(as CO 2 commonly does). If the temperature is higher than the critical tem-
perature, then the solid will “melt”into a supercritical fluid. The A →B line
in Figure 6.15 was intentionally selected to sample all three phases.]
Equation 6.22 is related to the vertical line in Figure 6.15 that connects
points C and D. As the pressure is increased at constant temperature, the chem-
ical potential also increases because for (almost) all substances, the relation
VsolidVliquidVgasapplies. That is, the volume of the solid is smaller than
the volume of the liquid, which is in turn smaller than the volume of the gas.
Therefore, as one increases the pressure, one tends to go to the phase that has
the smaller volume: this is the only way for the partial derivative in equation
6.22 to remain negative. If systems tend to go to lower chemical potential, then
the numerator () is negative. But if pis positive—the pressure is in-
creased—then the overall fraction on the left side of equation 6.22 represents
a negative number. Therefore, systems tend to go to phases that have smaller
volumes when the pressure is increased. Since solids have lower volumes than
liquids, which have smaller volumes than gases, increasing the pressure at con-
stant temperature takes a component from gas to liquid to solid: exactly what
is experienced.
Except for H 2 O. Because of the crystal structure of solid H 2 O, the solid
phase of H 2 O has a larger volume than the equivalent amount of liquid-phase
H 2 O. This is reflected in the negative slope of the solid-liquid equilibrium line
in the phase diagram of H 2 O, Figure 6.3. When the pressure is increased (at
certain temperatures), the liquid phase is the stable phase, not the solid phase.
H 2 O is the exception, not the rule. It’s just that water is so common, and its
behavior so accepted by us, that we tend to forget the thermodynamic impli-
cations.
There is also a Maxwell relationship that can be derived from the natural
variable equation for chemical potential . It is^
S
p
T^V
T
p (6.23)However, since this is the same relation as equation 4.37 from the natural vari-
able equation for G, it does not provide any new, usable relationships.6.8 Summary
Single-component systems are useful for illustrating some of the concepts of
equilibrium. Using the concept that the chemical potential of two phases of the
same component must be the same if they are to be in equilibrium in the same
system, we were able to use thermodynamics to determine first the Clapeyron
and then the Clausius-Clapeyron equation. Plots of the pressure and temper-
ature conditions for phase equilibria are the most common form of phase
diagram. We use the Gibbs phase rule to determine how many conditions we
need to know in order to specify the exact state of our system.
For systems with more than one chemical component, there are additional
considerations. Solutions, mixtures, and other multicomponent systems can
be described using some of the tools described in this chapter, but because of
the presence of multiple components, more information is necessary to de-
scribe the exact state. We will consider some of those tools in the next chapter.162 CHAPTER 6 Equilibria in Single-Component Systems