Thermodynamics of Condensed Phases
2.1 THERMODYNAMICS OF METALS AND ALLOYS
2.1.1 Phase Equilibria in Single-Component Systems
Table 2.1 Summary of Free Energy Effects on Process Spontaneity
G <0 Process proceeds spontaneously G >0 Process not spontaneous
G=0 Process at equilibrium
2.0.5 Chemical Potential
The final thermodynamic quantity for review is thechemical potential, which is rep- resented with the Greek letter mu,µ. The chemical potential can be defined in terms of the partial derivative of any of the previous thermodynamic quantities with respect to the number of moles of speciesi,ni, at constantnj (wherej indicates all species other thani) and thermodynamic quantities as indicated:
µi = ∂U
∂ni
S,V ,nj
= ∂H
∂ni
P ,S,nj
= ∂G
∂ni
T ,P ,nj
(2.13) The advantage of the chemical potential over the other thermodynamic quantities, U, H, andG, is that it is an intensive quantity—that is, is independent of the number of moles or quantity of species present. Internal energy, enthalpy, free energy, and entropy are all extensive variables. Their values depend on the extent of the system—that is, how much there is. We will see in the next section that intensive variables such asµ, T, andP are useful in defining equilibrium.
THERMODYNAMICS OF METALS AND ALLOYS 141
chemical characteristics. It need not be continuous. For example, a carbonated beverage consists of two phases: the liquid phase, which is continuous, and the gas phase, which is dispersed in the liquid phase as discrete bubbles. For the current discussion, it is easiest to visualize the two phases as a solid and a liquid, respectively, since we are pretty familiar with the processes of melting and solidification, but these phases could be two solids, two liquids, a liquid and a gas, or even a solid and a gas.
Phase α and phase β are in equilibrium with one another. What does this mean?
First of all, it means that although there is probably an exchange of atoms between the two phases; (i.e., some of the solid phaseαis melting to formβ, and some of the liquid phase β is solidifying to form α) these processes are occurring at essentially equal rates such that the relative amounts of each phase are unchanged. This is known as adynamic equilibrium. In terms of intensive variables, equilibrium means that
Tα =Tβ (2.14)
Pα =Pβ (2.15)
µα =µβ (2.16)
where Tα and Pα are the temperature and pressure of the solid phase α; Tβ and Pβ are the temperature and pressure of the liquid phase β, andµα and µβ are chemical potentials of each phase. Thus, we have six intensive quantities that establish equilib- rium between the two phases. However, these six variables are not all independent.
Changing any one of them can affect the others. This can be shown mathematically by assuming that the chemical potential is a function of temperature and pressure:
µα =µα(Tα, Pα) (2.17)
µβ =µβ(Tβ, Pβ) (2.18)
So, for a one-component system containing two phases in equilibrium, we have three thermodynamic conditions of equilibrium [Eqs. (2.14)–(2.16)] and four unknown parameters,Tα,Pα,Tβ, andPβ. If we arbitrarily assign a value to one of the parameters, we can solve for the other three (three equations, three unknowns).
Let us extend this analysis to the general case ofC independent, nonreacting com- ponents, so that we might arrive at a very useful, general conclusion. Instead of one component, we now have C; and instead of two phases (liquid and solid), we now have an arbitrary number, φ. The conditions of equilibrium are now analogous to Eqs. (2.14)–(2.16):
Tα=Tβ=Tγ =. . . Tφ (2.19) Pα =Pβ =Pγ =. . . Pφ (2.20) µ1α =µ1β =µ1γ =. . . µ1φ
µ2α =µ2β =µ2γ =. . . µ2φ (2.21) ...
µCα=µCβ =µCγ =. . . µCφ
There are a total of (φ−1) equalities for temperature in Eq. (2.19), (φ−1) equalities for pressure in Eq. (2.20), and (φ−1) equalities for chemical potential of each of the
components,C in Eq. (2.21). There areC equations in (2.21), so the total number of equations is (C+2). Thus, we have (φ−1)(C+2) total equalities, or restrictions, to be satisfied for equilibrium. The total number of intensive variables (excluding com- position) in the system is φ(C+2). The difference between the number of intensive variables and the number of independent restrictions is known as thedegrees of free- dom of the system, F. The degrees of freedom are the number of variables (including composition) that must be specified in order for the system to be defined in a strict, mathematical sense. So,
degrees of freedom=number of intensive variables
−number of independent restrictions
F =φ(C+1)−(φ−1)(C+2) (2.22)
F =C−φ+2 (2.23)
Equation (2.23) is a very important result. It is known as the Gibbs Phase Rule, or simply the “phase rule,” and relates the number of components and phases to the number of degrees of freedom in a system. It is a more specific case of the general case forN independent, noncompositional variables
F =C−φ+N (2.24)
We utilized only temperature and pressure as independent, noncompositional variables in our derivation (N=2), which are of the most practical importance.
Return now to the case of a single component and two phases, C=1 andφ=2, so that the phase rule for our element is
F =1−2+2=1 (2.25)
This means that when phasesα and β coexist at equilibrium, only one variable may be changed independently. For example, if temperature is changed, pressure cannot be changed simultaneously without affecting the balance of equilibrium. In those cases where F =2, both temperature and pressure can be changed without affecting the balance of equilibrium (number of phases present); and when F =0, none of the intensive variables can be changed without altering equilibrium between phases.
2.1.1.2 Unary Phase Diagrams. The phase rule provides us with a powerful tool for connecting the thermodynamics of phase equilibria with graphical representations of the phases known asphase diagrams.Phase diagrams tell us many things, but at a minimum, describe the number of components present, the number of phases present, the composition of each phase, and the quantity of each phase. There are several ways in which this information can be presented. We start with the free energy.
Take, for example, the plot ofGversus temperature for elemental sulfur, represented by the bottom diagram in Figure 2.1. We know from experiments and observation that there are four phases we have to consider for sulfur: two solid forms (a low-temperature orthorhombic form, R, and high-temperature monoclinic form, M), liquid (L), and vapor (or gas, V). The lines ofG versus T for each phase, which are partially solid and continue on as dashed lines, are constructed at constant pressure using Eq. (2.10),
THERMODYNAMICS OF METALS AND ALLOYS 143
P P′
at P′
a b
(M) (R)
(L)
(V)
T
c d e
(V) (L)
(M)
(R) a
c b
d
e
T G
Figure 2.1 Unary phase diagram (top) and Gibbs free energy plot (bottom) for elemental sulfur.
Reprinted, by permission, from D. R. Gaskell,Introduction to Metallurgical Thermodynamics, 2nd ed., p. 178, Copyright1981 by Hemisphere Publishing Corporation.
where each phase has a different enthalpy and entropy associated with it. The solid line in this figure represents the minimum free energy, regardless of which phase it represents, at a specified temperature. Hence, at this pressure (which is unspecified, but fixed atP), the orthorhombic form, R, is the most stable at low temperatures. As temperature is increased, monoclinic (M) becomes more stable at pointa, followed by the liquid, L, at point c, and finally the vapor phase (V) at pointe.
Plots of free energy as a function of temperature can be made at any pressure (in theory), but they are not particularly useful in and of themselves. Free energy is something we cannot generally measure directly, but temperature and pressure are.
Hence, plots of two intensive variables, such as pressure and temperature, are much more practical. Such plots are produced by simply translating the information in theG versusT plot at various pressures onto aP versusT plot, as shown in the top diagram of Figure 2.1. Point a in the bottom figure now becomes a point on the solid–solid equilibrium line between the orthorhombic and monoclinic forms of sulfur. Pointbis distinctly in the monoclinic phase field, M, since this form has the lowest free energy at this temperature and pressure. Point c is also on an equilibrium phase boundary between the monoclinic and liquid, L, forms of sulfur. Note that this is the melting (or solidification) point of sulfur at pressureP. Point d is distinctly in the liquid region, and point e is on the liquid–vapor equilibrium line. These plots of pressure versus temperature for a single-component system are calledunary phase diagrams. There is no limit to the number of phases that may be present on a unary diagram. The only constraint is that there be only one component—that is, no chemical transformations are taking place. Although our discussion here is limited to elements and metals, keep in mind that we are not limiting the number of atoms that make up the component, only the number of components. For example, water is a single component that is composed of hydrogen and oxygen. In a unary phase diagram for water, only the phase transformations of that component are shown—for example, ice, water, and steam. Two hydrogens are always bound to one oxygen. For now, we will continue to limit our discussion to metals.
The power of the phase rule is illustrated in a second example. Consider theT–P phase diagram for carbon shown in Figure 2.2. First of all, notice that pressure is plot- ted as the independent variable, instead of the usual temperature variable, but given what we know about the effects of pressure on carbon, this makes sense. Let us first examine the equilibrium between two solid forms of carbon: graphite and diamond.
This equilibrium is shown graphically by the line A–B in Figure 2.2. Application of Eq. (2.23) to any point on this line, or phase boundary, results in F =1 (C=1 component, carbon; φ=2, graphite and diamond). This means that any change in temperature requires a corresponding change in pressure to maintain this equilibrium.
Temperature and pressure are not independent. Similarly, the phase boundaries rep- resented by curves B–C (diamond–liquid), C–D (diamond–metallic carbon), C–E (metallic carbon–liquid), and J–B (graphite–liquid) represent lines ofF =1.
Any point within a single-phase region (φ=1) results in F =2. Point K is such a point. It is located in the diamondphase field, where both temperature and pressure can be changed independently without creating or destroying the phase. No equilibrium exists here—there is only one phase.
Since F =2 results in an area (phase field), and F =1 results in a line (phase boundary), we can predict that F =0 should occur at a point, and indeed it does.
Point B is such a point (φ=3; graphite, diamond, and liquid; C=1, carbon), as is
THERMODYNAMICS OF METALS AND ALLOYS 145
4000
3000
2000
1000
00 200 400 600 800
5000
K
A D
Diamond C
E B
J
Graphite
Liquid
Metallic carbon
P (katm)
T (K)
Figure 2.2 Temperature– Pressure unary phase diagram for carbon. From K. M. Ralls, T. H. Courtney, and J. Wulff, Introduction to Materials Science and Engineering. Copyright
1976 by John Wiley & Sons, Inc. This material is used by permission of John Wiley &
Sons, Inc.
point C. There are no degrees of freedom at this point—any variation in temperature or pressure will result in movement into a distinctly separate phase field, and at least one of the phases must necessarily be lost. In the cases where the three phases in coexistence are solid, liquid, and vapor, thisinvariant point is known as thetriple point.