230 5 Phase EquilibriumPROBLEMS
Section 5.5: Surfaces in One-Component Systems
5.36 Estimate the surface tension of ethanol as was done for
carbon tetrachloride in Example 5.10. The enthalpy change
of vaporization is roughly equal to 40 kJ mol−^1. State any
assumptions or approximations. Compare your answer to
the correct value, and explain any discrepancy.
5.37 Explain why interfacial tensions between two liquid
phases are generally smaller in magnitude than surface
tensions between liquid and vapor phases.
5.38 If the surface region is assumed to be two molecular
diameters thick, and if the average density in the surface
region is assumed to be half of that of the liquid, estimate
the fraction of the molecules of a sample of water that are
in the surface region if the water exists as (a) spherical
droplets with diameter 10. 0 μm and (b) a spherical “drop”
containing 1.00 mol.
5.39 The surface tension of water at 25◦C is equal to
0 .07197 N m−^1. Find the capillary rise of water in a glass
tube of diameter 0.095 mm at this temperature. State any
assumptions. Assume that the contact angle
equals 0.
5.40 Give an alternate derivation of the capillary rise formula
for zero contact angle beginning with the Laplace
equation, Eq. (5.5-14), and assuming that the meniscus is a
hemisphere.
5.41 Calculate the capillary rise of pure water in a glass
capillary of radius 0.350 mm at 298.15 K. Assume zero
contact angle.
5.42 Find the height of the column of pure water in a capillary
tube of diameter 0.125 mm at 25◦C. The surface tension of
water at this temperature is 0.0720 J m−^2.
5.43 One method of measuring the surface tension of a liquid is
to measure the force necessary to pull a fine wire ring out
of the surface of the liquid. For ethanol at 20◦C, calculate
the force for a ring 25.0 mm in diameter. Remember that
there is a surface on both the inner and outer diameter of
the ring.
5.44 The surface tension of mercury at 20◦C is equal to
0 .4355 N m−^1. The density of mercury is equal to
13 .56 g cm−^3. The contact angle of mercury against glass
is 180◦. Find the capillary depression (distance of the
meniscus below the liquid surface) for mercury in a glass
capillary tube of radius 0.35 mm at 20◦C.
5.45 a.If the barometric pressure is 760.00 torr, find the
pressure at 25◦C inside a droplet of water with diameter
1. 25 μm 1. 25 × 10 −^6 m.
b. Find the partial pressure of water vapor that would be
at equilibrium with the droplet of part a at 25◦C. The
normal vapor pressure of water at this temperature (for
a planar surface) is 23.756 torr.
5.46 Calculate the vapor pressure of a droplet of ethanol with a
radius of 0.00400 mm at 19◦C. The vapor pressure at a
planar surface is equal to 40.0 torr at this temperature.
5.47 Give an alternate derivation of the expression for the vapor
pressure of a spherical droplet, Eq. (5.5-16), using the
Laplace equation, Eq. (5.5-14), and the relation between
pressure on the liquid phase and the vapor pressure,
Eq. (5.3-18).5.6 Surfaces in Multicomponent Systems
In Example 5.10 we estimated the surface energy of a one-component liquid–vapor
surface as though a single layer of molecules had the normal liquid on one side and the
vapor on the other side. A solid surface might resemble this crude model, but a liquid
surface is more disordered and it is perhaps appropriate to call it a surface region or a
surface phase. Figure 5.19a shows schematically an average density profile through a
one-component liquid–vapor surface at equilibrium. The thickness of a liquid–vapor
interfacial region might be equal to several molecular diameters.
We assume that the surface is planar and horizontal and that the surface region
extends fromz 1 toz 2. We place a dividing plane at a locationz 0 inside the surface
region. The volume of each phase is assigned to be the volume that extends to the