THE PHASE RULE
9.2 COEXISTENCE CURVES
The thermodynamic state of water vapor, H2O(g), like that of any pure substance, has two degrees of freedom. Any state function, including the Gibbs free energy, can be expressed as a mathematical function of two independent variables, for example,µ= f ( p,T ). We can also write E = f ( p,T ) or Cp = f ( p,T ). In principle, equations
2By our conventional definition, pH≡ −log H+
; this leads us to say that an aqueous solution is neutral when its pH=7.
H2O(gas)
H2O(liquid) H2O(gas)
FIGURE 9.1 Pure water in one phase (left) and two phases (right). At some temperature, the phases coexist in equilibrium.
exist wherewith we can determine E and Cp from a knowledge of p and T even though we may not know exactly what these equations are. We could even write p= f (Cp,E). We are not restricted as to what the independent variables are, so long as there are precisely two of them.
The number of degrees of freedom for a pure, one-phase system is easy. Identifying some phases is easy as well.3We can see the phase separation in mixtures of carbon tetrachloride and water because they have different refractive indices. The two phases, one as the top layer and one as the bottom layer, look different. The trick is to express the behavior of a completely general system, which need not be pure, and may consist of many chemical entities distributed over many phases.
If we choose water vapor as our example of a pure phase, examine the Gibbs free energy as our state function, and take pressure and temperature as our independent variablesµ= f ( p,T ), the system as defined has a wide range of freedom. Within reasonable limits, we can change p and T arbitrarily. The state of the system can be represented as a point on a two-dimensional plot of p vs. T (or p vs. V, as in Chapter 1). Any point represents the system in some state.
Sooner or later though, we are bound to exceed reasonable limits; and the water, even in a closed container that previously contained only water vapor, is bound to condense to produce some liquid water in a state of equilibrium with gaseous water (Fig. 9.1)
We treat this physical equilibrium just as we do a chemical equilibrium. The vaporization equilibrium is
H2O(l)←→ H2O(g) The equilibrium constant is
Keq= pH2O(g)
aH2O(l) =pH2O(g)
where pure water is in its standard state, resulting in a denominator of aH2O(l)≡1.0 in the normal equilibrium expression. An equilibrium between pure liquid and pure
3Identification of some phases may not be easy. Solids can exist as phases that differ only in more subtle ways such as heat capacity or molar volume.
COEXISTENCE CURVES 127
p
T T*
p*
FIGURE 9.2 A liquid–vapor coexistence curve. Fixing the temperature at T*automatically fixes the pressure at p*for coexisting phases of a pure substance.
gas means that their free energies must be equal:
µH2O(l)=µH2O(g)
An equation restricting the variables to a fixed ratio reduces the number of inde- pendent variables by one. Now, by specifying either the temperature or the pressure, the other variable is no longer free. There is still an infinite number of possible free energies, but they are contiguous points on a line. Each specific T defines a coexisting state at pressure p. The locus of points at which liquid and vapor can coexist is called the coexistence curve. If T is changed by a small arbitrary amount, p automatically adjusts to an appropriate value to maintain the equilibrium and stay on the curve. The coexistence curve can now be completely described in two dimensions, p on the ver- tical axis as a function of an independent variable T on the horizontal axis (Fig. 9.2).
The point representing the system is no longer free to move over a two-dimensional p–V plane; it is restricted to the curve. The system has lost one degree of freedom.
The liquid–vapor coexistence curve is not unique, nor is its exponential shape.
Solids also exist in equilibrium with the vapor phase. That is why you can smell solids like naphthalene. Solids have an exponential (or approximately exponential) coexistence curve too, marking their equilibrium boundary with vapor, as seen at the left of Fig. 9.3. Of course, there is also a coexistence curve between solids and liquids (melting points), which normally has a positive slope as in Fig. 9.3. In the unusual case of water, the slope of the solid-liquid coexistence line is negative which is why you can ice skate.4The three curves taken together on a p–T surface constitute a phase diagram. As we shall see, phase diagrams are a very general way of representing all manner of coexistence curves.
At the critical point (Section 2.4), there is no longer a distinction between a liquid and a nonideal gas. Like the critical point, the triple point is unique to each pure substance. It cannot be changed by altering external conditions p and T. It has no degrees of freedom.
4How can there be a connection between a physical chemistry coexistence curve and ice skating? Think about it.
p
T
.
vapor liquid solid
FIGURE 9.3 A single-component phase diagram. The unusual solid–liquid coexistence curve for water is shown as a dotted line. The terminus (•) of the curve on the right is the critical point. The intersection of the three curves is the triple point.
9.3 THE CLAUSIUS–CLAPEYRON EQUATION