First-Order Phase Transitions in Single Component Systems

FIRST-ORDERPHASE TRANSITIONS

9.I FIRST-ORDER PHASE TRANSITIONS IN SINGLE COMPONENT SYSTEMS

Ordinary water is liquid at room temperatureand atmosphericpressure, but if cooledbelow 273.15K it solidifies;and if heatedabove 373.15 K it vaporizes.At each of these temperaturesthe material undergoesa pre- cipitous changeof properties-a "phase transition." At high pressures witer undergoesseveral additional phasetransitions from one solid form to another.These distinguishable solid phases,designated as "ice 1," "ice II," "ice fII," . . ., differ in crystalstructure and in essentiallyall thermodynamic properties (such as compressibility,molar heat capacity, and variousmolar potentialssuch as u or f). The "phasediagram" of wateris shownin Fig. 9.1. Each transition is associatedwith a linear region in the thermodynamic fundamentalrelation (such as BHF in Fig. 8.2),and eachcan be viewed as the result of failure of the stability criteria (convexity or concavity)in the underlying fundamentalrelation. In this section we shall consider systemsfor which the underlying fundamental relation is unstable.By a qualitativeconsideration of fluctuations in such systemswe shall see that the fluctuations are profoundly influencedby the detailsof the underlyingfundamental relatio,n. In contrast, the auerageualues of the extensiueparameters reflect only the stablethermo- dynamicfundamental relation. Consideration of the manner in which the form of the underlying fundamental relation influencesthe thermodynamicfluctuations will pro- vide a physical interpretation of the stability considerationsof Chapter 8 and of the construction of Fig. 8.2 (in which the thermodynamicfunda- mental relation is constructedas the envelopeof tangentplanes). A simple mechanicalmodel illustrates the considerationsto follow by an intuitively transparentanalogy. Consider a semicircularsection of pipe, closed at both ends.The pipe standsvertically on a table, in the form of

215 216 First- 0rder Phase Transitions

Criticalpoint 7..= 374.14"C P.,,= 22.09 MPa ll lzo II I 20 F18 ;16 dl5 t4 Liquid I2

10 -5

0 -50 0 50 t00 Gas Temperature('C) ---> tnc I ''0,'o il | ttl ZUU

7("C)---> FIGURE 9 Phasediagram of water. The region of gas-phasestability is representedby an indiscerni- bly narrow horizontal strip above the positive temperatureaxis in the phase diagram (small figure). The backgroundgraph is a magnificationof the vertical scaleto show the gas phaseand the gas-liquid coexistencecurve. an invertedU (Fig. 9.2).The pipe containsa freely-slidinginternal piston separatingthe pipe into two sections,each of which containsone mole of a gas. The symmetry of the systemwill prove to have important conse- quences,and to break this symmetrywe considerthat eachsection of the pipe containsa smallmetallic "ball bearing"(i.0., a smallmetallic sphere). The two ball bearingsare of dissimilarmetals, with different coefficientsof thermal expansion. At some particular temperature,which we designateas 7,, the two sphereshave equal radii; at temperaturesabove T, the right-hand sphere is the larger. The piston, momentarily brought to the apex of the pipe, can fall into either of the two legs,compressing the gas in that leg and expandingthe gas in the other leg. In either of thesecompeting equilibrium statesthe pressure difference exactly compensatesthe effect of the weight of the piston. In the absenceof the two ball bearingsthe two competingequilibrium states would be fully equivalent.But with the ball bearingspresent the First-Order Phase Transitions in Single Component Systems 217

Cylinder,

FIGURE 9.2 A simple mechanicalmodel. more stableequilibrium position is that to the left if T > 7,, and it is that to the right if T . 7,. From a thermodynamicviewpoint the Helmholtz potential of the system is F : U - TS, and the energyU containsthe gravitationalpotential energyof the piston as well as the familiar thermodynamicenergies of the two gases(and, of course,the thermodynamicenergies of the two ball bearings,which we assumeto be small and/or equal).Thus the Helmholtz potential of\he systemhas two local minima, the lower minimum correspondingto the piston beingon the sideof the smallersphere. As the temperature is lowered through T, the two minima of the Helmholtz potential shift, the absolute minimum changing from the left-hand to the right-handside. A similar shift of the equilibrium position of the piston from one side to the other can be inducedat a giventemperature by tilting the table-or, in the thermodynamicanalogue, by adjustmentof some thermodynamic parameterother than the temperature. The shift of the equilibrium statefrom one local minimum to the other constitutes a first-order phase transition, induced either by a changein temperatureor by a changein someother thermodynamicparameter. The two states betweenwhich a first-order phase transition occursare distinct, occurring at separateregions of the thermodynamicconfiguration space. To anticipate "critical phenomena" and "second-order phase transitions" (Chapter 10) it is useful briefly to consider the casein which the ball bearings are identical or absent.Then at low temperaturesthe two competing minima are equivalent. However as the temperature is increased the two equilibrium positions of the piston rise in the pipe, approaching the apex. Above a particular temperatute 7",, there is only one equilibrium position, with the piston at the apex of the pipe. In- versely, lowering the temperaturefrom T ) 7,, to T < 7,,, the single equilibrium state bifurcatesinto two (symmetric)equilibrium states.The 218 First-Order PhaseTransitions temperature7,, is the "critical temperature,"and the transition at 7:, is a " second-orderphase transition." The states betweenwhich a second-orderphase transition occurs are contiguousstates in the thermodynamicconfiguration space. In this chapter we considerfirst-order phasetransitions. Second-order transitions will be discussedin Chapter 10. We shall there also consider the "mechanical model" in quantitative detail, whereaswe here discussit only qualitatively. Returning to the caseof dissimilar sphelrs,consider the piston residing in the higher minimum-that is, in the sameside of the pipe as the larger ball bearing.Finding itself in sucha minimum of the Helmholtz potential, the piston will remain temporarily in that minimum though undergoing thermodynamic fluctuations ("Brownian motion"). After a sufficiently long time a giant fluctuation will carry the piston "over the top" and into the stable minimum. It then will remain in this deeperminimum until an even larger (and enormouslyless probable) fluctuation takesit back to the less stable minimum, after which the entire scenario is repeated.The probability of fluctuations falls so rapidly with increasingamplitude (as we shall seein Chapter 19) that the systemspends almost all of its time in the more stableminimum. All of this dynamicsis ignored by macroscopic thermodynamics,which concernsitself only with the stable equilibrium state. To discuss the dynamics of the transition in a more thermodynamic context it is convenientto shift our attention to a familiar thermodynamic systemthat again has a thermodynamicpotential with two local minima separatedby an unstableintermediate region of concavity.Specifically we consider a vesselof water vapor at a pressureof 1 atm and at a temperaturesomewhat above 373.15K (i.e., abovethe "normal boiling point" of water). We focus our attention on a small subsystem-a sphericalregion of such a (variable)radius that at any instant it contains one milligram of water. This subsystemis effectivelyin contact with a thermal reservoir and a pressurereservoir, and the condition of equilibrium is that the Gibbs potentialG(7, P, N) of the small subsystembe minimum. The two independentvariables which are determinedby the equilibrium conditions are the energy U and the volume V of the subsystem. If the Gibbs potentialhas the form shownin Fig. 9.3,where X, is the volume, the system is stable in the lower minimum. This minimum correspondsto a considerablylarger volume (or a smaller density) than does the secondarylocal minimum. Consider the behavior of a fluctuation in volume. Such fluctuations occur continually and spontaneously.The slopeof the curve in Fig. 9.3 representsan intensiveparameter (in the presentcase a differencein pressure)which actsas a restoring"force" driving the systemback toward density homogeneityin accordancewith Le Chatelier'sprinciple. Occa- First-Order Phase Transitions in Single Component Systems 219

I o c Eo o E o C B E o E F FIGURE 9 3 Thermodynamicpotential with multiple xi* minima. sionally d fluctuation may be so large that it takes the systemover the maximum, to the region of the secondaryminimum. The system then settlesin the region of this secondaryminimum-but only for an instant. A relatively small (and thereforemuch more frequent) fluctuation is all that is required to overcomethe more shallow barrier at the secondary minimum. The systemquickly returns to its stable state.Thus very small droplets of high density (liquid phase!)occasionally form in the gas,live briefly, and evanesce. If the secondaryminimum were far removed from the absolutemini- mum, with a very high intermediatebarrier, the fluctuations from one minimum to another would be very improbable. In Chapter 19 it will be shown that the probability of such fluctuations decreasesexponentially with the height of the intermediatefree-energy barrier. In solid systems(in which interaction energiesare high) it is not uncommon for multiple minima to exist with intermediatebarriers so high that transitions from one minimum to another take times on the order of the age of the universe! Systemstrapped in such secondary"metastable" minima are effectiuelyin stableequilibrium (as if the deeperminimum did not exist at all). Returning to the caseof water vapor at temperaturessomewhat above the " boiling point," let us supposethat we lower the temperatureof the entire system.The form of the Gibbs potential variesas shown schematically in Fig. 9.4. At the temperatureTo the two minima becomeequal, and below this temperaturethe high density(liquid) phasebecomes absolutely stable. Thus In is the temperatureof the phase transition (at the pre- scribedpressure). If the vapor is cooledvery gently through the transition temperaturethe system finds itself in a state that had been absolutely stable but that is now metastable. Sooner or later a fluctuation within the system will "discover" the truly stable state, forming a nucleusof condensedliquid. This nucleus then grows rapidly, and the entire systemsuddenly under- goesthe transition. In fact the time requiredfor the systemto discoverthe 220 Fi rs t - O rder P hase Transi ti ons

q j5 To t a FIGURE9.4 Schematic variation of Gibbs potential

N with volume (or reciprocal density) for 5 vanous temperatures (Tr < Tz < 4 < < To T). The temperature I is the transition temperature. The high density phase is stable below the transition temperature. preferablestate by an "exploratory" fluctuationis unobservablyshort in the caseof the vapor to liquid condensation.But in the transiiion from liquid to ice the delaytime is easilyobserved in a pure sample.The liquid so cooled below its solidification(freezing) temperature is said to be "supercooled."A slight tap on the container,however, sets up longitudi- nal waveswith alternatingregions of "condensation"and "rarefaJtion," and these externally induced fluctuations substitute for spontaneous fluctuationsto initiate a precipitoustransition. A usefulperspective emerges when the valuesof the Gibbs potentialat eachof its minima areplotted againsttemperature. The resultis as shown schematically_in_Fig.9.5. If theseminimum valueswere tdken from Fig. 9.4 therewould be only two suchcurves, but any numberis possible.At equilibrium the smallestminimum is stable,so the true GibbJpotentialis the lower envelopeof the curvesshown in Fig. 9.5.The discontinuitiesin the entropy (and hencethe latent heat)correspond to the discontinuities in slopeof this envelopefunction. Figure 9.5 should be extendedinto an additional dimension,the additional coordinateP playing a role analogousto z. The Gibbs potential is then representedby the lower envelopesurface, as each of the three

FIGURE 9 5 Minima of the Gibbs potential as a T--+ function of 7. First-Order Phase Transitions in Single Component Slstems 221 single-phasesurfaces intersect. The projection of thesecurves of intersection onto the P-T planeis the now familiar phasediagram (e.g., Fig. 9.1). A phase transition occurs as the state of the systempasses from one envelopesurface, across an intersectioncurve, to anotherenvelope surface. The variableX,, or V inFig.9.4, can be any extensiveparameter. In a transition from p-aramagneticto ferromagneticphases X, is the magnetic moment.In transitionsfrom one crystalform to another(e.g., from cubic to hexagonal)the relevantparameter X, is a crystal symmetryvariable. In a solubility transitionit may be the mole numberof one component.We shall see examplesof such transitions subsequently.All conform to the generalpattern described. At a first-order phasetransition the molar Gibbs potential of the two phasesaqe equal, but othermolar potentials(u, f , h, etc.)are discontinu- ous acrossthe transition, as are the molar volume and the molar entropy. The two phasesinhabit different regionsin " thermodynamicspace," and equality of any propertyother than the Gibbs potentialwould be a pure coincidence. The discontinuity in the molar potentials is the defining property of a first-ordertransition. As shown in Fig. 9.6, as one movesalong the liquid-gas coexistence curve away from the solid phase(i.e., toward higher temperature),the discontinuities in molar volume and molar energy becomeprogressively smaller.The two phasesbecome more nearly alike. Finally, at the terminus of the liquid-gas coexistencecurve, the two phasesbecome indistinguish- able.The first-ordertransition degenerates into a more subtletransition, a second-ordertransition, to which we shall return in Chapter 10. The terminus of the coexistencecurve is called a criticalpoint. The existenceof the cntical point precludesthe possibilityof a sharp distinction betweenthe generic term liquid and the genericterm gas. In crossing the liquid-gas coexistencecurve in a first-order transition we distinguish two phases,one of which is "clearly" a gasand one of which is

T--> V-----+ FIGURE 9,6 The two minima of G corresponding to four points on the coexistence curve. The minima coalesce at the critical point D. 222 First-Order Phase Trcnsitions

PROBLEM

9.1-1. The slopesof all three curvesin Fig. 9.5 are shown as negative.Is this necessary?Is there a restriction on the curvatureof thesecurves?

9.2 THE DISCONTINUITY IN THE ENTROPY_LATENT HEAT

Phasediagrams, such as Fig. 9.1, are divided by coexistencecurves into regions in which one or another phase is stable. At any point on such a curve the two phases have precisely equal molar Gibbs- potentials, and both phases can coexist.

temperature increases at an approximately constant rate. But when the temperature reaches the "melting temperature," on the solid-liquid coexistence line, the temperature ceases to rise. As additional heat is supplied ice melts, forming liquid water at the same temperature. It requires roughly 335 kJ to melt each kg of ice. At any moment the amount of liquid water in the container depends on the quantity of heat that has entered the container since the arrival of the system at the coexistence curve (i.e., at the melting temperature). when finally the requisite amount of heat has been supplied, and the ice has been entirely melied, continued heat input again results in an increase in temperature-now at a The Discontinuity in the Entropy-I-atent Hest 223

rate determinedby the specificheat capacityof liquid water (= 4.2kJ/ ke-K). The quantity of heat required to melt one mole of solid is the heat of fusion (or the latentheat of fusion).It is relatedto the differencein molar entropiesof the liquid and the solid phaseby

/r":TIs{z)-"

where 7 is the melting temperatureat the given pressure. More generally,the latent heat in any first-order transition is

/: TA,s (e.2)

where Z is the temperatureof the transition and As is the differencein molar entropies of the two phases.Alternatively, the latent heat can be written as the differencein the molar enthalpiesof the two phases

/: Lh (9.3)

which follows immediately from the identity h : Ts * p (and the fact that p,, the molar Gibbs function, is equal in each phase). The molar enthalpiesof eachphase are tabulatedfor very many substances. If the phasetransition is betweenliquid and gaseousphases the latent heat is called the heat of uaporization,and if it is between solid and gaseousphases it is called the heat of sublimation. At a pressureof one atmospherethe liquid-gas transition (boiling) of water occursat 373.75K, and the latent heat of vaporizationis then 40.7 kJ/mote (540 cal/g). In each casethe latent heat must be put into the systemas it makes a transition from the low-temperaturephase to the high-temperaturephase. Both the molar entropy and the molar enthalpy are greater in the high-temperaturephase than in the low-temperaturephase. It should be noted that the method by which the transition is inducedis irrelevant-the latent heat is independentthereof. Instead of heating the ice 'at constant pressure(crossing the coexistencecurve of Fig. 9.1a "horizontally"), the pressurecould be increasedat constant temperature (crossingthe coexistencecurve " vertically"). In either casethe samelatent heat would be drawn from the thermal reservoir. The functional form of the liquid-gas coexistencecurve for water is given in "saturatedsteam tables"-the designation"saturated" denoting that the steam is in equilibrium with the liquid phase. ("Superheated steam tables" denote compilations of the properties of the vapor phase alone, at temperaturesabove that on the coexistencecurve at the given pressure).An exampleof such a saturatedsteam table is given in Table 9.1, from Sonntagand Van Wylen. The propertiess, u, u and h of each 224 First- Order Phase Transitions

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phase are conventionallylisted in such tables; the latent heat of the transition is the differencein the molar enthalpiesof the two phases,or it can alsobe obtainedas ?"As. similar data are compiled in the thermophysicaldata literature for a wide variety of other materials. The molar volume, like the molar entropy and the molar energy, is discontinuousacross the coexistencecurve. For water this is particiiarly interestingin the caseof the sotd-liquid coexistencecurve. Itls common experiencethat ice floats in liquid water. The molar volume of the solid (ice) phase accordingly is greater than the molar volume of the liquid phase-an uncommonattribute of Hro. The much more commonsitua- tion is that in which the solid phaseis hore compact,with a smallermolar volume. one mundaneconsequence of this peculiarproperty of Hro is the proclivity of frozen plumbing to burst. A compeniatingconsequence, to which we shall return in Section 9.3, is the poisibility of ice s-kating. And,_underlyingall, this peculiarproperty of waier is essentialto the ver:y possibility of life on earth.If ice weremore densethan liquid water thl frozen winter surfacesof lakesand oceanswould sink to the bottom; new surfaceliquid, unprotectedby an ice layer, would again freeze(and sink) until the entire body of water would be frozen solid (,, frozenundert, insteadof "frozen over").

PROBLEMS

latent heat at the point Po* p, To + t. 9.2-2. Discuss the equilibrium that eventually results if a solid is placed in an initially evacuated closed container and is maintained at a given temperature. .. Explain why the solid-gas coexistence curve is said to define the vapor pressure of the solid" at the given temperature.

9.3 THE SLOPE OF COEXISTENCE CURVES; THE CLAPEYRON EQUATION

The coexistencecurves illustrated in Fig. 9.1 are less arbitrary than is immediately evident; the slope dP/dr of a coexistencecurve is fullv determinedby the propertiesof the two coexistingphases. The Slope of Coexistence Curues; The Clapeyron Equation 229

The slope of a coexistencecurve is of direct physicalinterest. Consider cubesof ice at equilibrium in a glassof water. Given the ambientpressure, the temperature of the mixed systemis determinedby the liquid-solid coexistencecurve of water; if the temperaturewere not on the coexistence curve some ice would melt, or some liquid would freeze, until the temperaturewould againlie on the coexistencecurve (or one phasewould becomedepleted). At 1 atm of pressurethe temperaturewould be 273.15 K. If the ambient pressurewere to decrease-perhaps by virtue of a changein altitude (the glassof water is to be servedby the flight attendant in an airplane),or by a variation in atmosphericconditions (approach of a storm)-then the temperatureof the glassof water would appropriately adjust to a new point on the coexistencecurve. If AP were the changein pressurethen the changein temperaturewould be AT : LP/(dP/dT),,, where the derivative in the denominator is the slope of the coexistence curve. Ice skating, to which we havemade an earlier allusion,presents another interesting example.The pressureapplied to the ice directly beneath the blade of the skate shifts the ice acrossthe solid-liquid coexistencecurve (vertically upward in Fig. 9.la), providing a lubricating film of liquid on which the skate slides. The possibility of ice skating dependson the negative slope of the liquid-solid coexistencecurve of water. The existenceof the ice on the upper surfaceof the lake, rather than on the bottom, reflectsthe larger molar volume of the solid phaseof water as comparedto that of the liquid phase.The connectionof thesetwo facts, which are not independent,lies in the Clapeyron equation,to which we now turn. Consider the four statesshown in Fig. 9.7. StatesA and A' areon the coexistencecurve, but they correspondto different phases(to the left-hand and right-hand regions respectively.)Similarly for the states B and B'. The pressuredifference Pn - Pn (or, equivalently, P", - Pn) is assumed to be infinitesimal (: dP), and similarly for the temperaturedifference T" - Tn (: dT). The slopeof the curveis dP/df.

FIGURE 9.7 T+ Four coexistencestates. 230 First-Order PhaseTransitions

Phaseequilibrium requiresthat

lt,s : P,a, (e.4) and

lla: PB, (e.5) whence

lra-P,q:Ps,-ltr (e.6) But -sdT FB- lre: * udP (e.7) and

Fa,- lLt,: -s' dT + u'dP (9.8) in which s and .r' are the molar entropies and u and u,are the molar volumes in each of the phases.By inserting equations 9.7 and 9.8 in equation 9.6 and rearrangingthe terms,we easilyfind

dP J'- ,t (e.e) dr: u,- u dP As (e.10) dr: Lu in which As and Au are the discontinuitiesin molar entropy and molar volume associatedwith the phasetransition. According to equation9.2 the latent heat is /: TLs (e.11) whence

dP / (e.72) dT TA,u This is the Clapeyron equation. The Stopeol CoexistenceCuroes; The ClapeyronEqntion 231 the pressureincrease), and an increasein temperaturetends to drive the system to the more entropic (liquid) phase. Conversely,if s, > s" but t)/ 1 u", then the slopeof the coexistencecurve is negative,and an increase of the pressure(at constant T) tends to drive the system to the liquid phase-again the more densephase. In practical problemsin which the Clapeyron equation is applied it is often sufficient to neglectthe molar volume of the liquid phaserelative to the molar volume of the gaseousphase (u, - ur= ug),and to approximate the molar volumeof the gasby the ideal gasequation (u. = RT/P).This "Clapeyron-Clausius approximation" may be used wheie appropriate in the problems at the end of this section.

Example A lightrigid metallic bar of rectangularcross section lies on a blockof ice,extend- ingslightly over each end. The width of thebar is 2 mmand the length of thebar in contactwith the ice is 25 cm. Two equal masses, each of massM, arehung from the extendingends of the bar.The entiresystem is at atmosphericpressure and is maintainedat a temperatureof | - - 2'C. Whatis theminimum value of. M fot whichthe bar will passthrough the block of iceby "regelation"? The given data are thatthe latent heat of fusionof wateris 80callgram, that the density of liquidwater is I gram/cm3,and that ice cubes float with =4/5of theirvolume submerged. Solution TheClapeyron equation permits us to find the pressureat whichthe solid-liquid transitionoccurs at T = -zoc. Howeverwe must first use the "ice cube data" to obtainthe differenceAu in molarvolumes of liquidand solid phases. The data givenimplythatthedensityoficeis0.8g/cm3. Furthermor€u110 =lScm3/mole, and thereforeu*ria=22.5 x 10-6m3/mole. Thus

dP\ / (80 x 4.2 x 18)J/mole -t :-: : -5 x 106Pa/K dr 1," TLu 271x (-4.5 x 10-6)K-m3/mole so that the pressuredifference required is

P = -5 x 106x (-2) -107Pa

Thispressure is to beobtained by aweight2Mgacting on the areaA : 5 x 10-sm2,

u:)*4

PROBLEMS

9.3-1. A particular liquid boils atl^zToc at a pressureof 800 mm Hg. It has a heat of vaporization of 1000 car/mole. At what temperaturewill it boil if the pressureis raised to 810 mm Hg? 9.3-2. A long vertical column is closedat the bottom and open at the top; it is partially filled with a particular liquid and cooled to -5"c. At this temperature the fluid solidifiesbelow a particular level,remaining liquid abovethis level.If the temperature is further lowered to -5.2oC the solid-liquid interface moves upward by 40 cm. The latent heat (per unit mass)is2 cal/g, and the density of the liquid phaseis 7 g/crrt. Find the density of the solid phase.Neglect theimal expansionof all materials. Ifint: Note that the pressureat the original position of the interface remains constant.

Answer: 2.6g/crrf

9.3-3. It is found that a certainliquid boils at a temperatureof 95" c at the top of a hill, whereasit boils at a temperatureof 105oc at the bottom. The latent heat is 1000 callmole. What is the approximateheight of the hill? 9.3-4. Two weightsare hung on the endsof a wire, which passesover a block of ice. The wire graduallypasses through the block of ice, but the block remains intacteven after the wire haspassed completely through it. Explainwhy lessmass is requiredif a semi-flexiblewire is used,rather than a rigid baias in the Example. 9.3-5. In the vicinity of the triple point the vapor pressureof liquid ammonia(in Pascals)is representedby

lnP:2438-30!3 T

This is the equation of the liquid-vapor boundary curve in a p-T diagram. Sirnilarly, the vapor pressureof solid ammoniais

lnP:27.92-37!4 T what are the temperatureand pressureat the triple point? what are the latent heats of sublimation and vaporization?what is the latent heat of fusion at the triple point? 9.3-6. Let x be the mole fraction of solid phase in a solid-liquid two-phase system.If the temperatureis changedat constant total volume, find the rate of change of x; that is, find dx/dr. Assume that the standard parameters 1,, d, K71cpare known for eachphase. Problems 2 3 3

9.3-7. A particular material has a latent heat of vaporizationof 5 x r03 J/mole, constantalong the coexistencecurve. One mole of this materialexists in two-phase (liquid-vapor) equilibriumin a containerof volumev: r0-z nf , at a temperature of 300 K and a pressureof 105 Pa. The systemis heatedat constantvolume, increasingthe pressureto 2.0 x 105 Pa. (Note that this is nor a small Ap.) The vapor phasecan be treatedas a monatomicideal gas,and the molar volumeof the liquid can be neglectedrelative to that of the gas.Find the initial and final mole fractions of the vapor phasefx = Nr/(N, + trr)]. 9.3-8. Draw the phase diagram, in the B"-T plane, for a simple ferromagnet; assumeno magnetocrystallineanisotropy and assumethe external field B" to be alwaysparallel to a fixed axisin space.What is the slopeof the coexistencecurve? Explain this slopein terms of the Clapeyronequation. 93-9. A systemhas coexistencecurves similar to those shown in Fig. 9.6a, but with the liquid-solid coexistencecurve having a positive slope. sketch the isothermsin the P-u plane for temperatureZ such that : (a) T < 7,, (b) T 4, (c) ?i < T < 4.i,, (d) ?i < T S T*it, (e) T : ?in,,(D r)?i.',. HereI and dn, denotethe triplepoint andcritical temperatures, respectively.

,4 UNSTABLE ISOTHERMS AND FIRST.ORDER PHASE TRANSITIONS

form of its isotherms.For many gasesthe shapeof the isothermsis well represented(at leastsemiquantitatively) by the van der Waalsequation of state(recall Section3.5)

P:,RT._A (e.13) \u-b) D2

The shape of such van der waals isotherms is shown schematicallv in the P-u diagram of Fig. 9.8. As pointed out in Section 3.5 the van der waals equation of state can be viewed as an " underlying equation of state," obtained by curve fitting, by inference based on plausible heuristic reasoning, or by statisticil mechanical calculations based on a simple molecular model. other em- pirical or semiempirical equations of stite exist, and they all have isotherms that are similar to those shown in Fig. 9.8. _ we now explore the manner in which isotherms of the general form shown reveal and define a phase transition. 234 First - O rder P hase Transitions

FIGURE 9 8 van der Waalsisotherms (schematic). ?i < Tz < Tt...

It should be noted immediatelythat the isothermsof Fig. 9.8 do not satisfythe criteriaof intrinsicstability everywhere, for oneof thesecriteria (equation8.21) is Kr ) 0, or (#),<0 (e.14)

This condition clearly is violated over the portion FKM of a typical isotherm (which, for clarity, is shownseparately in Fig. 9.9). Becauseof this violation of the stability condition a portion of the isotherm must be unphysical, supersededby a phase transition in a manner which will be explored shortly. The molar Gibbs potential is essentiallydetermined by the form of the isotherm.From the Gibbs-Duhem relationwe recallthat

dP: -tdT + udP (e.15) whence, integrating at constant temperature

r: IrdP+ QQ) (e.16) where f(7) is an undetermined function of the temperature, arising as the "constant of integration." The integrand u(P), for constant temperature, is given by Fig. 9.9, which is most conveniently represented with P as Unstable Isotherms and First-Order Phase Transitions 235

P-----> FIGURE 9 9 A particular isotherm of the van der Waals shape.

abscissaand u as ordinate.By arbitrarily assigninga value to the chemical potentialat the point A, we can now computethe valueof ,.rat any other point on the sameisotherm, such as B, f.orfrom equation9.16 : Fn- Fe tBu(r)dr (e.17)

In this way we obtain Fig. 9.10.This figure,representing p versusP, can be consideredas a planesection of a three-dimensionalrepresentation of F versusP and 7, as shownin Fig. 9.11.Four differentconstant-temperature sectionsof the p-surface,corresponding to four isotherms,are shown. It is alsonoted that the closedloop of the p versusP curves,which results from the fact that u(P) is triple valuedin P (seeFig. 9.9),disappears for high temperaturesin accordancewith Fig. 9.8. Finally, we note that the relation p: p(7, P) constitutesa fundamental relation for one mole of the material,as the chemicalpotential p is the Gibbs function per mole. It would then appearfrom Fig. 9.11 that we have almost succeededin the construction of a fundamental equation from a single given equation of state, but it should be recalled that dthough each of the traces of the p-surface (in the various constant temperatureplanes of Fig. 9.11)has the proper form, eachcontains an additive "constant" +(T), which variesfrom one temperatureplane to another. Consequently, we do not know the complete form of the (7, P)-surface, although we certainly are able to form a rather good tal picture of its essentialtopological properties. With this qualitative picture of the fundamental relation implied by the van der Waals equation, we return to the question of stability. 236 Firs t - 0 rder P hase Trans itions

P-----> FIGURE9.10 Isothermal dependence of the molar Gibbs potential on pressure. t It

FIGURE 91I Functional dependenceof the molar Gibbs potential. Unstable Isotherms and First-Order Phase Transitions 237

Consider a systemin the stateA of Fig. 9.9 and in contact with thermal d pressure reservoirs. Suppose the pressure of the reservoir to be quasi-statically,maintaining the temperatureconstant. The sys- proceedsalong the isotherm in Fig. 9.9 from the point A in the ion of point B. For pressuresless than P" we seethat the volume of system (for given pressure and temperature) is single valued and ique. As the pressureincreases above Ps, however,three statesof equal and 7 become available to the system, as, for example, the states nated by C, L, and i/. Of thesethree states Z is unstable,but at C and N the Gibbs potential is a (local) minimum. Thesetwo local values of the Gibbs potential (or of p,) are indicated by the ints C and N in Fig. 9.10.Whether the systemactually selectsthe state or the state N dependsupon which of thesetwo local minima of the iibbs potential is the lower, or absolute,minimum. It is clear from Fig. 10 that the state C is the true physicals-tate for this value of the pressure temperature. As the pressure is further slowly increased,the unique point D is rched.At this point the p-surfaceintersects itself, as shownin Fig. 9.10, d the absolute minimum of p or G thereafter comes from the other ch of the curve. Thus at the pressuraPn: Pg, which is greaterthan the physical state is Q. Below Po the righfhand branch of the in Fig. 9.9a is the physically significant branch, whereasabove the left-hand branch is physically significant. The physical isotherm deducedfrom the hypotheticalisotherm of Fig. 9.9 is thereforeshown in .9.12. The isotherm of Fig. 9.9 belongsto an "underlying fundamental lation"; that of Fig.9.72 belongsto the stable"thermodynamic funda- relation."

, u- physical van der waals isotherm.The "underlying" isothermis s7MKFDA, but the constructionconverts it to the physicalisotherm SOKDA. 238 First-Order Phase Transitions

The points D and O are determined by the condition that p'o: !1o or, from equation 9.17 t u(r)ar : o (e.18) where the integral is taken along the hypothetical isotherm. Referring to Fig. 9.9, we see that this condition can be given a direct graphical interpretation by breaking the integral into several portions + o (e.1e) I',* . I:,dP + I:udP loudP: and rearrangingas follows

I',ar- IidP: IKudP-toudP (e.20)

determinedby the graphicalcondition

areal: areaII (e.2r)

It is only after the nominal (non-monotonic) isothermhas beentruncated by this equal area construction that it representsa true physical isotherm. Not only is there a nonzero change in the molar volume at the phase transition, but there are associated nonzero changes in the molar energy and the molar entropy as well. The change in the entropy can be computed by integrating the quantity

ds: (H),^ (e.22)

along the hypothetical isotherm OMKFD. Alternatively, by the thermodynamic mnemonic diagram, we can write As:so-so: oJ*,o(#).* (e.23) A geometrical interpretation of this entropy difference, in terms of the area between neighboring isotherms,is shown in Fig. 9.13. Unstahle Isotherms and First-Order Phase Transitions 239

= - = .lshaded As sa so or-J-fnp4u = uf areai

FIGURE 9.13 The discontinuity in molar entropy. The areabetween adjacent isotherms is related to the entropy discontinuity and thenceto the latent heat.

As the systemis transformedat fixed temperatureand pressurefrom the pure phaseO to the pure phaseD, it absorbsan amountof heat per mole equal to loo: ZAs. The volumechange per mole is Au : up - D6t and this is associatedwith a transferof work equal to PAu. Consequently, the total changein the molar energyis ^u : uD- uo: TAs- PAu (e.24) Each isotherm,such as that of Fig. 9.12,has now been classifiedinto three regions.The region SO is in the liquid phase.The region Dl is in the gaseousphase. The flat region OKD correspondsto a mixture of the two phases.Thereby the entire P-u plane is classifiedas to phase, as shown in Fig. 9.14. The mixed liquid-plus-gasregion is bounded by the inverted parabola-like curvejoining the extremitiesof the flat regions of eachisotherm. Within the two-phaseregion any given point denotesa mixture of the two phasesat the extremitiesof the flat portion of the isotherm passing through that point. The fraction of the systemthat existsin each of the two phasesis governedby the "lever rule." Let us supposethat the molar volumesat the two extremitiesof the flat regionof the isotherm are u, and u" (suBgestingbut not requiring that the two phasesare liquid and gas,for definiteness).Let the molar volume of the mixed systembe u : V/N. Then if x, and x s are the mole fractions of the two phases V: Nu : Nxrur* Nxru, (e.2s) from which one easily finds un- u - L?:' (e.26) ur- Dl 240 Firs t - Orde r P hase Transitions

,- FIGURE 9.14 Phaseclassification of the P - u plane. and

u-Ut Xo: - (e.27) " ur- u,

That is, an intermediatepoint on the flat portion of the isothermimplies a mole fraction of eachphase that is equal to the fractional distanceof the point from the oppositeend of the flat region. Thus the point Z in Fig. 9.14 denotes a mixed liquid-gas systemwith a mole fraction of liquid phase equal to the "length" ZD diided by the "length" OD. This is the very convenientand pictorial lever rule. The vertex of the two-phaseregion, or the point at which O,, and D,, coincide in Fig. 9.14,corresponds to the critical poinl-the termination of the gas-liquid coexistencecurve in Fig. 9.1a. For temperaturesabove the critical temperaturethe isothermsare monotonic(Fig. 9.1a)and the molar Gibbs potentialno longeris reentrant(Fig. 9.10). Just as a P-u diagramexhibits a two-phaseregion, associatedwith the discontinuity in the molaivolume, so a z-s diagram exhibits a two-phase region associatedwith the discontinuity in the molar entropy.

Example1 Find the critical temperature7", andcritical pressure {, for a systemdescribed I by the van der Waals equation of state. Write the van der Waals equation of state I in terms of the reduced variables i = T/7",, F = p/p", and D = u/u",. ! 241

Solution The critical statecoincides with a point of horizontal inflection of the isotherm,or (#).:(#).-0 (Why?) Solving thesetwo simultaneousequations gives

u",: 3b P.,:' J- , RT.,: * 27b' " /' tb from which we can write the van der Waals equationin reducedvariables: 8r 'b_ _3 3fr-I it2

Example 2 Calculatethe functional form of the boundaryof the two-phaseregion in the p-? plane for a systemdescribed by the van der Waals equationof state. Solution we work in reducedvariables, as definedin the precedingexample. we considera fixed temperature and we carry out a Gibbs equal area construction on the correspondingisotherm. Let the extremitiesof the two-phaseregion, corresponding to the reduced temperaturei, be t" and Ar. The equal area construction correspondingto equations9.20 and 9.21-is Fr(ar-ar,1 J,,["Paa: where Pr: P, is the reduced_pressureat which the phasetransition occurs(at the given reduced temperaturer). The reader should draw the isotherm, identify the significanceof eachside of the precedingequation, and reconcilethis form of lhe statementwith that in equations9.20 and 9.27; he or she should alsojustify the use of reduced variablesin the equation. Direct evaluation of the integral gives rn(:a,- r) + :h(3a,- #+-4= r)+ #I,- ;- Simultaneous solution of this equation and of the van der waals equations for iJF,f) and tr(P,f ; gives 6r, 0, and P for each value of f.

l. Show that the difference in molar volumes across a coexistence curve is byAu: -P-rAf. 2. Derive the expressions for u., P. artd f given in Example 1. 242 First-Order Phsse Transitions

9.4-3. Using the van der Waals constantsfor HrO, as given in Table 3.1, calculate the critical temperature and pressure of water. How does this compare with the observed value T" : 641.05K (Table 10.1)? 9.4-4. Show that for sufficiently low temperature the van der Waals isotherm intersects the P : 0 axis, predicting a region of negative pressure. Find the temperature below which the isotherm exhibits this unphysical behavior. 9.4 Hint: Let P : O in the reduced van der Waals equation and consider the flu - I condition that the resultant quadratic equation for the variable D have two real av( roots. H2 9.4 Answer: an 7:4=0.84 anl 9.4-5. Is the fundamentalequation of an ideal van der Waals fluid, as given in 9.4 Section3.5, an "underlying fundamentalrelation" or a "thermodynamicfunda- pr( mental relation?" Why? pr( 9.4-6. Explicitly derive the relationship among D' D, and f, as given in latr Example2. Era res 9.4-7. A particular substancesatisfies the van der Waals equation of state. The coexistencecurve is plottedin the P, i plane,so that the criticalpoint is at (1,1). 9.4 Calculate the reducedpressure of the transition for i :0.95. Calculatethe f= reducedmolar volumesfor the correspondinggas and liquid phases. phu

Answer: 9-5

( sha F= 0.81 chu per A I tial I ( la, fun UI col] mu bel, ( 0.6 08 L8 20 22 tol U will FIGURE 9 T5 abc The T: 0.95isotherm. o. The f : 0.95 isotherm is shown in Fig. 9.15. me( Counting squares permits the equal area construction for Ceneral Attributes ol First-Order Phase Transitions 243

shown, giving the approximate roots indicated on the figure. Refinement of these roots by the analytic method of Example 2 yields ^P: 0.8t+, Ds: 7.77and tr: 0.683

9.4-8. Using the two points at i :0.95 and f : 1 on the coexistencecurve of a fluid obeying the van der waals equation of state (Problem 9.4-7), calculate the average latent heat of vaporization over this range. Specifically apply this result to Hro. 9.4-9. Plot the van der Waalsisotherm, in reducedvariables, for T :0.9{. Make an equal area constructionby counting squareson the graph paper. Corroborate and refine this estimateby the method of Example2. 9.4-lO. Repeatproblem 9.4-8 in the range0.90 < i <0.95, using the resultsof problems 9.4-'7 and 9.4-9. Does the latent heat vary as the temperature ap- proachesd? What is the expectedvalue of the latent heat preciselyat TJ The latent heat of vaporizationof water at atmosphericpressure is = 540 caloriesper gram. Is this value qualitatively consistentwith the trend suggestedby your results? 9.4'll. Two moles of a van der waals fluid are maintainedat a temperature T : 0.957"in a volume of 200cm3. Find the mole number and volume of each phase.Use the van der Waalsconstants of oxygen.

9-5 GENERAL ATTRIBUTES OF FIRST.ORDER PHASE TRANSITIONS

our discussionof first-ordertransitions has beenbased on the general shapeof realisticisotherms, of which the van der Waals isothermis a characteristicrepresentative. The problemcan be viewedin a moregeneral perspectivebased on the convexityor concavityof thermodynamicpotentials. Considera generalthermodynamic potential, U[p,,..., p,l, that is a functionof S, Xr, Xr,..., X,-r,P",... ,P,.Thecriterion of stabilityis that UIP",. . . , P,f must be a convexfunction of its extensiveparameters and a concavefunction of its' intensiveparameters. Geometrically, the function must lie aboveits tangenthyperplanes in the Xr,..., X,_r subspaceand below its tangenthyperplanes in the P,,.. ., P, subspace. Considerthe functionUfP",..., P,f asa functionof X,, and supposeit to havethe form shownin Fig. 9.16a.A tangentline Do'is alsoshown. It will be noted that the function lies abovethis tangentline. It also lies above all tangent lines drawn at points to the left of D or to the right of o. The function doesnot lie abovetangent lines drawn to points inter- mediatebetween D and o. The local curvatureof the potential is positive for all points exceptthose between points F and M. Neverthelessa phase tr 244 First-Order P hase Transitions

A I

: ^t \

xi xf xf P"+ xi- FIGURE 9 T6 Stabitty reconstructionfor a generalpotential.

transition occursfrom the phaseat D to the phaseat O. Global curvature fails (becomesnegative) at D beforelocal curvaturefails at F. The "amended" thermodynamicpotential U\P,,..., P,l consistsof the segmentAD in Fig. 9.15a, the straight line two-phasesegrnent DO, and. the original segmentOR. An intermediate point on the straight line segment,such as Z, cofie- sponds to a mixture of phasesD and O. The mole fraction of phaseD varies linearly from unity to zero as Z movesfrom point D to point O, from which it immediatelyfollows that (xi - xf) I/- .4- \x: - xi) This is again the "lever ruIe." The value of the thermodynamicpotential U[{, ..., P,l in the mixed state (i.e., at Z) clearlyis lessthan that in the pure state(on the initial curve correspondingto X,r). Thus the mixed state given by the straight line constructiondoes miriimize UIP,,. . ., P,l and doescorrespond to the physical equilibrium stateof the system. The dependenceof U\P",..., P,l on an intensiueparameter { is subject to similar considerations,which should now appear familiar. The Gibbs potential UlT, Pl: Np(T, P) is the particularexample studied in the preceding section. The local curvature is negative except fqr the segment MF (Fig. 9.16b). But the segmentMD hes above,rather than below, the tangent drawn to the segment ADF at D. Only the curve ADOR lies everywherebelow the tangent lines, thereby satisfying the conditions of global stability. Thus the particular resultsof the precedingsection are of very general applicability to all thermodynamicpotentials. First-Order Phase Transitions in Multicomponent Systems-Gibbs Phase Rule 245

9-6 FIRST.ORDER PTIASE TRANSITIONS IN MULTICOMPONENT SYSTEMS_GIBBS PTIASE RULE

If a systemhas more than two phases,as doeswater (recall Fig. 9.1),the phase diagram can becomequite elaborate.In multicomponent systems the two-dimensional phase diagram is replaced by a multidimensional space, and the possible complexity would appear to escalaterapidly. Fortunately, however,the permissiblecomplexity is severelylimited by the "Gibbs phase rule." This restrictionon the form of the boundariesof phase stability applies to single-componentsystems as well as to multicomponent systems,but it is convenient to explore it directly in the generalcase. The criteria of stability, as developedin Chapter 8, apply to multicomponent systemsas well as to single-componentsystems. It is necessary only to considerthe variousmole numbersof the componentsas extensive parameters that are completely analogous to the volume V and the entropy S. Specifically,for a single-componentsystem the fundamental relation is of the form

U: U(S,V,N) (e.28) or. in molar form u: u(s,u) (e.2e)

For a multicomponent systemthe fundamentalrelation is

U : U(5,V, N1,Nr,..., N,) (e.30) and the molar form is

u : u(s, u, x11x2;..., xr-t) (e.31)

The mole fractions x, : N,/N sum to unity, so that only r - 1 of the x, are independent, and-only-r - 1 of the mole fractions appear as indepenl dent variables in equation 9.31. All of this is (or should be) familiar, but it is repeated here to stress that the formalism is completely symmetric in the variables s, u, x1,..., x,_1, and that the stability criteria can be interpreted accordingly. At the equilibrium state the energy, the enthalpy, and the Helmholtz and Gibbs potentials are convex functions of the mole fractions x1, x22. . ., xr_r (seeProblems 9.6-1 and 9.6-2). If the stability criteria are not satisfied in multicomponent systems a phase transition again occurs. The mole fractions, like the molar entropies end the molar volumes, differ in each phase. Thus the phases generally are different in gross composition. A mixture of salt (NaCl) and water r

246 First - O rder Phas e Trunsitions

brought to the boiling temperatureundergoes a phasetransition in which the gaseousphase is almost pure water, whereasthe coexistentliquid phasecontains both constituents-the differencein compositionbetween the two phasesin this caseis the basisof purification by distillation. Given the fact that a phasetransition does occur, in either a single or multicomponent system,we are faced with the problem of how such a multiphase systemcan be treatedwithin the frameworkof thermodynamic theory. The solution is simple indeed, for we need only consider each separatephase as a simplesystem and the given systemas a composite system.The "wall" betweenthe simple systemsor phasesis then completely nonrestrictiveand may be analyzedby the methodsappropriate to nonrestrictivewalls. As an exampleconsider a containermaintained at a temperature7 and a pressureP and enclosinga mixture of two components.The systemis observedto containtwo phases:a liquid phaseand a solidphase. We wish to find the composition of eachphase. The chemicalpotential of the first componentin the liquid phaseis p\t)(T, P, *l.t)), and in the solidphase it is ptsr(2,P,"{")); it shouldbe noted that differentfunctional forms for;.r, areappropriate to eachphase. The condition of equilibrium with respectto the transfer of the first componentfrom phaseto phaseis

p\u(T,p,rl')): rr1")(7,P, xts)) (e.32)

Similarly, the chemical potentials of the second component are pf)(T,P,r{t)) and p!sl(Z,P,"{t)); we can write thesein iermsof x, rather than x, becausexr l x, is unity in eachphase. Thus equatingplt') and pf) givesa secondequation, which, with equation9.32, determines x{t) and y{s). Let us supposethat three coexistentphases are observedin the foregoing system. Denoting these by I, II, and III, we have for the first component

p'r(r,P, xl) : p\t(T,p, : pltt(7,P. xltr) (e.33) ",tt) and a similar pair of equations for the second component. Thus we have four equations and only three composition variables: x!, xlt, and xrIII. This means that we are not free to specify both Z and P a priori, but if 7 is specifled then the four equations determine P, xl, xfl, and xrIII. Although it is possible to select both a temperature and a pressure arbitrarily, and then to find a two-phase state, a three-phase state can exist only for one particular pressure if the temperature is specifled. In the same system we might inquire about the existence of a state in which four phases coexist. Analogous to equation 9.33, we have three First-Order Phase Transitions in Multicomponent Systems-Gibbs Phase Rute 247 equationsfor the first componentand three for the second.Thus we have six equationsinvolvin1T, P, xI, ,It, xfII, and xlIV.This meansthat we can have four coexistentphases only for a uniquely defined temperature and pressure, neither of which can be arbitrarily preselectedby the experimenterbut which are unique propertiesof the system. Five phasescannot coexist in a two-componentsystem, for the eight resultant equations would then overdetermine the seven variables (7, P, xl, . . . , xf ), and no solutionwould be possiblein general. We can easily repeat the foregoing counting of variablesfor a multicomponent, multiphase system.In a system with r componentsthe chemical potentials in the first phase are functions of the variables, T, P, x!, xrr,. . . , xr,-r. The chemical potentials in the second phase are functionsof T, P, xlr, x|,..., *f,t-r.If there areM phases,the complete set of independentvariables thus consistsof Z, P, and M(r - 1) mole fractions; 2 + M(r - 1) variablesin all. There are M - 1 equationsof chemicalpotential equalityfor eachcomponent, or a total of r(M - 1) equations.Therefore the number f of variables,which can be arbitrarily assigned,is [2 + M(r - L)] - r(M - 1), or

f:r-M+2 (e.34) The fact that r - M + 2 variablesfrom the set (2, P,xl,xrr,...,x!_r) can be assignedarbitrarily in a systemwith r componentsand M phases is the Gibbsphase rule. The quantity f can be interpreted alternatively as the number of thermodynamicdegrees of freedom, previously introduced in Section 3.2 and defined as the number of intensiueparameters capable of independent variation. To justify this interpretationwe now count the number of thermodynamicdegrees of freedomin a straightforwardway, and we show that this number agreeswith equation9.34. For a single-componentsystem in a singlephase there are two degrees of freedom, the Gibbs-Duhem relation eliminating one of the three variablesT,P,p.For a single-componentsystem with two phasesthere are threeintensive parameters (7, P, and p, eachconstant from phaseto phase) and there are two Gibbs-Duhem relations.There is thus one degreeof freedom.In Fig. 9.1 pairs of phasesaccordingly coexist over one-dimensionalregions (curves). If we have three coexistentphases of a single-componentsystem, the three Gibbs-Duhem relations completely determine the three intensive parametersT, P, and p. The three phasescan coexist only in a unique zero-dimensionalregion, or point; the several"triple points" in Fig. 9.1. For a multicomponent, multiphase system the number of degreesof freedom can be counted easily in similar fashion. If the system has r components,there ate r + 2 intensiveparameters: 7,P,F,F2.,...,11,. Each of theseparameters is a constantfrom phaseto phase.But in eachof 248 First-Order Phase Transitions

the M phasesthere is a Gibbs-Duhem relation. TheseM relationsreduce the number of independentparameters to (r * 2) - M.The number of degreesof freedom / is therefore r - M + 2, as given in equation 9.34. The Gibbs phase rule therefore can be stated as follows. In a system with r componentsand M coexistentphases it is possiblearbitrarily to - preassignr M * 2 uariablesfrom theset (7, P, xl, xrr,..., x!-r) or from theset (7, P, Fp Fz,.. ., p,). It is now a simple matter to corroboratethat the Gibbs phaserule gives the sameresults for single-componentand two-componentsystems as we found in the precedingseveral paragraphs. For single-componentsystems r : 1 and f : 0 1f.M :3. This agreeswith our previousconclusion that the ffiple point is a unique statefor a single-componentsystem. Similarly, for the two-componentsystem we saw that four phasescoexist in a unique point (/: 0, r :2, M: 4), that the temperaturecould be arbitrarily assignedfor the three-phasesystem (f :1, r:2, M :3), and that both T and P could be arbitrarily assignedfor the two-phasesystem (f : 2, f :2, M :2).

PROBLEMS

9.6-1.In a particular system,solute I and solute B are each dissolvedin solventC. a) What is the dimensionalityof the spacein whichthe phaseregions exist? b) What is the dimensionalityof the regionover which two phasescoexist? c) What is the dimensionalityof the regionover which three phases coexist? d) What is the maximumnumber of phasesthat cancoexist in this system? 9.6-2.It g, themolar Gibbs function, is a convexfunction of xyx2,...,x,-1s show that a cha.pgeof variablesto x2; x3,.. .t x, resultsin g being a convex function of x2, x3,. . . , x,.That is, showthat the convexitycondition of the molar Gibbs potential is independentof the choiceof the "redundant" mole fraction. 9.6-3. Show that the conditions of stability in a rnulticomponentsystem require that the partial molar Gibbs potential Fi of any componentbe an increasing function of the mole fraction x, of that component,both at constant u and at constant P, and both at constant s and at constant T.

9.7 PHASE DIAGRAMS FOR BINARY SYSTEMS

The Gibbs phase rule (equation 9.34) provides the basis for the study of the possible forms assumed by phase diagrams. These phase diagrams, particularly for binary (two-component) or ternary (three-component) systems, are of great practical importance in metallurgy and physical chemistry, and much work has been done on their classification. To PhaseDiagrams for BinarySystems 249 illustrate the application of the phase rule, we shall discusstwo typical diagramsfor binary systems. For a single-componentsystem the Gibbs function per mole is a function of temperatureand pressure,as in the three-dimensionalrepre- sentationin Fig. 9.11.The "phase diagram"in the two-dimensionalT-P plane (such as Fig. 9.1) is a projection of the curve of intersection(of the p-surfacewith itself; onto the T-P plane. For a binary system the molar Gibbs function G/(N, + Nr) is a function of the three variablesT, P, and xr. The analogueof Fig.9.11 is then four-dimensional,and the analogueof the T-P phase diagram is three-dimensional.It is obtained by projection of the "hypercurve" of intersectiononto the P,T, xr"hyperplane." The three-dimensionalphase diagram for a simple but common type of binary gas-liquid systemis shown in Fig. 9.17.For obviousreasons of graphic conveniencethe three-dimensionalspace is representedby a series of two-dimensional constant-pressuresections. At a fixed value of the mole fraction xr and fixed pressurethe gaseousphase is stable at high temperature and the liquid phase is stable at low temperature.At a temperaturesuch as that labeledC in Figure 9.77 the systemseparates into two phases-a liquid phaseat A and a gaseousphase at 8. The

r1+ xl+

xl-+ fl+ FIGURE 9.17 The three-dimensional phase diagram of a typical gas-liquid binary system. The two- dimensional sections are constant pressure planes, with P, . P, . P, < Po. 250 First-OrderPhase Transitions composition at point C in Figure 9.17is analogousto the volume at point Z in Figure 9.14 and a form of the leverrule is clearly applicable. The region marked "gas" in Figure 9.17 is a three-dimensionalregion, and T, P, and xl can be independentlyvaried within this region. This is true alsofor the regionmarked "liquid." In eachcase r : 2, M: 1, and f:3. The staterepresented by point C in Figure 9.77is really a two-phase state, composedof A and,,B.Thus only A and B are physicalpoints, and the shadedregion occupiedby point C is a sort of nonphysical"hole" in the diagram. The two-phaseregion is the surface enclosing the shaded volume in Figure 9.17.This surfaceis two-dimensional(r : 2, M : 2, f : 2).Specifying7 and P determinesx1 and xrBuniquely. If a binary liquid with the mole fraction xl is heated at atmospheric pressure,it will follow a vertical line in the appropriate diagram in Fig. 9.17.When it reachespoint A, it will beginto boil. The vaporthat escapes will have the compositionappropriate to point B. A common type of phasediagram for a liquid-solid, two-component system is indicated schematicallyin Fig. 9.18 in which only a single constant-pressuresection is shown.Two distinct solid phases,of different crystal structure, exist: One is labeled c and the other is labeled B. The curve BDHA is called the liquidus curve, and the curves BEL and ACJ are called solidus curves. Point G corresponds to a two-phase system-some liquid at H and somesolid at F. Point K correspondsto a-solid at ,I plus B-solidat t.

f,l-

FIGURE 9 8 Typical phasediagram for a binary systemat constantpressure.

If a liquid with composition x., is cooled, the first solid to precipitate out has composition xr. If it is desiredto have the solid precipitatewith the samecomposition as the liquid, it is necessaryto start with a liquid of Phase Diagrams for Binary Systems 2 51 compositionx D. A liquid of this compositionis calleda eutecticsolution. A eutectic solution freezessharply and homogeneously,producing good alloy castingsin metallurgicalpractice. The liquidus and solidus curves are the traces of two-dimensional surfacesin the completeT-xr-P space.The eutecticpoint D is the trace of a curve in the full T-xr-P space.The eutecticis a three-phaseregion, in which liquid at D, B-solid at E, and a-solid at C can coexist.The fact that a three-phasesystem can exist over a one-dimensionalcurve follows from the phaserule (r : 2, M : 3, f : l). Supposewe start at a statesuch as N in the liquid phase.Keeping Z and x, constant,we decreasethe pressureso that we follow a straight line perpendicularto the planeof Fig. 9.18in the T-xr-P space.We eventually come to a two-phasesurface, which representsthe liquid-gas phase transition. This phasetransition occursat a particular pressurefor the given temperatureand the given composition.Similarly, there is another particularpressure which corresponds to the temperatureand composition of point Q and for which the solid B is in equilibrium with its own vapor. To eachpoint 7, xl we can associatea particularpressure P in this way. Then a phasediagram can be drawn, as shownin Fig. 9.19.This phase diagramdiffers from that of Fig. 9.18in that the pressureat eachpoint is different,and eachpoint representsat leasta two-phasesystem (of which one phaseis the vapor).The curveB'D' is now a one-dimensionalcurve (M : 3, f : 1), and the eutecticpoint D' is a unique point (M : 4, | : 0). Point ,B' is the triple point of the pure first componentand point ,{' is the triple point of the pure secondcomponent. Although Figs. 9.18 and 9.19 are very similar in generalappearance, they are clearly very different in meaning,and confusion can easily arise

D' C'

Yapor+a+p

URE 919 diagram for a binary system in equilibrium with its vapor phase. 252 First-Order Phase Transitions from failure to distinguish carefully between these two types of phase diagrams.The detailed forms of phasediagrams can take on a myriad of differencesin detail, but the dimensionality of the intersectionsof the various multiphaseregions is determinedentirely by the phaserule.

PROBLEMS

9.7-1.The phasediagram of a solutionof A in B, at a pressureof 1 atm,is as shown.The upperbounding curve of the two-phaseregion can be representedby

T:To-(ro-rr)xj

Ne .a - w^TE

The lower bounding curve can be representedby

T:To-(4-Tr)xn(2-xn)

A beakercontaining equal mole numbersof A and,B is brought to its boiling temperature.What is the compositionof the vapor as it first begins to boil off? Does boiling tend to increaseor decreasethe mole fraction of A in the remaining liquid? Answer: xr(vaPor): 0.866

9.7-2. show that if a small fraction (-dN/N) of the material is boiled off the system referred to in Problem 9.7-1, the change in the mole fraction in the remaining liquid is

dxn:-lQ,^ - *1)'- -^l(#) Prohlems 253

9.7-3. The phase diagram of a solution of ,4 in B, at a pressureof 1 atm and in the region of small mole fraction (x, << 1), is as shown. The upper bounding

T=To-Cxt

xA+ curve of the two-phaseregion can be representedby

T: To- C*n and the lower bounding curve by

T: To- Dxn in which C and D arepositive constants (D > C). Assumethat a liquid of mole fraction xi is brought to a boil and kept boiling until only a fraction (Nf/N) of the materialremains; derive an expressionfor the final mole fraction of l. Show that if D : 3C and if N/ry: t, the final mole fraction of component,4 is one fourth its initial value.