the phase diagram and observe which phase transitions occur for that change.
Single-component phase diagrams are especially easy to interpret.Example 6.10
Use the phase diagram of CO 2 , Figure 6.4, to describe the changes in phase
as one makes the following changes in conditions.
a.50 K to 350 K at a pressure of 1.00 bar
b.50 K to 350 K at a pressure of 10 bar
c.1 bar to 100 bar at a temperature of 220 KSolution
a.Figure 6.11 shows the change in conditions for this isobaric process.
Starting at point A, the temperature is increased as we move from left to right,
indicating that we are warming the solid CO 2 , until we reach the line at point
B indicating the equilibrium between solid and gas phases. At this point, the
solid CO 2 sublimes directly into the gas phase. (This occurs at about 196 K,
or 77 °C.) As the temperature increases to 350 K, we are warming gaseous
CO 2 until we reach point C, the final conditions.
b.Figure 6.12 shows the change in conditions for the isobaric warming of
CO 2 at 10 bar. In this case, we start with a solid at point A, but since we are
above the critical point for CO 2 , at point B we are in an equilibrium with
solid and liquid CO 2 present. As we add heat, solid melts until all solid be-
comes liquid, and then the liquid CO 2 warms. We continue warming until
point C is reached, which represents the conditions where CO 2 liquid is in
equilibrium with CO 2 gas. When all of the liquid is converted to gas, the gas
warms until the final conditions at point D are reached.
c.Figure 6.13 illustrates the isothermal process. The starting point A is at low
enough pressure that the CO 2 is in the gas phase. However, as the pressure is
increased, the CO 2 passes into the liquid phase (briefly) and then into the
solid phase. Note that if the temperature were only a few degrees lower, this
change would have occurred on the other side of the triple point and the
phase transition would have been a direct gas-to-solid condensation.Phase diagrams of single-component systems are useful in illustrating a
simple idea that answers a common question: How many variables must be
specified in order to determine the phase(s) of the system when it’s at equilib-
rium? These variables are called degrees of freedom.What we need to know is
how many degrees of freedom we need to specify in order to characterize the
state of the system. This information is more useful than one might think.
Because the position of phase transitions (especially transitions that involve
the gas phase) can change quickly with pressure or temperature, knowing how
many state variables mustbe defined is important.
Consider the two-dimensional phase diagram for H 2 O. If you knew that
only H 2 O was in the system at equilibrium and that it was in the solid
phase, then any point in the shaded region of Figure 6.14 would be possi-
ble. You would have to specify both the temperature and the pressure of the
system. However, suppose you knew that you had solid andliquid H 2 O in
the system at equilibrium. Then you know that the condition of the system
must be indicated by the line in the phase diagram that separates the solid
and liquid phase. You need only specify temperature orpressure, because
knowing one gives you the other (because the system—with two phases in158 CHAPTER 6 Equilibria in Single-Component Systems
1Temperature (°C)- 78.5
5.1173- 56.4 31.1
CO 2 (s)CO 2 (g)CO 2 ( )To
350 K
CPressure (atm)
AB1Temperature (°C)- 78.5
5.1173- 56.4 31.1
CO 2 (s)CO 2 (g)CO 2 ( )To
350 K
Pressure (atm)ABC D1Temperature (°C)- 78.5
5.1173- 56.4 31.1
CO 2 (s)CO 2 (g)CO 2 ( )To 100 barPressure (atm)
ABCDFigure 6.11 An illustration of the isobaric
change for CO 2 specified in Example 6.10a.
Compare this to Figure 6.12.
Figure 6.12 An illustration of the isobaric
change for CO 2 specified in Example 6.10b.
Compare this to Figure 6.11.
Figure 6.13 An illustration of the change
specified in Example 6.10c.