Page 249 - Physical Chemistry

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Chapter 7 As the temperature of a liquid in equilibrium with its vapor is raised, the two
One-Component Phase Equilibrium phases become more and more alike until at the critical temperature T the
and Surfaces c
liquid–vapor interface disappears and only one phase is present. At T , the value of g
c
must therefore become 0, and we expect that g of a liquid will continually decrease as
T is raised to the critical temperature. The following empirical equation (due to
Katayama and Guggenheim) reproduces the g(T) behavior of many liquids:

g g 11 T>T 2 11>9 (7.28)
0
c
where g is an empirical parameter characteristic of the liquid. Since 11/9 is close to
0
1, we have g g g T/T , and g decreases approximately linearly as T increases.
0
0
c
Figure 7.18 plots g versus T for some liquids.
The quantity P in (7.26) is the pressure in each of the bulk phases a and b of the
system. However, because of the surface tension, P is not equal to the pressure exerted
by the piston in Fig. 7.19 when the system and piston are in equilibrium. Let the sys-
tem be contained in a rectangular box of dimensions l , l , and l , where the x, y, and
y
z
x
z axes are shown in Fig. 7.19. Let the piston move a distance dl in the process of doing
y
work dw rev on the system, and let the piston exert a force F pist on the system. The work
done by the piston is dw rev F pist dl [Eq. (2.8)]. Use of (7.26) gives F pist dl PdV
y
y
g d . The system’s volume is V l l l , and dV l l dl . The area of the interface
x z
y
x y z
Figure 7.18 between phases a and b is l l , and d l dl . Therefore F pist dl Pl l dl y
y
x
x y
x z
y
gl dl and
y
x
Temperature dependence of the
surface tension of some liquids. g F pist Pl l gl x (7.29)
x z
becomes zero at the critical point. exerted by the piston is F / F /l l , where
C H is naphthalene. The pressure P pist pist pist pist x z pist is the
10
8
piston’s area. F pist is in the negative y direction and so is negative; pressure is a posi-
tive quantity, so the minus sign has been added. Division of (7.29) by pist l l gives
x z
P pist P g>l z (7.30)
z
The term g/l is ordinarily very small compared with P.For the typical values l 10 cm
z z
and g 50 mN/m, one finds g/l 5 10 6 atm (Prob. 7.51).
z
y Since the force exerted by body A on body B is the negative of the force of B on
A (Newton’s third law), Eq. (7.29) shows that the system exerts a force Pl l gl on
x x z x
the piston. The presence of the interface causes a force gl to be exerted by the system
x
l y on the piston, and this force is in a direction opposite that associated with the system’s
pressure P. The quantity l is the length of the line of contact of the interface and the
x
piston, so g is the force per unit length exerted on the piston as a result of the exis-

tence of the interphase region. Mechanically, the system acts as if the two bulk phases
l z were separated by a thin membrane under tension. This is the origin of the name “sur-
face tension” for g. Insects that skim over a water surface take advantage of surface
tension.
In the bulk phases a and b in Figs. 7.15 and 7.19, the pressure is uniform and
Figure 7.19
equal to P in all directions. In the interphase region, the pressure in the z direction
A two-phase system confined by a equals P, but the pressure in the x and y directions is not equal to P. Instead, the fact
piston. that the pressure (7.30) on the piston is less than the pressure P in the bulk phases tells
us that P (the system’s pressure in the y direction) in the interphase region is less than
y
P. By symmetry, P P in the interphase region. The interphase region is not homo-
x y
geneous, and the pressures P and P in this region are functions of the z coordinate.
x y
Because the interphase region is extremely thin, it is an approximation to talk of a
macroscopic property like pressure for this region.
Measurement of the surface tension of a solid is very difficult.
One can modify the thermodynamic equations of Chapter 4 to allow for the ef-
fects of the interface between phases. The most common way to do this was devised
by Gibbs in 1878. Gibbs replaced the actual system by a hypothetical one in which the