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 thermo- dynamic 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 fluctua- tions 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 sys- tem 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 corre- spondingto 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 transi- tions" (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 in- creased 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 equi- librium 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 subsys- tem. 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