UNITS, GRAPHS, AND PLOTS
Chapter 1 Kinetics and Materials Science and Engineering 3
1.7 REVIEW OF THERMODYNAMIC CONCEPTS .1 P urPoSe
1.7.8 P HASe r ule AnD P HASe D iAGrAmS
1.7.8.5 Phase Diagrams and Gibbs Energy-Composition Diagrams
One of the most useful things about phase diagrams is that there is a one-to-one correspondence between them and the molar Gibbs energies of the phases that in the diagram. Consider the Gibbs energy versus composition diagram in Figure 1.17. The upper straight line represents the Gibbs energy
* The liquidus is the line separating the liquid phase region from the liquid + solid region. The solidus is the line separating the two-phase liquid plus solid region and the solid region. The solvus is the line separating the two solid phase region from the single-phase solid region. The names for these lines were given in the late 1800s when a Latin sounding name was assumed to lend more scientific credibility to something.
Temperature
A B
Mole % B
α Liquid + β β
Liquid + α
α + β Liquid
20 40 60 80
T5 T4 T3 T2 T1
Point 2 Point 1
Point 3 Liquidus Solidus
Solvus
FIGURE 1.16 Binary phase diagram of two components A and B that exhibits a eutectic and point 3 and terminal solid solutions alpha, α, and beta, β. The Gibbs energy-composition plots in Figure 1.18a–e are constructed at the temperatures labeled 1–5.
A B
Mole % B
20 40 60 80
Temperature T
Gibbs energy (G0, G*) GA0
GB0 G0 = GA0 + XB (GB0 – GA0)
Physical mixture
0
Solution G* = G – G0 = 0
RT ln aA
RT ln aB
X1
FIGURE 1.17 Schematic diagram showing how a new function G* can be calculated that removes the standard state part of the equation for mixing two components A and B. Plotting G*= −G Go versus composition is often convenient since the tangent intercepts at the two vertical axis give µA−µAo and µB−µBo.
of two components A and B that only form a mechanical mixture such as sand and water, salt and pepper, or a polymer and a solid filler material. In these cases, there is no reaction between the two phases, and there is simple linear relation between any molar quantity of A and B, Gin the figure, namely,
G GAo X G G
B Bo Ao
= +
(
−)
(1.49)where the molar Gibbs energies of A and B remain in their standard states. It is sometimes use- ful, however, to remove this part involving the standard states when working with Gibbs energy- composition diagrams, so that it is easier to evaluate the role of solutions. Therefore, if Equation 1.49 is relabeled Goand a new Gibbs function G* is defined as G*= −G Go. Then when the two compo- nents do not form a solution, G* = 0 for all XB, the horizontal dashed line in Figure 1.17. If A and B do form a solution, as shown in the bottom curve of Figure 1.17, then the intersection of the tangent line at a point X1 with the two vertical axes gives RT a and RT aln A ln B, which can be useful, where aA
and aB are the thermodynamic activities of A and B, respectively.
Example: Gibbs Energy-Composition Diagrams for the Simple Eutectic System of Figure 1.16.
From the phase diagram in Figure 1.16, the Gibbs energy, G*, composition diagrams can be con- structed. At T1, the system is liquid for any composition. Therefore, Figure 1.18a shows the G*-composition diagram for this temperature. The liquid has the lowest Gibbs energy of the three
(a)A B
Mole % B
20 40 60 80
Liquid α β
Temperature T1
Gibbs energy (G*)
Phases present = Liquid GA – GA° =
RT In aA
GB – GB° = R T InaB
A
(b) B
Mole % B
20 40 60 80
Liquid
β
α
α α + liquid
Temperature T2
Liquid
Gibbs energy (G*)
Phases present 1
2
FIGURE 1.18 (a) Gibbs energy-composition diagram constructed from the phase diagram in Figure 1.16 at T1. (b) Gibbs energy-composition diagram constructed from the phase diagram in Figure 1.16 at T2. (Continued)
phases (liquid, solid alpha, and solid beta) so it is the stable phase at all compositions with the tan- gent line at some composition giving the G* for A and B at the intersection of the two vertical axes.
At T2, Figure 1.18b, the α phase has a lower energy than the liquid over a finite region. A com- mon tangent line to the two phases, α and liquid, at points 1 and 2, respectively, gives the G* values for A and B again at the intersection of the tangent line with the two vertical axes. Of course, if these two phases are in equilibrium, then the Gibbs energies of both A and B must be the same in both phases; hence, the common tangent. However, between the contacts, points 1 and 2 of the common
Gibbs energy (G*)
(c)A B
Mole % B
20 40 60 80
Liquid
β + liquid β α β
α α + liquid
Temperature T3
Liquid Phases present
1 2 3
4
Gibbs energy (G*)
(d)A B
Mole % B
20 40 60 80
Liquid
β β α
α α + β + liquid Temperature T4
Phases present
1 2 3
Gibbs energy (G*)
A
(e) B
Mole % B
20 40 60 80
Liquid
β β α
α α + β
Temperature T5
Phases present
1 2
FIGURE 1.18 (Continued) (c) Gibbs energy-composition diagram constructed from the phase diagram in Figure 1.16 at T3. (d) Gibbs energy-composition diagram constructed from the phase diagram in Figure 1.16 at T4. (e) Gibbs energy-composition diagram constructed from the phase diagram in Figure 1.16 at T5.
tangent with the two phases, a mixture of two phases with compositions given by the tangent points has a lower energy, the tangent line, than either of the two phases in equilibrium. Therefore, these two tangent points determine the composition range of a two-phase region at this temperature in Figure 1.16. For compositions to the left of the tangent point 1 on the α phase, the α phase has the lowest Gibbs energy and so this tangent point determines the maximum composition of single phase alpha on the phase diagram in Figure 1.16.
At T3, Figure 1.18c, two tangent lines can be drawn: one intersecting the alpha and liquid phases at points 1 and 2, and one intersecting the liquid and beta phases at points 3 and 4. As at T2, G* is lower for the two-phase mixtures of alpha + liquid (points 1 and 2) and liquid + beta (points 3 and 4) between these common tangent points on the tangent lines. Again, the extensions of the tangent lines to the vertical axes give the activities (actually RT ln a) of A and B for the compositions of the three phases at the tangent points. Again, to the left of the α + liquid, two-phase region (point 1) the single phase α has the lowest energy and is the stable phase. Similarly, for compositions to the right of point 4, tangent to the beta phase, beta has the lowest energy and is the stable phase.
At T4, a single tangent line intersects all three phases at points 1, 2, and 3. This is the invariant eutectic temperature where the three phases are in equilibrium so all three must have the same G and G*A
B*
where the tangent line intersects the left and right vertical axes, respectively. To the left of point 1, alpha has the lowest Gibbs energy and is the stable phase while to the right of point 3, beta has the lowest energy and is the stable phase.
Finally, at T5, both solid phases are lower in energy than the liquid as given by the tangent line between them intersecting at points 1 and 2. Between points 1 and 2, the tangent has the lowest energy so these two points give the limits of the two-phase α + β region. To the left of point 1, α has the lowest energy and is the stable phase while to the right of point 2, β has the lowest Gibbs energy and is the stable phase.
Conversely, the phase diagram shown in Figure 1.16 could be constructed from the Gibbs energy- composition diagrams in Figure 1.18. This example demonstrates the very close relationship between Gibbs energy-composition diagrams and phase diagrams. It is possible to create either from the other.
Understanding this relationship is invaluable in being able to use phase diagrams to initiate develop- ment of kinetic models for the rates of reactions and phase transitions.