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, DG*, necessary to form a stable nuclei, in terms of the surface tension, g, and latent heat of melting, DEv.
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, g = 200 ergs/cm2, density = 10 g/cm3, and DEv = -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 DG*, 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?)