Here begins the Tutorial in Binary phase equilibria to accompany the demonstration applet

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).

## Tutorial

### 1. What's this all about?

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.

### 2. First and second laws, equilibrium, Gibbs free energy

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 < 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 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, Xo, 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 Go, Vo and So are the (constant) reference Gibbs free energy, volume, and entropy, repsectively. This form was adopted not because it has any basis in the thermodynamics of solutions (it does not) but because it is analytically very tractable and all the phase relations in the applet can be found algebraically.

Note that the adopted form has a constant second derivative d2 /dX2 = 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)]/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. 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.

### 3. Chemical potential

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 = 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 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 the Applet uses to find equilibria.

### 4. The graphical representation of equilibrium conditions 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 Go, C, and Xo. 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 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) = fA A(XA(B)) + fB B(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 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 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).

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.