H.2 Euler-Maclaur in Sum Formula
7.6 Principles of Le Chatlier and Le Chatlier-Braun
Before leaving the subject of stability, we mention some general principles that govern the approach of systems to equilibrium. The first, due to Le Chatlier, states that if some extensive variable fluctuates from its equilibrium value, its conjugate intensive variable will change in such a way as to restore that extensive variable to its equilibrium value.
The second, due to Le Chatlier-Braun, states that if some extensive variable fluctuates and also produces changes in non-conjugate intensive variables, secondary induced processes occur in such a way as to oppose the change in the conjugate intensive variable associated with the original extensive variable. Thus, any fluctuations of a stable state will tend to decay in such a way as to restore equilibrium values. For formal treatments, see Landau and Lifshitz [7, p. 63] or Callen [2, p. 212].
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8
Monocomponent Phase Equilibrium
In this Chapter, we examine equilibrium for a monocomponent system for the simple case in which the solid phase has only a single crystal structure. The situation can be described by means of a phase diagram in the T,p plane, such as sketched in Figure 8–1. This diagram divides the plane into regions where the phases solid (S), liquid (L), and vapor1(V) are stable. Therefore, the only lines that appear on the diagram are curves where pairs of these phases are in equilibrium. These are called coexistence curves and we shall proceed to develop equations that describe them.
According to the thermodynamics of open monocomponent systems, the conditions for phases to be in equilibrium (see Chapter 6) are for them to have the same temperature T, the same pressure p, and the same chemical potentialμ. But according to Eq. (5.45), the Gibbs-Duhem equation, these variables are not independent and one can regard the chemical potentialμ(T,p)to be a function of temperature and pressure. This function is not the same for different phases, so thecoexistence curvesare given by the following equations:
μS(T,p)=μL(T,p), solid-liquid coexistence curve, (8.1) μS(T,p)=μV(T,p), solid-vapor coexistence curve, (8.2) μL(T,p)=μV(T,p), liquid-vapor coexistence curve. (8.3) According to the Gibbs phase rule for a monocomponent system, the number of ther- modynamic degrees of freedom is 3−nwherenis the number of phases. A single phase region, such as the solid, is represented by an area; accordingly,n=1 and there are two degrees of freedom,p andT, that may be chosen independently throughout this area.
Along each of the coexistence curves,p=2 so there is one degree of freedom along these curves. Thus, ifT is specified,pis known from the curve. For either solid-vapor or solid- liquid equilibrium, the corresponding pressure of the vapor for a given value ofTis known as the vapor pressure. Ifn=3, there are no degrees of freedom; this happens at a point known as thetriple pointwhere solid, liquid, and vapor are in mutual equilibrium with each other. Thus we have
μS(T,p)=μL(T,p)=μV(T,p), triple point. (8.4) Equation (8.4) represents two equations in two unknowns; their solution determinesTt
and pt, the unique coordinates of the triple point. It turns out that the liquid-vapor coexistence curve actually ends at a pointTc and pc known as thecritical point. Thus,
1A vapor is a gaseous phase that can be condensed to form a liquid or solid. Sometimes the word “gas” is used interchangeably with “vapor,” but an ideal gas cannot undergo a phase transformation.
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110 THERMAL PHYSICS
p
T Tt, pt
Tc, pc
S L
V
FIGURE 8–1 Sketch (not to scale) of a phase diagram for a monocomponent system. The curves are coexistence curves for pairs of the phases solid (S), liquid (L), and vapor (V). All three phases coexist in mutual equilibrium at the triple pointTt,pt. The liquid-vapor coexistence curve ends at the critical pointTc,pc. This diagram pertains to the usual case in which the molar volume of the solid is less than that of the liquid from which it freezes. SeeFigure 8–3 for the unusual case.
forT >Tcorp>pc, liquid and vapor become indistinguishable. In Chapter 9 we will see how such a behavior follows from the van der Waals model of a fluid.
Phase diagrams for monocomponent systems can have great variety because the crystalline solids can have different crystal structures, each considered to be a phase. For example, if the solid can have two crystal structures, sayαandβ, there can be more than one triple point, for example, for equilibrium among(α,L,V)and(α,β,L). See deHoff [21, chapter 7] for some specific examples as well as geometrical details of chemical potential surfaces.