PHASE DIAGRAMS OF ONE-COMPONENT SYSTEMS

4. Reversible adiabatic compression of the gas from state D to state A

6.1 PHASE DIAGRAMS OF ONE-COMPONENT SYSTEMS

Figure 6.1

follow Henry’s law at low concentrations, and activity coefﬁcients can also be cal- culated from deviations from Henry’s law. Ideal solubility, freezing-point depres- sion, and osmotic pressure are also discussed in this chapter.

The – – surface for a one-component system was introduced in Section 1.6, and we saw that it is convenient to use projections onto the – plane and the – plane. Projections onto the – plane are shown for water, carbon dioxide, and carbon in Fig. 6.1.

The phase diagram for water at low pressures is given in Fig. 6.1 , and the phase diagram at high pressures is given in Fig. 6.1 . In Fig. 6.1 , the vapor pres-

Liquid

Tm Tb

Tm′ Tb′ T

T Solid

Solid Gas

Gas à

(a)

(b)

Comment:

P T b

F C p

F p

p F

T P

We represent a critical point with a dot in a phase diagram, but in the neighbor- hood of that dot there are some phenomena, such as the turbidity mentioned in Section 1.7, that are ery interesting in their own right. We do not ha e the space to pursue these remarkable phenomena, but we should remember that they also occur in connection with critical points of mixtures of liquids, which we will encounter later in this chapter.

T T T

T T

179

*According to the phase rule, the intensive state of liquid water is characterized by two intensive variables. It is all right if we pick temperature and pressure. However, there is a problem if we pick pressure and molar volume since the temperature is not uniquely determined by these variables. In the case of water, the molar volume has a minimum in the neighborhood of 4 C. The lesson from this is that we should pick one conjugate variable from each pair, and not two conjugate variables from the same pair.

⬚

m b m

m b

( ) Dependence of the chemical potentials of solid, liquid, and gas phases on temperature at constant pressure. The dashed lines in ( ) are for a lower pressure.

The plots should be slightly concave downward, since entropy increases with increasing temperature, but they have been drawn as straight lines here for simplicity.

6.1 Phase Diagrams of One-Component Systems

⬚

v v

⫺

⫺ ⫺

Figure 6.2 sure of water is given as a function of temperature by the line between the liquid

and vapor areas. The dashed extension of this line gives the vapor pressure of su- percooled water. The curve for the sublimation pressure of ice goes down to zero at 0 K. At higher pressures, four other crystal forms of ice are formed. Starting at the triple point, the freezing point of ice I is lowered to 22 C when the pressure is raised to 2000 bar, but higher pressures lead to other crystal forms of ice for which d /d is positive, as shown in Fig. 6.1 .

The phase diagram for carbon dioxide in Fig. 6.1 shows the equilibrium between solid and gas at 1 bar at 78 C. Liquid carbon dioxide is produced only above 5.1 bar.

The phase diagram for carbon in Fig. 6.1 shows that graphite and diamond are in equilibrium at room temperature only at pressures above 10 000 bar. Dia- monds for industrial use are produced at high pressures and temperatures using catalysts. The details of this phase diagram are not well known because of the difﬁculty in obtaining equilibrium.

In Section 5.9, we saw that the number of degrees of freedom for the descrip- tion of the intensive state of the system is 2 if only pressure–volume work is involved. For a one-component system, 3 , so more than three phases cannot be in equilibrium. In the areas of Fig. 6.1, 1, so 2 and the intensive state of the system is completely described by specifying and .*

Along a line, there are two phases at equilibrium, so the system can be completely described by specifying either or . At a triple point, three phases are in equilibrium, so 0. If the temperature or pressure is changed, two phases will dis- appear because the point representing the system will then lie in the solid, gas, or liquid area of the phase diagram.

To understand the change from solid to liquid to gas phase when a solid is heated at constant pressure, we may consider a plot of chemical potential versus temperature at constant pressure for the various phases, as shown in Fig. 6.2 . The stable phase is that with the lowest value of the chemical potential.

If two or three phases of a single component have the same chemical potential at a certain temperature and pressure, they will coexist at equilibrium as at the melting point , boiling point , or triple point. Below the melting point the solid has the lowest chemical potential and is therefore the stable phase. Between and the liquid is the stable phase. It may be seen from Fig. 6.2 that the

P T

b a

S .

S S S

T b

V .

V V V

b,d

g l s

ﬁrst- order phase transitions

冢冣

⭸ ⫺

⭸

⬎ ⬎

⭸

⬎⬎

⭸ ⭸

␮

phase transitions are sudden, but there are no indications of drastic change in the properties of the system as the temperature approaches the transition point.

For a single phase of a pure substance, the chemical potential is a function of temperature and pressure. Thus, the chemical potential can be represented as a surface in – – space. There are chemical potential surfaces of this type for each phase of a substance: gas, liquid, and one or more solid phases.

The surfaces representing any two phases will intersect along a line, and three surfaces will intersect at a point, called the triple point. The phase diagram for a one-component system is the projection of these intersections onto the

– plane.

Rather than looking at surfaces in three dimensions, it is more convenient to consider the chemical potential as a function of temperature at speciﬁed pressures, as in Fig. 6.2 .

The slopes of the lines giving the chemical potentials of solid, liquid, and gas in Fig. 6.2 are given by (see equation 4.84)

(6 1) Since the entropy is positive, the slopes are negative, and since , the slope is more negative for the gas than for the liquid and more negative for the liquid than for the solid.

At a lower pressure the plots of versus are displaced, as shown in Fig. 6.2 . The effect of pressure on the chemical potential of a pure substance at constant temperature is given by (see equation 4.85)

(6 2) Since the molar volume is always positive, the chemical potential decreases as the pressure is decreased at constant temperature. Since , , this effect is much greater for a gas than for a liquid or solid. As shown in Fig. 6.2 , reducing the pressure lowers the boiling point and normally lowers the melting point. The effect on the boiling point is much greater because of the large differ- ence in the molar volumes of gas and liquid. As a result, the range of temperature over which the liquid is the stable phase has been reduced. It is evident that at a sufﬁciently low pressure the curve for the chemical potential of the gas will in- tercept the solid curve below the temperature where the solid and liquid have the same chemical potential. At this low pressure the solid will sublime instead of melt; that is, it passes directly into the vapor state without going through the liquid state, as illustrated by dry ice in Fig. 6.1 .

At some particular pressure the solid, liquid, and vapor curves will intersect at a point; the temperature and pressure at which these three phases coexist is referred to as the triple point. If a substance can exist in more than one solid phase, the phase diagram will have more than one triple point, as illustrated in Fig. 6.1 .

The transitions we discussed in the preceding section are referred to as because there is a discontinuity in the ﬁrst derivatives of the chemical potential. Since ( / ) is different on the two sides of the transition temperature, the two phases have different entropies, and thus differ-

α Phase β

Phase P

dT T

2 1

C q T

C T T

P T

G V P S T

V P S T V P S T .

S S

P S H

T V V V T V .

S V

P T

H H H .

P P

181

T P

2 2

1 1 1

vap

sub fus

Coexistence curve for a one-component system. Along this line . If the temperature is changed by d , the pressure has to be changed by d as indicated to maintain equilibrium. At point 1,

. At point 2, d d , so that d d .

␮ ␮

␮ ␮ ␮ ␮

second-order phase transition

coexistence curve

Clapeyron equation,