CH353 Homework #4
Answer the following two questions, based on the posted reading “Kinetics of Phase
Transformations” (from The Principles of Engineering Materials, Barrett, Nix and
Tetelman):
1. Suppose that nuclei formed during a phase transformation (for instance in forming a
solid crystal from a liquid) are cubic in shape rather than spherical. If a is the
dimension of the cube, compute a value for the critical nucleus size, a*, and critical
free energy, G*, necessary to form a stable nuclei, in terms of the surface tension, ,
and latent heat of melting, Ev.
The reasoning for the formation of the cubic critical nucleus is the same as for the
spherical critical nucleus, except that the volume and surface area terms that appear in the
free energy cost of forming the nucleus are now a
2. A small droplet (1 mg) of liquid copper is cooled until it solidifies. Calculate the
amount of undercooling (i.e., the temperature at which at least one stable nucleus
forms) if Tmelt = 1083°C,  = 200 ergs/cm2, density = 10 g/cm3, and Ev = -50 cal/g.
(Hint: You will need to numerically solve an implicit equation (by iteration or other
method) for T such that n*=1.) You can assume that the density of nuclei sites is 1020
per cm3. (This is the value given in the example in the reading. Actually, it is also
possible to estimate the density of potential nuclei sites as (r*) -3, where r* is the
critical nuclei radius. This expression for r* can then be used to solve for n*, but this
will lead to a negligible change in the final value for n*, since the temperature
dependence occurs primarily in G*, not s.)
Answer the following, based on the posted reading “Surface Phenomena” (from Physical
Chemistry, 6th ed., Barrow):
3. Suppose that water in a certain container boils when the liquid is at a temperature at
which bubbles of 10-7 m diameter can be formed by the equilibrium vapor. Estimate
from approximate relations and data given in the reading (a) the vapor pressure that
must be reached by the liquid for these vapor bubbles to be formed, and (b) the
temperature at which boiling will occur at a pressure of 1 bar.
4. What is the minimum size of the droplets that could grow spontaneously from a vapor
with twice the equilibrium vapor pressure at 20°C if the vapor is (a) benzene, and
(b) water. You can use a value of .9 g/cm3 for benzene. (At 20°C the vapor pressure
of benzene is about 0.1 bar, and for water it is about 0.023 bar. Do we need this
information to solve the problem?)