Page 251 - Physical Chemistry

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Chapter 7 To allow for this pressure difference, we rewrite the definition (7.26) of g as
One-Component Phase Equilibrium
b
a
a
b
and Surfaces dw rev P dV P dV g d (7.31)*
a
V V V b
a
a
b
a
where P dV is the P-V work done on the bulk phase a, V and V are the volumes
of phases a and b, and V is the total volume of the system. Since the volume of the
interphase region is negligible compared with that of a bulk phase, we have taken
a
b
V V V.
b
a
To derive the relation between P and P , consider the modified setup of Fig. 7.21b.
We shall assume the interface to be a segment of a sphere. Let the piston be re-
versibly pushed in slightly, changing the system’s total volume by dV. From the def-
inition of work as the product of force and displacement, which equals (force/area)
(displacement area) pressure volume change, the work done on the system by
†
†
the piston is P dV, where P is the pressure at the interface between system and
†
b
surroundings, where the force is being exerted. Since P P , we have
b
b
a
a
b
b
b
dw rev P dV P d1V V 2 P dV P dV b (7.32)
Equating (7.32) and (7.31), we get
b
b
a
b
P dV P dV P dV P dV g d
a
a
b
b
a
a
b
P P g1d >dV 2 (7.33)
Let R be the distance from the apex of the cone to the interface between a and b in
Fig. 7.21b, and let the solid angle at the cone’s apex be . The total solid angle around
4
a
3
a point in space is 4p steradians. Hence, V equals /4p times the volume pR of a
3
2
sphere of radius R, and equals /4 times the area 4 R of a sphere. (In Fig. 7.21b,

p
p
all of phase a is within the cone.) We have
2
3
a
V R >3, R
2
a
dV R dR, d 2 R dR
a
Hence d /dV 2/R and (7.33) for the pressure difference between two bulk phases
separated by a spherical interface becomes
2g
b
P P spherical interface (7.34)
a
R
Equation (7.34) was derived independently by Young and by Laplace about 1805. As
R → q in (7.34), the pressure difference goes to zero, as it should for a planar inter-
face. The pressure difference (7.34) is substantial only when R is small. For example,
b
a
for a water–air interface at 20°C, P P is 0.1 torr for R 1 cm and is 10 torr for
R 0.01 cm. The pressure-difference equation for a nonspherical curved interface is
more complicated than (7.34) and is omitted.
One consequence of (7.34) is that the pressure inside a bubble of gas in a liquid
is greater than the pressure of the liquid. Another consequence is that the vapor pres-
sure of a tiny drop of liquid is slightly higher than the vapor pressure of the bulk liq-
uid; see Prob. 7.72.

Equation (7.34) is the basis for the capillary-rise method of measuring the surface

tension of liquid–vapor and liquid–liquid interfaces. Here, a capillary tube is inserted
in the liquid, and measurement of the height to which the liquid rises in the tube allows
(a) (b) calculation of g.You have probably observed that the water–air interface of an aque-
ous solution in a glass tube is curved rather than flat. The shape of the interface de-
Figure 7.22 pends on the relative magnitudes of the adhesive forces between the liquid and the
glass and the internal cohesive forces in the liquid. Let the liquid make a contact angle
Contact angles between a liquid
and a glass capillary tube. u with the glass (Fig. 7.22). When the adhesive forces exceed the cohesive forces, u lies