FIRST-ORDER PHASE TRANSITIONS

FIRST-ORDER PHASE TRANSITIONS

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

Ordinary water is liquid at room temperature and atmospheric pressure, but if cooled below 273.15 K it solidifies; and if heated above 373.15 K it vaporizes. At each of these temperatures the material undergoes a pre- cipitous change of properties-a "phase transition." At high pressures witer undergoes several additional phase transitions from one solid form to another. These distinguishable solid phases, designated as "ice 1," "ice II," "ice fII," . . ., differ in crystal structure and in essentially all thermo- dynamic properties (such as compressibility, molar heat capacity, and various molar potentials such as u or f). The "phase diagram" of water is shown in Fig. 9.1.

Each transition is associated with a linear region in the thermodynamic fundamental relation (such as BHF in Fig. 8.2), and each can be viewed as the result of failure of the stability criteria (convexity or concavity) in the underlying fundamental relation.

In this section we shall consider systems for which the underlying fundamental relation is unstable. By a qualitative consideration of fluctua- tions in such systems we shall see that the fluctuations are profoundly influenced by the details of the underlying fundamental relatio,n. In contrast, the auerage ualues of the extensiue parameters reflect only the stable thermo- dynamic fundamental relation.

Consideration of the manner in which the form of the underlying fundamental relation influences the thermodynamic fluctuations will pro- vide a physical interpretation of the stability considerations of Chapter 8 and of the construction of Fig. 8.2 (in which the thermodynamic funda- mental relation is constructed as the envelope of tangent planes).

A simple mechanical model illustrates the considerations to follow by an intuitively transparent analogy. Consider a semicircular section of pipe, closed at both ends. The pipe stands vertically on a table, in the form of

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FIGURE 9

Phase diagram of water. The region of gas-phase stability is represented by an indiscerni- bly narrow horizontal strip above the positive temperature axis in the phase diagram (small figure). The background graph is a magnification of the vertical scale to show the gas phase and the gas-liquid coexistence curve.

an inverted U (Fig. 9.2). The pipe contains a freely-sliding internal piston separating the pipe into two sections, each of which contains one mole of a gas. The symmetry of the system will prove to have important conse- quences, and to break this symmetry we consider that each section of the pipe contains a small metallic "ball bearing" (i.0., a small metallic sphere).

The two ball bearings are of dissimilar metals, with different coefficients of thermal expansion.

At some particular temperature, which we designate as 7,, the two spheres have equal radii; at temperatures above 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 expanding the gas in the other leg. In either of these competing equilibrium states the pressure difference exactly compensates the effect of the weight of the piston.

In the absence of the two ball bearings the two competing equilibrium states would be fully equivalent. But with the ball bearings present the

t 4 I 2 1 0

First-Order Phase Transitions in Single Component Systems 217

FIGURE 9.2

A simple mechanical model.

more stable equilibrium position is that to the left if T > 7,, and it is that to the right if T . 7,.

From a thermodynamic viewpoint the Helmholtz potential of the sys- tem is F : U - TS, and the energy U contains the gravitational potential energy of the piston as well as the familiar thermodynamic energies of the two gases (and, of course, the thermodynamic energies of the two ball bearings, which we assume to be small and/or equal). Thus the Helmholtz potential of\he system has two local minima, the lower minimum corre- sponding to the piston being on the side of the smaller sphere.

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-hand side.

A similar shift of the equilibrium position of the piston from one side to the other can be induced at a given temperature by tilting the table-or, in the thermodynamic analogue, by adjustment of some thermodynamic parameter other than the temperature.

The shift of the equilibrium state from one local minimum to the other constitutes a first-order phase transition, induced either by a change in temperature or by a change in some other thermodynamic parameter.

The two states between which a first-order phase transition occurs are distinct, occurring at separate regions of the thermodynamic configuration space.

To anticipate "critical phenomena" and "second-order phase transi- tions" (Chapter 10) it is useful briefly to consider the case in which the ball bearings are identical or absent. Then at low temperatures the 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 temperature from T ) 7,, to T < 7,,, the single equilibrium state bifurcates into two (symmetric) equilibrium states. The

Cylinder,

218 First-Order Phase Transitions

temperature 7,, is the "critical temperature," and the transition at 7:, is a

" second-order phase transition."

The states between which a second-order phase transition occurs are contiguous states in the thermodynamic configuration space.

In this chapter we consider first-order phase transitions. Second-order transitions will be discussed in Chapter 10. We shall there also consider the "mechanical model" in quantitative detail, whereas we here discuss it only qualitatively.

Returning to the case of dissimilar sphelrs, consider the piston residing in the higher minimum-that is, in the same side of the pipe as the larger ball bearing. Finding itself in such a 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 deeper minimum until an even larger (and enormously less probable) fluctuation takes it back to the less stable minimum, after which the entire scenario is repeated. The probability of fluctuations falls so rapidly with increasing amplitude (as we shall see in Chapter 19) that the system spends almost all of its time in the more stable minimum. All of this dynamics is ignored by macroscopic thermodynamics, which concerns itself only with the stable equilibrium state.

To discuss the dynamics of the transition in a more thermodynamic context it is convenient to shift our attention to a familiar thermodynamic system that again has a thermodynamic potential with two local minima separated by an unstable intermediate region of concavity. Specifically we consider a vessel of water vapor at a pressure of 1 atm and at a temperature somewhat above 373.15 K (i.e., above the "normal boiling point" of water). We focus our attention on a small subsystem-a spherical region of such a (variable) radius that at any instant it contains one milligram of water. This subsystem is effectively in contact with a thermal reservoir and a pressure reservoir, and the condition of equi- librium is that the Gibbs potential G(7, P, N) of the small subsystem be minimum. The two independent variables which are determined by the equilibrium conditions are the energy U and the volume V of the subsys- tem.

If the Gibbs potential has the form shown in Fig. 9.3, where X, is the volume, the system is stable in the lower minimum. This minimum corresponds to a considerably larger volume (or a smaller density) than does the secondary local minimum.

Consider the behavior of a fluctuation in volume. Such fluctuations occur continually and spontaneously. The slope of the curve in Fig. 9.3 represents an intensive parameter (in the present case a difference in pressure) which acts as a restoring "force" driving the system back toward density homogeneity in accordance with Le Chatelier's principle. Occa-

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First-Order Phase Transitions in Single Component Systems 219

FIGURE 9 3

Thermodynamic potential with multiple minima.

sionally d fluctuation may be so large that it takes the system over the maximum, to the region of the secondary minimum. The system then settles in the region of this secondary minimum-but only for an instant.

A relatively small (and therefore much more frequent) fluctuation is all that is required to overcome the more shallow barrier at the secondary minimum. The system quickly 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 secondary minimum were far removed from the absolute mini- mum, with a very high intermediate barrier, the fluctuations from one minimum to another would be very improbable. In Chapter 19 it will be shown that the probability of such fluctuations decreases exponentially with the height of the intermediate free-energy barrier. In solid systems (in which interaction energies are high) it is not uncommon for multiple minima to exist with intermediate barriers so high that transitions from one minimum to another take times on the order of the age of the universe! Systems trapped in such secondary "metastable" minima are effectiuely in stable equilibrium (as if the deeper minimum did not exist at all).

Returning to the case of water vapor at temperatures somewhat above the " boiling point," let us suppose that we lower the temperature of the entire system. The form of the Gibbs potential varies as shown schemati- cally in Fig. 9.4. At the temperature To the two minima become equal, and below this temperature the high density (liquid) phase becomes absolutely stable. Thus In is the temperature of the phase transition (at the pre- scribed pressure).

If the vapor is cooled very gently through the transition temperature the 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 nucleus of condensed liquid.

This nucleus then grows rapidly, and the entire system suddenly under- goes the transition. In fact the time required for the system to discover the

x i *

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To 220 Fi rs t - O rder P h ase Transi ti ons

a t

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5

FIGURE 9.4

Schematic variation of Gibbs potential with volume (or reciprocal density) for vanous temperatures (Tr < Tz < 4 <

To < T). The temperature I is the transition temperature. The high density phase is stable below the transition tem- perature.

preferable state by an "exploratory" fluctuation is unobservably short in the case of the vapor to liquid condensation. But in the transiiion from liquid to ice the delay time is easily observed 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 waves with alternating regions of "condensation" and "rarefaJtion,"

and these externally induced fluctuations substitute for spontaneous fluctuations to initiate a precipitous transition.

A useful perspective emerges when the values of the Gibbs potential at each of its minima are plotted against temperature. The result is as shown schematically_in_Fig. 9.5. If these minimum values were tdken from Fig.

9.4 there would be only two such curves, but any number is possible. At equilibrium the smallest minimum is stable, so the true GibbJpotential is the lower envelope of the curves shown in Fig. 9.5. The discontinuities in the entropy (and hence the latent heat) correspond to the discontinuities in slope of this envelope function.

Figure 9.5 should be extended into an additional dimension, the ad- ditional coordinate P playing a role analogous to z. The Gibbs potential is then represented by the lower envelope surface, as each of the three

FIGURE 9 5

Minima of the Gibbs potential as a function of 7.

T--+

First-Order Phase Transitions in Single Component Slstems 221

single-phase surfaces intersect. The projection of these curves of intersec- tion onto the P-T plane is the now familiar phase diagram (e.g., Fig. 9.1).

A phase transition occurs as the state of the system passes from one envelope surface, across an intersection curve, to another envelope surface.

The variable X,, or V inFig.9.4, can be any extensive parameter. In a transition from p-aramagnetic to ferromagnetic phases X, is the magnetic moment. In transitions from one crystal form to another (e.g., from cubic to hexagonal) the relevant parameter X, is a crystal symmetry variable. In a solubility transition it may be the mole number of one component. We shall see examples of such transitions subsequently. All conform to the general pattern described.

At a first-order phase transition the molar Gibbs potential of the two phases aqe equal, but other molar potentials (u, f , h, etc.) are discontinu- ous across the transition, as are the molar volume and the molar entropy.

The two phases inhabit different regions in " thermodynamic space," and equality of any property other than the Gibbs potential would be a pure coincidence. The discontinuity in the molar potentials is the defining property of a first-order transition.

As shown in Fig. 9.6, as one moves along the liquid-gas coexistence curve away from the solid phase (i.e., toward higher temperature), the discontinuities in molar volume and molar energy become progressively smaller. The two phases become more nearly alike. Finally, at the terminus of the liquid-gas coexistence curve, the two phases become indistinguish- able. The first-order transition degenerates into a more subtle transition, a second-order transition, to which we shall return in Chapter 10. The terminus of the coexistence curve is called a critical point.

The existence of the cntical point precludes the possibility of a sharp distinction between the generic term liquid and the generic term gas. In crossing the liquid-gas coexistence curve in a first-order transition we distinguish two phases, one of which is "clearly" a gas and one of which is

FIGURE 9,6

The two minima of G corresponding to four points on the coexistence curve. The minima coalesce at the critical point D.

T--> V---+

222 First-Order Phase Trcnsitions

PROBLEM

9.1-1. The slopes of all three curves in Fig. 9.5 are shown as negative. Is this necessary? Is there a restriction on the curvature of these curves?

9.2 THE DISCONTINUITY IN THE ENTROPY_LATENT HEAT Phase diagrams, such as Fig. 9.1, are divided by coexistence curves 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 determined by the specific heat capacity of liquid water (= 4.2kJ/

ke-K).

The quantity of heat required to melt one mole of solid is the heat of fusion (or the latent heat of fusion).It is related to the difference in molar

entropies of the liquid and the solid phase by / r " : T I s { z ) - " < s l ]

where 7 is the melting temperature at the given pressure.

More generally, the latent heat in any first-order transition is

(e.1)

/: TA,s

(e.2)

where Z is the temperature of the transition and As is the difference in molar entropies of the two phases. Alternatively, the latent heat can be written as the difference in the molar enthalpies of the two phases

/ : L h ( 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 enthalpies of each phase are tabulated for very many substances.

If the phase transition is between liquid and gaseous phases the latent heat is called the heat of uaporization, and if it is between solid and gaseous phases it is called the heat of sublimation.

At a pressure of one atmosphere the liquid-gas transition (boiling) of water occurs at 373.75 K, and the latent heat of vaporization is then 40.7 kJ /mote (540 cal/g).

In each case the latent heat must be put into the system as it makes a transition from the low-temperature phase to the high-temperature phase.

Both the molar entropy and the molar enthalpy are greater in the high-temperature phase than in the low-temperature phase.

It should be noted that the method by which the transition is induced is irrelevant-the latent heat is independent thereof. Instead of heating the ice 'at constant pressure (crossing the coexistence curve of Fig. 9.1a

"horizontally"), the pressure could be increased at constant temperature (crossing the coexistence curve " vertically"). In either case the same latent heat would be drawn from the thermal reservoir.

The functional form of the liquid-gas coexistence curve for water is given in "saturated steam 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 temperatures above that on the coexistence curve at the given pressure). An example of such a saturated steam table is given in Table 9.1, from Sonntag and Van Wylen. The properties s, u, u and h of each

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