MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Thermodynamics of Materials

3.00 Fall 2001

W. Craig Carter

Department of Materials Science and Engineering

Massachusetts Institute of Technology

77 Massachusetts Ave.

Cambridge, MA 02139

Problem Set 6: Due Fri. Nov. 2, Before 5PM in 4-047

Exercise 6.1

Consider a ternary system composed of independent chemical species $A$, $B$, and $C$. Suppose there are three phases ($\alpha$, $\beta$, and $\gamma$) in a closed system that has $N_A$, $N_B$, and $N_C$ molecules of $A$-, $B$-, and $C$-type. Find an expression for the phase fractions in terms of the compositions of the system and the various phases.

Generalize your result to a system with $C$ independent components and $f$ phases.

Exercise 6.2

The bulk modulus, $B$, is a material property that relates the change in volume with a change in pressure.

$$B \equiv -V \ensuremath{ \left( \frac{\partial{P}}{\partial{V}} ... ...math{ \left( \frac{\partial{P}}{\partial{\log V}} \right)_{T=\mbox{constant}} }$$

A typical bulk modulus of medium density polyethylene is about $10^{6}$ pascals. At one atmosphere pressure, the ratio of the density of this polyethylene to water is about $0.93$.

The deepest part of the ocean is located 338 km southwest of Guam at the Marianas trench and is 10.3 km below sea level.

Calculate the reversible work required to sink 1 kg of medium density polyethylene to the deepest part of the ocean.

List some of the approximations that you have made to obtain your solution and try to justify these approximations.

You will want to save your work, because there may be a follow-up to this problem.

Exercise 6.3

In the lectures, the molar free energies plotted against temperature at constant pressure had negative slopes and negative curvatures (see Figure 16-3).

Why?

If a chemical species is completely soluble in a liquid and not soluble at all in a solid, what effect will the addition of that soluble species have on the molar Gibbs free energies of the liquid and solid phases?

Using Figure 16-3, what effect will the addition of a species that is soluble in one phase and not soluble in the other have on the equilibrium transition temperature?