MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Thermodynamics of Materials

3.00 Fall 2000

W. Craig Carter

Department of Materials Science and Engineering

Massachusetts Institute of Technology

77 Massachusetts Ave.

Cambridge, MA 02139

Problem Set 7: Not to be turned in or graded (Intended as Practice for Exam 2)

This problem set is designed to simulate the second exam that will take place Friday Nov. 17. I expect that each student could answer all the questions in about 45 minutes. You need not turn it in, but I encourage you to compare your answers with your working group. You should study until are satisfied with your answers--this would be much better preparation for the exam than just asking for the answers. Don't expect the TA or the instructor to just tell you the answer. It is also good practice to design your own questions in a similar vein. If you think of a good question that would take 15 minutes to answer and tests something that everyone in the class should know by now, then send it to me; who knows, it may appear on the exam.

Note that the questions ask you to demonstrate your understanding of the lecture material from Lecture 12 up to and including the material that gets covered in the lecture notes for Lecture 26 which takes place on the Monday before the exam.

When answering questions like these, please think through your response carefully. You may wish to write your answer down on scratch paper and then refine it until you are satisfied with it.

Accuracy is valued above all. Clarity is nearly as important. A brief answer is almost invariably better than an incoherent rambling answer. Furthermore, you should consider that all of your answer will be graded--including the parts that are incorrect.

This is how I intend to grade the exams. For each question, I will read every exam and sort them into about five piles. I will iterate between reading and ranking until I have ranked them according to my best judgment. I will assign points according to the ranking.

Exercise 7.1

Please ascertain whether the following statements are true or false under the assumptions of the laws of thermodynamics. If the statement is true, indicate which of the thermodynamic laws or thermodynamic principles pertain. If it is false, indicate your thermodynamic reasoning or with the presentation of a physical counter-example.

1. The entropy of a material can never decrease.

2. A body in equilibrium and in thermal and mechanical contact with a reservoir at constant pressure and temperature will have the lowest possible value of Gibbs Free energy.

3. If two phases are in equilibrium at constant pressure, then they must have the same value of Gibbs free energy.

4. If two phases are in equilibrium at constant pressure and temperature, then they must have the same composition.

5. Melting of a fixed amount of a pure material at constant pressure is an endothermic process when the entropy of a liquid is greater than the entropy of a solid.

6. Pure gold does not melt at room temperature at one atmosphere because the internal energy of the universe would not be conserved.

7. For a system composed of $f$ phases, and $C$ components with chemical potentials $\mu_i^j$ for the $i$-th component in the $j$-th phase, $\sum_i^C \sum_j^f \mu_i^j$ will always have its smallest possible value.

Exercise 7.2

A pure substance at constant pressure is observed to melt at $T_m$ and boil at $T_v$. For a temperature range $T_{low} < T_m < T_v < T_{high}$ and constant pressure, sketch plots of the following:

1. versus $T$.

2. $\ensuremath{{G}_{\mbox{total}}}$ versus $T$.

3. Phase fractions versus $\ensuremath{{H}_{\mbox{total}}}$.

4. $\ensuremath{\overline{G}}$ versus $\ensuremath{{H}_{\mbox{total}}}$.

Exercise 7.3

Write out the differential expressions for the thermodynamic energies $dU$, $dF$, $dH$, and $dG$ for a closed system that does only $PV$ work. Relate the derivatives of the energies to thermodynamic quantities. Write out all resulting Maxwell relations.