Concise Physical Chemistry

c09 JWBS043-Rogers September 13, 2010 11:26 Printer Name: Yet to Come

COEXISTENCE CURVES 127

p

T

T*

p*

FIGURE 9.2 A liquid–vapor coexistence curve. Fixing the temperature atT*automatically

fixes the pressure atp*for coexisting phases of a pure substance.

gas means that their free energies must be equal:

μH 2 O(l)=μH 2 O(g)

An equation restricting the variables to a fixed ratio reduces the number ofinde-

pendentvariables 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 specificTdefines a coexisting

state at pressurep. The locus of points at which liquid and vapor can coexist is called

thecoexistence curve.IfTis changed by a small arbitrary amount,pautomatically

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,pon the ver-

tical axis as a function of an independent variableTon the horizontal axis (Fig. 9.2).

The point representing the system is no longer free to move over a two-dimensional

p–Vplane; 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.^4 The three curves taken together on ap–Tsurface constitute aphase

diagram. As we shall see, phase diagrams are a very general way of representing all

manner of coexistence curves.

At thecritical point(Section 2.4), there is no longer a distinction between a liquid

and a nonideal gas. Like the critical point, thetriple pointis unique to each pure

substance. It cannot be changed by altering external conditionspandT. It hasno

degrees of freedom.

(^4) How can there be a connection between a physical chemistry coexistence curve and ice skating? Think
about it.