THE PHASE RULE

CHAPTER 13 THE PHASE RULE

The mathematical basis of classic thermodynamics was developed by J. Willard Gibbs in his essay [1], “On the Equilibrium of Heterogeneous Substances,” which builds on the earlier work of Kelvin, Clausius, and Helmholtz, among others. In par- ticular, he derived the phase rule, which describes the conditions of equilibrium for multiphase, multicomponent systems, which are so important to the geologist and to the materials scientist. In this chapter, we will present a derivation of the phase rule and apply the result to several examples.

13.1 DERIVATION OF THE PHASE RULE The phase rule is expressed in terms of ‘, the number of phases in the system; C, the

number of components; and F, the number of degrees of freedom or the variance of the system.

The number of phases is the number of different homogeneous regions in the system. Thus, in a system containing liquid water and several chunks of ice, only two phases exist. The number of degrees of freedom is the number of intensive vari- ables that can be altered freely without the appearance or disappearance of a phase. First we will discuss a system that does not react chemically, that is, one in which the number of components is simply the number of chemical species.

Chemical Thermodynamics: Basic Concepts and Methods, Seventh Edition . By Irving M. Klotz and Robert M. Rosenberg Copyright # 2008 John Wiley & Sons, Inc.

THE PHASE RULE

Nonreacting Systems If we express the composition of a phase in terms of the mole fractions of all the

components, then (C 2 1) intensive variables are needed to describe the composition, if every component appears in the phase, because the mole fractions must sum to 1. In

a system of ‘ phases, ‘(C 2 1) intensive variables are used to describe the compo- sition of the system. As was pointed out in Section 3.1, a one-phase, one-component

system can be described by a large number of intensive variables; yet the specification of the values of any two such variables is sufficient to fix the state of such a system. Thus, for example, two variables are needed to describe the temperature and pressure of each phase of constant composition or any alternative convenient choice of two intensive variables. Therefore, the total number of variables needed to describe the state of the system is

(13 :1) To calculate the number of degrees of freedom, we need to know the number of

‘ (C

constraints placed on the relationships among the variables by the conditions of equilibrium.

Mechanical Equilibrium. For a system of fixed total volume and of uniform temperature throughout, the condition of equilibrium is given by Equation (7.9) as

dA ¼0

If phase I of the system changes its volume, with a concurrent compensating change in the volume of phase II, then at constant temperature, it follows from Equation (7.39) that

dA ¼ dA I þ dA II

(13 :2) As the total volume is fixed,

(13 :3) The equilibrium constraint of Equation (13.3) can be met only if P I ¼P II , which is

the condition for mechanical equilibrium. (We will discuss several special cases to which this requirement does not apply.) Or, to put the argument differently, if the pressures of two phases are different, the phase with the higher pressure will spon- taneously expand; and the phase with the lower pressure will spontaneously contract, with a decrease in A, until the pressures are equal. Thus, for ‘ phases, ‘ 2 1 inde- pendent relationships among the pressures of the phases can be written as follows:

P I ¼P II ‘

305 Thermal Equilibrium. For an isolated system, one at constant total energy and

13.1 DERIVATION OF THE PHASE RULE

constant total volume, the condition of equilibrium follows from Equation (6.108) as

(13 :4) If an infinitesimal amount of heat DQ is transferred reversibly from phase I to phase

dS ¼0

II, it follows from Equation (6.49) that

The constraint of Equation (13.5) can be met only if T I ¼T II , which is the condition for thermal equilibrium. Or, to put the argument differently, if the temperatures of two phases differ, heat will flow irreversibly from the phase at higher temperature to the phase at lower temperature, with an increase in entropy, until the temperatures are equal. Thus, for ‘ phases, ‘ 2 1 independent relationships among the tempera- tures of the phases can be written as follows:

T I ¼T II ‘

Transfer Equilibrium. For a system at constant temperature and pressure, the con- dition of equilibrium is given from Equation (7.18) as

(13 :6) If dn moles of a substance are transferred from phase I to phase II, then it follows from

dG ¼0

Equation (9.15) that

dG I dn þm II dn ¼0

(13 :7) The condition of Equation (13.7) can be met only if m I ¼m II , which is the condition

of transfer equilibrium between phases. Or, to put the argument differently, if the chemical potentials (escaping tendencies) of a substance in two phases differ, spon- taneous transfer will occur from the phase of higher chemical potential to the phase of lower chemical potential, with a decrease in the Gibbs function of the system, until the chemical potentials are equal (see Section 10.5). For each component present in all ‘ phases, (‘ 2 1) equations of the form of Equation (13.7) provide constraints at transfer equilibrium. Furthermore, an equation of the form of Equation (13.7) can

be written for each one of the C components in the system in transfer equilibrium between any two phases. Thus, C(‘ 2 1) independent relationships among the

chemical potentials can be written. As chemical potentials are functions of the mole fractions at constant temperature and pressure, C(‘ 2 1) relationships exist

among the mole fractions. If we sum the independent relationships for temperature,

THE PHASE RULE

pressure, and composition in the system, we find (‘

(13 :8) independent relationships or constraints exist among the variables.

The Phase Rule. The number of degrees of freedom is the difference between the number of variables needed to describe the system and the number of independent relationships or constraints among those variables:

(13 :9) In a system in which one component is absent from a phase, the number of variables

needed to describe the system decreases by one. As the number of independent relationships also decreases by one, the number of degrees of freedom remains the same.

Reacting Systems For a system undergoing R independent chemical reactions among N chemical

species, R equilibrium expressions are to be added to the relationships among the intensive variables. From Equation (13.1), the total number of intensive variables in terms of N becomes

(13 :10) If we add R to the number of independent relationships specified by Equation (13.8),

‘ (N

we obtain

(13 :11) Thus, the number of degrees of freedom for a reacting system is

(N

F (13 :12)

We define the number of components in a system as N 2 R, which is also the minimum number of chemical species from which all phases in the system can be

prepared. Each equilibrium relationship decreases by one the number of species required to prepare a phase. Thus, the quantity (N 2 R) in Equation (13.12) is equiv-

alent to C in Equation (13.9). For example, water in equilibrium with its vapor at room temperature and atmospheric pressure is a one-component system. Water in

equilibrium with H 2 and O 2 at a temperature and pressure at which dissociation

13.2 ONE-COMPONENT SYSTEMS

307 takes place is a two-component system unless the mole ratio of H 2 /O 2 is exactly 2;

2 then C ¼ 1. Water in equilibrium with OH þ and H ions at room temperature and atmospheric pressure is a one-component system because the requirement for electri-

cal neutrality in ionic solutions imposes an additional relationship on the system.

13.2 ONE-COMPONENT SYSTEMS The number of degrees of freedom for a one-component system [C ¼ 1 in Equation

(13.9)] is

F (13 :13)

F is at most two because the minimum value for ‘ is one. Thus, the temperature and pressure can be varied independently for a one-component, one-phase system and the system can be represented as an area on a temperature versus pressure diagram.

If two phases of one component are present, only one degree of freedom remains, either temperature or pressure. Two phases in equilibrium are represented by a curve on a T 2 P diagram, with one independent variable and the other a function of the first. When either temperature or pressure is specified, the other is determined by the Clapeyron Equation (8.9). If three phases of one component are present, no degrees of freedom remain, and the system is invariant. Three phases in equilibrium are represented on a T 2 P diagram by a point called the triple point. Variation of either temperature or pressure will cause the disappearance of a phase.

An interesting example of a one-component systems is SiO 2 , which can exist in five different crystalline forms or as a liquid or a vapor. As C ¼ 1, the maximum number

of phases that can coexist at equilibrium is three. Each phase occupies an area on the T 2 P diagram; the two-phase equilibria are represented by curves and the

three-phase equilibria by points. Figure 13.1 (2, p. 123), which displays the equili- brium relationships among the solid forms of SiO 2 , was obtained from calculations of the temperature and pressure dependence of DG (as described in Section 7.3) and from experimental determination of equilibrium temperature as a function of equi- librium pressure.

A one-phase system that is important in understanding the geology of diamonds as well as the industrial production of diamonds is that of carbon, which is shown in Figure 13.2. The phase diagram shows clearly that graphite is the stable solid phase at low pressure. Thus, diamond can spontaneously change to graphite at atmos- pheric pressure (

5 Pa). Diamond owners obviously need not worry, however; the transition in the solid state is infinitely slow at ordinary temperatures. Some uncertain-

ties about the phase diagram for carbon are discussed by Bundy [3]. Although phase diagrams such as Figure 13.1 and Figure 13.2 describe the con- ditions of T and P at which different phases are stable, they do not describe the prop- erties of the system. As the specification of two intensive variables is sufficient to fix all other intensive variables, the variation of any other intensive variable can be described in terms of a surface above the T 2 P plane, and the height of any point

THE PHASE RULE

Figure 13.1. Phase diagram for SiO 2 for a range of pressures and temperatures. From data in the sources in Ref. 2 (p. 123).

Figure 13.2. Phase diagram for carbon. From C. G. Suits, Am. Sci. 52, 395 (1964).

13.3 TWO-COMPONENT SYSTEMS

Figure 13.3. A P – V – T surface for a one-component system in which the substance contracts on freezing, such as water. Here T 1 represents an isotherm below the triple-point temperature,

T 2 represents an isotherm between the triple-point temperature and the critical temperature, T c

is the critical temperature, and T 4 represents an isotherm above the triple-point temperature. Points g, h, and i represent the molar volumes of solid, liquid, and vapor, respectively, in equi- librium at the triple-point temperature. Points e and d represent the molar volumes of solid and

liquid, respectively, in equilibrium at temperature T 2 and the corresponding equilibrium pressure. Points c and b represent the molar volumes of liquid and vapor, respectively, in equi- librium at temperature T 2 and the corresponding equilibrium pressure. From F. W. Sears and G. L. Salinger, Thermodynamics, Kinetic Theory, and Statistical Thermodynamics. 3rd ed., Addison-Wesley, Reading, MA, 1975, p. 31.

in the surface about the T 2 P plane represents the value of the intensive variable. Figure 13.3 shows such a surface for the molar volume V m as a function of T and P.

When two or more phases are present at equilibrium, V m is a multivalued function of T and P. Similar surfaces can be constructed to describe other thermodynamic properties, such as G, H, and S, relative to some standard value.

13.3 TWO-COMPONENT SYSTEMS From Equation (13.9), it follows that the number of degrees of freedom for a two-

component system

F (13 :14)

THE PHASE RULE

has a maximum value of three. As a complete representation of such a system requires three coordinates, we can decrease the variance by fixing the temperature and leaving pressure and composition as the variables of the system, or by fixing the pressure and leaving temperature and composition as the variables of the system. Then,

F (13 :15) In a reduced-phase diagram for a two-component system, F ¼ 2 for a single phase and

an area is the appropriate representation. F ¼ 1 for two phases in equilibrium, and a curve that relates the two variables is the appropriate representation. As the compo-

sition of the two phases generally is different, two conjugate curves are required. Figure 13.4 [4] is a reduced two-component diagram for the mineral feldspar, which is a solid solution of Albite (NaAlSi 3 O 8 ) and anorthite (CaAlSi 2 O 8 ). Above the liquidus curve, the system exists as a single liquid phase. Between the two curves, liquid and solid phases are in equilibrium, and their compositions are given by the intersections of a constant temperature line with liquidus and solidus curves.

A point in the region between the two curves represents only the overall composition of the systemm and not the composition of either phase. This value is not described by the phase rule, which is concerned omly with the number of phases and their com- position. Although the region between the curves is frequently called a “two-phase area,” only the curves correspond geometrically to a value of F ¼ 1 [5].

Other two-component systems may exhibit either limited solubility or complete insolubility in the solid state. An example with limited solubility is the silver – copper system, of which the reduced-phase diagram is shown in Figure 13.5.

Region L represents a liquid phase, with F ¼ 2, and S 1 and S 3 represent solid-solution phases rich in Ag and Cu, respectively, so they are properly called “one-phase areas.”

S 2 is a two-phase region, with F ¼ 1, and the curves AB and DF represent the compositions of the two solid-solution phases that are in equilibrium at any

Figure 13.4. Phase diagram of the system feldspar, a solid solution of albite (NaAlSi 3 O 8 ) and anorthite (CaAlSi 2 O 8 ). Data from Ref. 4.

13.3 TWO-COMPONENT SYSTEMS

Figure 13.5. The phase diagram for the system Ag-Cu at constant pressure. With permission, from R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, and K. K. Kelley, Selected Values of Thermodynamic Properties of Binary Alloys , American Society of Metals, Metals Park, OH, 1973, p. 46.

temperature. At 1052 K, liquid of composition C is in equilibrium with solid sol- utions of composition B and D. With three phases, F ¼ 0, and the three compositions

are represented by points B, C, and D. Between 1052 K and 1234 K, lines EB and EC represent the compositions of solid solution and liquid solution, respectively, in equi- librium, with overall system composition richer in Ag than at C. Curves are appropri- ate because F ¼ 1. Between 1052 K and 1356.55 K, lines GC and GD represent the compositions of liquid solution and solid solution, respectively, in equilibrium, with overall system composition richer in Cu than at C.

In two-component phase diagrams such as Figure 13.5 and Figure 13.6, the dis- tinction between two-phase regions and one-phase regions is made on the basis of the number of degrees of freedom represented by each. The one-phase region has two degrees of freedom, two variables can be varied freely, and the region is truly an “area.” The two-phase region has only one degree of freedom, only one variable can be varied freely, and the second variable is a function of the first, with the func- tion represented by a curve. Because two phases of different compositions in equili- brium exist, two different curves exist, which bound a region. The two kinds of region can be distinguished because the two-phase region is bounded by curves that rep- resent a function; therefore, the curves must have continuous first derivatives in the range of temperature characteristic of the region. The same is not required of the boundaries of one-phase regions, which are true areas and are not bounded by curves that represent functions [5].

THE PHASE RULE

Figure 13.6. The phase diagram at constant pressure of the system Na-K. With permission, from R. Hultgren, D. T. Hawkins, M. Gleiser, and K. K. Kelley, Selected Values of Thermodynamic Properties of Binary Alloys , American Society of Metals, Metals Park, OH, 1973, p. 1057.

Figure 13.6 is the phase diagram of the system Na-K at constant pressure, which is

a system that exhibits limited solubility and the occurrence of a solid compound Na 2 K (b), which melts at 280.06 K to form a solid solution of composition rich in Na and liquid of composition H.

Although the phase rule is concerned with the number of relationships among system variables that are represented by the equilibrium curves, it provides no infor- mation about the nature of those relationships. We will consider the dependence of the chemical potential on the system variables for various systems in later chapters.

Two Phases at Different Pressures When we derived the phase rule, we assumed that all phases are at the same pressure.

In mineral systems, fluid phases can be at a pressure different from the solid phases if the rock column above them is permeable to the fluid. Under these circumstances, the system has an additional degree of freedom and the equilibrium at any depth depends

on both the fluid pressure P F and the pressure on the solid P S at that level. Each pressure is determined by r, the density of the phase, and h, the height of the column between the surface and the level being studied.

The equations required to calculate the effect of pressure and temperature on DG are modified from Equation (7.43) to include a term for each pressure at any tempera- ture T. For example, for the gypsum – anhydrite equilibrium,

CaSO 4 2 O(s, P S , T) ¼ CaSO 4 (s, P S , T) þ 2H 2 O(l, P F , T) (13 :16) gypsum

anhydrite

313 the change in the Gibbs function can be computed by resolving the transformation

13.3 TWO-COMPONENT SYSTEMS

into the following four steps: Step 1.

CaSO 4 2 O(s, P ¼ P8, T) ¼ CaSO 4 (s, P ¼ P8, T) þ 2H 2 O(l, P ¼ P8, T),

DG m ¼ DG m (P ¼ P8, T)

Step 2. CaSO 4 2 O(s, P s , T) ¼ CaSO 4 2 O(s, P ¼ P8, T),

DG m ¼V m,CaSO 4 2 O(s) (P8 S ) S V m,CaSO 4 2 O(s)

Step 3. CaSO 4 (s, P ¼ P8, T) ¼ CaSO 4 (s, P s , T),

DG m ¼V m,CaSO 4 (s) (P s

ffiP S V m,CaSO 4 (s)

Step 4. 2H 2 O(l, P ¼ P8, T) ¼ 2H 2 O(l, P F , T),

DG m ¼ 2V m,H 2 O(l) (P F ffi 2P F V m,H 2 O(l)

The approximations for Steps 2 through 4 are reasonable because P8, atmospheric pressure, is small compared with P F and P S , the high pressures found in geologic formations. The sum of the transformations in Steps 1 through 4 leads to the change shown in Equation (13.16). The sum of the DG m ’s for Steps 1 through 4 produces Equation (13.17)

DG m (P F ,P S , T) ¼ DG m (P ¼ P8, T) þ P S (DV mS ) þP F (DV mF ) (13 :17) in which DV mS represents the molar volume change of the solid phases in the trans-

formation and DV mF represents the molar volume change of the fluid phase in the transformation. (See Table 2.1 for the definition of one mole for a chemical transformation.) That is,

DV mS ¼V m,CaSO 4 (s)

m,CaSO 4 2 O(s)

3 mol (13 :18)

THE PHASE RULE

and

m,H 2 O(l) ¼ 36:14 cm mol (13 :19) The equilibrium diagram (2, p. 274; 6) for the species in Equation (13.16) is

DV m,F ¼ 2V

shown in Figure 13.7. If gypsum and anhydrite are both under liquid water at 1 bar, then equilibrium can be attained only at 408C (see Fig. 13.7). If the liquid pressure is increased, and the rock formation is completely impermeable to the liquid phase, so that the pressure on the fluid phase is equal to the pressure on the solid phase, then the temperature at which the two solids, both subject to this liquid pressure, are in equilibrium is given by the curve with positive slope on the right side of Figure 13.7. Thus, the right curve applies to any situation in which P F is equal to P S . Under these conditions, the net DV m for the transformation of

Equation (13.16) is 36.14 2 29.48 ¼ 6.66 cm 3 [see Equations (13.18) and (13.19)], and 6.66

F makes a positive contribution to the DG m of Equation (13.17). Thus, an increase in pressure should shift the equilibrium from anhydrite to gypsum, as indicated in Figure 13.7.

Figure 13.7. Equilibrium diagram for transformations in Equation (13.16). Gibbs function data from K. K. Kelley, J. Southard, and C. T. Anderson, Bureau of Mines Technical Paper 625 , 1941.

315 In contrast, if the rock above the layer being studied is completely permeable to the

13.3 TWO-COMPONENT SYSTEMS

fluid, the pressure on the solid phases is that of the overlying rock r s hg , whereas the pressure on the fluid phase is r F hg , in which g is the acceleration caused by gravity. Under these circumstances, the equilibrium temperature for the transformation in Equation (13.16) varies with pressure according to the curve with negative slope at

S ) exceeds P F DV mF (36.14

the left side of Figure 13.7. In this case, P S DV mS

F ) in magnitude, and the net contribution to DG of the P DV terms in Equation (13.17) is negative. Hence, increased pressure shifts the equilibrium from gypsum to anhydrite. If the rock is partially permeable, the equilibrium curve falls

between the two curves shown, the exact position depending on the ratio of P F to P S . At some value of the ratio between unity and r F /r S , the equilibrium temperature becomes independent of the pressure. Whatever the position of the equilibrium curve, gypsum is the stable solid phase at low temperatures (to the left of the curve) and anhydrite is the stable solid phase at high temperatures (to the right of the curve).

Phase Rule Criterion of Purity Equation (13.9) is written as if the number of degrees of freedom of a system were

calculated from known values of the number of phases and the number of com- ponents. In practice, an experimentalist often determines F and ‘ from his or her observation and then calculates C, the number of components.

The determination of the purity of a homogeneous solid from solubility measure- ments is an example of this application of the phase rule. The experimental procedure

Figure 13.8. Solubility curves of chymotrypsinogen A in two different solvents. Adapted from J. H. Northrop, M. Kunitz, and R. M. Herriot, Crystalline Enzymes, 2nd ed., Columbia University Press, New York, 1948. Originally published in J. Gen. Physiol. 24, 196 (1940); reproduced by copyright permission of the Rockefeller University Press.

THE PHASE RULE

is to measure the concentration of dissolved material in equilibrium with excess solid, at a fixed temperature and pressure, as a function of the amount of solid added. If the solid is pure, the solid and solvent constitute a two-component system. At constant temperature and pressure, F ¼ 2 2 ‘ and a two-phase system of solid and saturated solution has zero degrees of freedom. If the solid contains more than one species, the system has three components and F ¼ 1 when two phases are present. In the former case, the solubility is independent of the amount of excess solid; in the latter case, the solubility increases with the amount of added solid. Figure 13.8 shows the data of Butler [7] on the solubility of chymotrypsinogen A, which was the precursor of the pancreatic enzyme chymotrypsin. These data represent one of the earliest rigorous demonstrations of the purity of a protein.

EXERCISES

13.1. What would be the number of degrees of freedom in a system in which pure

H 2 O was increased to a temperature sufficiently high to allow dissociation into H 2 and O 2 ?

13.2. At atmospheric pressure (101.325 kPa), a quartz is in equilibrium with b quartz at 847 + 1.5 K [8]. The enthalpy change of the transition from a

quartz to b quartz is 728 + 167 J mol 21 . Berger et al. [9] measured the volume change by an X-ray diffraction method, and they reported a value of 0.154 + 0.015 cm 3 mol 21 . Use these values to calculate the slope at atmos- pheric pressure of the equilibrium curve between a quartz and b quartz in Figure 13.1. Compare your results with the value of the pressure derivative

of the equilibrium temperature dT /dP, which is equal to 0.21 K (Mpa) 21 [10].

13.3. For Equation (13.16) [2] DG m (P

þ (685:8 J mol K )T The value of DV

mol 21 and the value of DV

¼ 36.14 cm (2H 2 O). Calculate the ratio of P S to P F at which DG m (P F ,P S , T ) is

mol 21

independent of pressure.

13.4. In Figure 13.6, identify the phases in equilibrium and the curves that describe the composition of each phase in Regions I, II, III, IV, V, VI, VII, and VIII. Identify the phases in equilibrium and the composition of those phases at 260.53 K along line BDE and at 280.66 K along line HGM.

REFERENCES 1. J. Willard Gibbs, Trans. Conn. Acad. Sci. 3, 108 – 248 (1876); 3, 343-542 (1878).

Reprinted in The Collected Works of J. Willard Gibbs, Vol. I, Yale University Press, New Haven, CT, 1957.

2. R. Kern and A. Weisbrod, Thermodynamics for Geologists, Freeman, Cooper and Co., San Francisco, CA, 1967.

3. F. P. Bundy, J. Geophys. Res. 85B, 6930 (1980).

317 4. N. L. Bowen and J. R. Schairer, Am. J. Sci. 229, 151 – 217 (1935).

REFERENCES

5. R. M. Rosenberg, J. Chem. Educ. 76, 223 (1999). 6. G. J. F. McDonald, Amer. J. Sci. 251, 884 – 898 (1953). 7. J. A. V. Butler, J. Gen. Physiol. 24, 189 (1940); M. Kunitz and J. H. Northrop, Compt.

Rend. Trav. Lab. Carlsberg, Ser. Chim. 22 , 288 (1938). Adapted from J. Gen. Physiol. 1940, 24, p. 196, by copyright permission of the Rockefeller University Press.

8. M. W. Chase, Jr., C. A. Davies, J. R. Downey, Jr., D. J. Frurip, R. A. McDonald, and A. N. Syverud, JANAF thermochemical tables, 3rd ed., J. Phys. Chem. Ref. Data 14, Supplement No. 1, 1675 (1985).

9. C. Berger, L. Eyraud, M. Richard, and R. Riviere, Bull. Soc. Chim. France 628, 1966. 10. R. E. Gibson, J. Phys. Chem. 32, 1197 (1928).