the Clapeyron or the Clausius-Clapeyron equations.Single-component phase
diagrams are nothing more than plots of the Clapeyron equation or the Clausius-
Clapeyron equation for a substance. This is true for pressure-temperature phase
diagrams, which is what we will consider almost exclusively here. For a phase
diagram where volume as well as pressure and temperature varies, a three-
dimensional plot would be necessary and the equation of state for all phases
would be needed.Example 6.9
The line between the solid and liquid phases for the H 2 O phase diagram in
Figure 6.3 is a fairly straight line, indicating a constant slope. Use the answers
to Example 6.4, the melting of ice, to calculate the value for the slope of that
line.Solution
Recall that one definition of the slope of a line is y/x. The y-axis repre-
sents pressure and the x-axis represents temperature, so for p/Twe expect
a slope where the units are bar/K or atm/K. Example 6.4 showed that it
takes 1.35
103 bar to change the melting point of water by 10 °C,
which is 10 K. Therefore,p/Tis equal to (1.35
103 bar)/(10 K) or
1.35
102 bar/K. This is a fairly large slope.One other thing to notice from the example is that the slope is negative.
Almost all compounds have a positive slope for the solid-liquid equilibrium
line, because solids have less volume than the same amount of liquid. The neg-
ative slope is a consequence of the increasein volume experienced by H 2 O
when it solidifies.
The solid-gas equilibrium line represents those conditions of pressure and
temperature where sublimation occurs. For H 2 O, obvious sublimation occurs
at pressures lower than those that are normally experienced. (Sublimation of
ice does occur slowly at normal pressures, which is why ice cubes get smaller
over time in your freezer. The so-called freezer burn of frozen foods is caused
by sublimation of ice from the food. This is why it’s important to wrap frozen
food tightly.) However, for carbon dioxide, normal pressures are low enough
for sublimation. Figure 6.4 shows a phase diagram for CO 2 , with the 1-atm
position marked. Liquid CO 2 is stable only under pressure. Some gas cylin-
ders of carbon dioxide are high enough in pressure that they actually con-
tain liquid CO 2.
The liquid-gas equilibrium line represents conditions of pressure and tem-
perature where those phases can exist at equilibrium. Notice that it has the
form of an exponential equation; that is,p eT. This is consistent with equa-
tion 6.16. The vaporization line in the phase diagram is a plot of the Clapeyron
equation or the Clausius-Clapeyron equation. Notice, however, that this line
ends at a particular pressure and temperature, as shown in Figure 6.5. It is the
only line that doesn’t have an arrow on its end to indicate that it continues.
That’s because beyond a certain point, the liquid phase and the gas phase be-
come indistinguishable. This point is called the critical pointof the substance.
The pressure and temperature at that point are called the critical pressure pC
and critical temperature TC.For H 2 O,pCand TCare 218 atm and 374°C. Above
that temperature, no pressure can force the H 2 O molecules into a definite liquid
state. If the H 2 O in the system exerts a pressure higher than pC, then it cannot6.6 Phase Diagrams and the Phase Rule 1551Temperature (°C)–78.55.1173–56.4 31.1LiquidSolidPressure (atm) Gas215Temperature (°C)374Pressure (bar)
0.006110.01SolidLiquidGasTriple
pointCritical
pointFigure 6.5 The triple point and the critical
point for H 2 O. The liquid-gas equilibrium line is
the only one that ends at a certain set of condi-
tions for all substances. For H 2 O, the line ends at
374 °C and 215 bar. At higher temperatures or va-
por pressures, there is no distinction between a
“liquid”and a “gas”phase.
Figure 6.4 A phase diagram for carbon diox-
ide, CO 2. Notice that as the temperature of solid
CO 2 is increased at standard pressure, the solid
goes directly into the gas phase. Liquid CO 2 is
stable only at increased pressure.