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Chapter 8 tension, which corresponds to a negative pressure. For water, negative pressures of
Real Gases hundreds of atmospheres have been observed. Sap in plants is at a negative pressure
(P. G. Debenedetti, Metastable Liquids, Princeton, 1996, sec. 1.2.3). According to the
cohesion–tension theory of sap ascent in plants, water in plants is pulled upward by
negative pressures created by evaporation of water from the leaves; the term cohesion
refers to intermolecular hydrogen bonding that holds the water molecules together in
the liquid, allowing for large tensions. Direct measurements of negative pressures in
plants support the cohesion–tension theory [M. T. Tyree, Nature, 423, 923 (2003)].

8.5 CALCULATION OF LIQUID–VAPOR EQUILIBRIA

At any given temperature T, an equation of state can be used to predict the vapor pres-
l
v
sure P, the molar volumes V and V of the liquid and vapor in equilibrium, and the
m
m
enthalpy of vaporization of a substance.
For the 200°C isotherm in Fig. 8.3, the points J and N correspond to liquid and
vapor in equilibrium. The phase-equilibrium condition is the equality of chemical po-
v
l
v
tentials of the substance in the two phases: m m or G l m,J G m,N , since m G for
N
J
m
v
l
a pure substance. Dropping the J and N subscripts, we have G G or in terms of
m
m
the Helmholtz function A:
v
l
l
A PV A PV v m
m
m
m
v l v l
P1V V 2 1A A 2 (8.23)
m
m
m
m
The Gibbs equation dA S dT PdV at constant T gives dA PdV and
m
m
m
m
m
integration from point J to N along the path JKLMN gives
v
A A V m v P dV const. T
l
m
m
m
eos
l
V m
where eos indicates that the integral is evaluated along the equation-of-state isotherm
JKLMN. Equation (8.23) becomes
v
l
P1V V 2 V m v P dV const. T (8.24)
m
eos
m
m
l
V m
The left side of (8.24) is the area of a rectangle whose top edge is the horizontal line
l
v
JLN of length (V V ) in Fig. 8.3 and whose bottom edge lies on the P 0 (hori-
m
m
zontal) axis. The right side of (8.24) is the area under the dotted line JKLMN. This
M
area will equal the rectangular area only if the areas of the regions labeled I and II in
II Fig. 8.5 are equal (Maxwell’s equal-area rule).
J N For the Redlich–Kwong equation (8.3), Eq. (8.24) becomes
I L v
v
l
P1V V 2 V m c RT a d dV const. T
m
m
m
m
l V b V 1V b2T 1>2
m
m
V m
v
l
v
1 V b a V 1V b2
m
m
m
P cRT ln ln d (8.25)
v
v
K V V m V b bT 1>2 1V b2V l m
l
l
m
m
m
Figure 8.5 where the identity [v(v b)] 1 dv b 1 ln [v/(v b)] was used. In addition to sat-
isfying (8.25), the Redlich–Kwong equation (8.3) must be satisfied at point J for the
JKLMN is a cubic-equation-of- liquid and at point N for the vapor, giving the equations
state isotherm in the liquid–vapor
region on a P-versus-V plot RT a RT a
m
(Fig. 8.3). Areas I and II must be P l and P v (8.26)
v
l
v
l
equal. V b V 1V b2T 1>2 V b V 1V b2T 1>2
m
m
m
m
m
m