
We learn in elementary
algebra that if we have the same number of unknowns (variables) and equations (constraints) then
we should expect to find a unique solution, i.e. there are no remaining degrees of freedom in
the unknown variables. On the other hand, if we have more unknowns than equations, there is usually
a family of solutions, and the dimension of the solution space is the number of extra unknowns. Finally,
if there are more equations than unknowns, then unless we are very lucky there will be no solution at all.
In the case of the phase rule, the unknowns are P, T, and (c-1)* independent
phase compositions. The constraints are the (-1)*c statements of equality of
chemical potential among phases. Hence the dimension of the allowed solution space is
f = 2 + (c-1)* - (-1)*c = c + 2
- . Once it is given that we have an equilibrium assemblage in a system of
c components containing particular phases, the variance is the number
of remaining variables we have to fix to determine the state of the system.
Consider, e.g., the one component system H2O. If one phase — say, liquid water — is present, then f = 1 + 2 - 1 = 2, and indeed we can freely vary both temperature and pressure within a two-dimensional but bounded stability region without change of phase. However, if we insist that water and ice are coexisting (so f = 1 + 2 - 2 = 1) and we fix one more variable — say make the pressure 1 atm — then the temperature becomes fixed (it must be 0 °C). Ice+liquid in the H2O system is an example of a univariant assemblage, which is restricted to a one-dimensional array (i.e., a line or curve) in P-T-X space. Furthermore, although it is slightly more remote from everyday experience, there is a single point in (P,T) space at P = 6 mbar, T = 0.01 °C where ice, liquid water, and water vapor (steam) can all coexist. This is called the triple point and is an example of an invariant condition f = 1 + 2 - 3 = 0 where merely declaring the number of phases and components has entirely determined the state.
There can be other constraints in the phase rule that lower the variance without adding more
phases. For example, if we require that two phases are equal in composition or that three phases
are collinear in a three-or-more component system, this takes away one degree of freedom. Hence in
the binary system if a solid phase coexists with a liquid of the same composition as the solid (a
congruent melting point) this assemblage is univariant: f = 2 + 2 - 2 - one extra
restriction = 1. Likewise, specifying that one of the phases is at its
critical point
reduces the variance by two extra restrictions. And there can be situations where some of the phases
are restricted to specific compositions or subspaces of the composition space; this leads to
degenerate equilibria where
the variance is actually higher than if the phases were free to vary in composition.
II. Binary systems
We now begin specifically applying the concepts we have learned so far to binary systems, with the help
of the binary visualization applet. We will work upwards in complexity from systems with one phase through
systems of four or more phases. In each case we will focus on the special situations that can arise as the
number of phases increase. Thus, when we get to two phases the possibility of the two phases coexisting with
equal composition arises, so we will look at
coincidences. When we get to three phases, we can have
univariant assemblages, so we will focus on such cases. Finally, when we get four phases we can have an
invariant point. We begin, however,
with one phase by itself, but already some interesting phenomena can arise.
A. One phase
1. The situation of a binary system with a single phase that contains a critical point is explored
using the new binary visualization applet on the Example 1 page.
B. Two phases
2. The simplest situation that can arise with two phases is a binary phase loop, with degenerate coincidences at the bounding pure systems, but only a divariant field in the binary. Go to the Example 2 page to learn more.
3. When the phases are not ideal solutions, the possibility can arise that they have a coincidence point or azeotrope in the binary system. We first look at the case where this coincidence point forms a minimum in temperature: Example 3.
4. Next comes the case where this coincidence point forms a maximum in temperature: Example 4.
5. Perhaps the most complicated situation that can arise involving only two phases occurs when the two-phase loop has a coincidence point as in example 3 and
furthermore one of the phases has a critical point and miscibility gap as in example 1. The intersection of these phenomena generates three new features: a univariant curve
involving two instances of the phase with a miscibility gap and one instance of the other phase; a critical end-point where the univariant line terminates against the critical line
and the critical line becomes metastable; and a singular point where the coincidence encounters the univariant, at which point the coincidence becomes metstable and the
unvariant changes from eutectic-type to peritectic-type. All this is visible in Example 5.
C. Three phases
6. Example 6. Explanation to come.
7. Example 7. Explanation to come.
D. Four phases
8. Example 8. Explanation to come.
9. Example 9. Explanation to come.
E. Degeneracy and final remarks
10. Example 10. Explanation to come.
III. Ternary systems
Under construction.
Send suggestions, whines, and flames to Paul Asimow.