This Tutorial begins at the beginning. To just get started using the Applet, see instead the Instructions for a quick start. In reading this tutorial, click on any highlighted word to jump to the glossary entry; be aware that some words that have common meanings in everyday life are used in very particular ways in thermodynamics, so check the glossary if you don't understand a usage.

This is still under construction (it begins at the beginning but it does not yet reach the end).

This applet demonstrates the geometric constructions that allow a visual interpretation
of some basic principles of phase equilibria, a branch of chemical
thermodynamics. *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.

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 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

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

and ,which allows us to calculate G at any P and T from calorimetric and volumetric data.

The intensive Gibbs free energy of a phase 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
is apparent from a graph of its intensive Gibbs free energy vs. composition. In the demonstration
applet we make the particularly simple (but not very realistic) assumption that all the phases
have quadratic dependence on composition, which one parameter, X_{o}, describing the lowest
energy composition and one parameter, C, describing how easily its composition can be varied.
Putting together this simple composition dependence with the simplest possible pressure and temperature
dependences (i.e., constant entropy and volume for each phase, also not realistic), leads to a
general form for the intensive Gibbs free energy of each phase in the model:

,where G

Note that the adopted form has a constant second derivative
d^{2}/dX^{2} = 2C. For a 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)]/e^{2},
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. So all the phases we consider will have C > 0.
A more complex and realistic expression for (X) would allow for a curve with a
concave-down region between two concave-up intervals, which would lead to
exsolution, the only aspect of binary phase equilibria we are not going to cover here.

Now that we have a binary system 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.

In the binary system, there are two components (say, 1 and 2) and we may write the intensive Gibbs free energy either of the system or of a phase as the sum of the mass fractions of the components times their partial specific Gibbs free energies or chemical potentials:

.Since the two mass fractions sum to unity, however, let us define X = x

,which is the equation of a line in -X space with intercept

Consider the -X diagram in Figure 1. This diagram always shows the properties of
phases as a function of composition (X) at contant temperature and
pressure. In Figure 1 we show the individual (X) curves for two phases, with different
values of G_{o}, C, and X_{o}. If we imagine fixing P, T and X, 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 X_{A(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 X_{B(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 f_{A} and f_{B} of the
system, respectively, and having composition X_{A(B)} and X_{B(A)}. Then the bulk
composition of the system is X = f_{A}X_{A(B)} + f_{B}X_{B(A)}, and
the intensive Gibbs free energy of the system is (X) =
f_{A}^{A}(X_{A(B)}) +
f_{B}^{B}(X_{B(A)}). Now look again at Figure 2: these
two equations define a line segment that connects the two points (X_{A(B)},
^{A}(X_{A(B)})) [when f_{A}=1 and f_{B}=0] and
(X_{B(A)}, ^{B}(X_{B(A)})) [when f_{A}=0 and
f_{B}=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 woul have to be negative.

Now we have what we need to read the stable phase assemblage off a -X
diagram. At any X, the lowest available is represented either by the
(X) curve of a phase or by the interior segment of a tangent line connecting
two such curves. 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 and 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

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).

Now would be a good time to go back to the applet page and 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.

Now, proceed to Page 2 of the tutorial.

Send suggestions, whines, and flames to Paul Asimow.