Physics 406
Name____________________________________
April 29, 1998
Final Exam

Please show all of your work (formulas used and intermediate steps) on these pages. Students showing the most work will receive the most credit. All you will need for this exam are a pencil, a calculator, and your three formula pages.

1. (20 points) Water Molecule
The water molecule (H₂O) is non-linear. The hydrogen atom has an atomic mass of 1 while the oxygen atom has an atomic mass of 16. Consider a system of water molecules in the gas phase at 450 Kelvins.
a) How many degrees of freedom does a single water molecule have?
b) What is the average translational kinetic energy of a single water molecule?
c) What is the average velocity of a single water molecule?
d) What is the average total rotational kinetic energy of a single water molecule?
e) What is the average vibrational energy of a single water molecule?
f) What is the average potential energy associated with vibrations of a single molecule?

2. (20 points) Carnot Cycle
a) The set of axes drawn below has temperature on the vertical axis and entropy on the horizontal axis. On this set of axes sketch in the four steps of the Carnot cycle. Label each step clearly. Assume that the high temperature is T_h and the low temperature is T_l.
b) What is the general connection between heat and entropy?
c) How does the above connection apply to the steps of the Carnot cycle?
d) How would Q_h, the heat added at the high temperature, be represented on this set of axes?
e) How would the net work done in one cycle be represented on this set of axes?

T 0
	0 S

3. (20 points) Orbitals
Consider a system with four particles in five orbitals of distinct energy.
a) How many microstates does the system have if the particles are distinguishable?
b) How many microstates does the system have if the particles are bosons?
c) How many microstates does the system have if the particles are fermions?
d) What would be your answers to a), b), and c) if the particles satisfied classical statistics only?

4. (20 points) Equipartition Theorem
We used the Equipartition Theorem extensively in the early portions of this course to get a feeling for the concept of temperature and to help us understand thermal conductivity processes. We now know, I hope, that it is of limited usefulness.
a) First, state the Equipartition Theorem. Remember that there are two distinct aspects to this Theorem.
b) Second, explain, on the basis of the Equipartition theorem, why energy flows from high temperature to low temperature when two blocks, at initially different temperatures, are placed in direct contact. Also explain, on the same basis, why the energy flow stops when the two blocks are at the same temperature.
c) Third, state the limitations of the Equipartition Theorem and explain why it fails in some systems. You should be able to find at least three limitations.

5. Maxwell Relations
a) Write down the First Law of Thermodynamics in differential form.
b) Solve for dN in terms of other differentials.
c) What are the appropriate independent variables for N, the particle?
d) From your answers to b) and c) you can construct three partial derivatives of N. Express each of these derivatives in terms of thermodynamic variables.
e) Now construct two Maxwell relations using your answer to d).

6. (20 points) Indium Phosphide
The semiconductor indium phosphide (InP) has a band gap of 1.35 eV, the effective mass of the holes in the valence band is 0.42 times the mass of an electron and the effective mass of the electrons in the conduction band is 0.073 times the mass of an electron. In this problem measure all energies with respect to the top of the valence band.
a) Find the quantum concentration of the electrons in the conduction band at room temperature.
b) Find the quantum concentration of the holes in the valence band at room temperature.
c) Find the Fermi energy at zero temperature. Give your answer in electron volt (eV) units.
d) What is the Fermi energy at room temperature?
e) Find the concentration of electrons in the conduction band at room temperature.
f) Find the concentration of holes in the valence band at room temperature.
g) Qualitatively, how do your answers to d) and e) change when acceptor atoms are added to the material?

7. (20 Points) Ideal Gases
A large room is partitioned so that one third of the room is separated from the remaining two thirds. Identical gas molecules are introduced into each side. The total energy of gas molecules on the smaller side (side #1) is ½ that of the gas molecules on the larger side (side #2), and the total particle number of side #1 is four times that of side #2. The partition is then changed to one that is thermally conducting and moveable.
a) Which side gains in temperature and why?
b) Which side expands at the expense of the other and why?
c) What is the final temperature of the two systems. Express your answer in terms of the initial temperature of side #2, T₂.
d) What is the final pressure of the two systems. Express your answer in terms of the initial pressure of side #2, p₂.
e) Will the entropy of this system change when the gases interact? Explain your answer.

8. (20 points) Partition Function
The partition function for a certain system is given by Z(T,V,N) = C V^N (kT)^7N/2, where C is a constant.
a) What is the average energy of this system as a function of T, V, and N?
b) What is the average pressure of this system as a function of T, V, and N?
c) What is the average chemical potential of this system as a function of T, V, and N?
d) What is the Helmholtz Free Energy of this system as a function of T, V, and N?
e) Identify this system and give the number of degrees of freedom it possesses.

9. (20 points) Energy and Particle Spectrum. Consider a small system which is undergoing a thermal and diffusive process with a large environment at temperature T and chemical potential µ. The energy and particle number spectrum is given below, where is a positive constant.

s	Es	Ns
1	e	1
2	2e	1
3	2e	2

a) Find the partition function of this system.
b) Find the probability of having microstate number 2.
c) Find the probability of having energy 2.
d) Find the probability of having particle number 1.
e) Find the average energy of the system.
f) Find the average particle number.

10. (20 points) Mini-Problems
a) Where does the relationship Z(T,V,N) = Z(T,V,1)^N/N! come from? What are the assumptions used to obtain it and what are its limitations.
b) What role does the fermi energy play in determining whether a material is a solid or an insulator. You may draw a sketch or two to make your point.
c) What is statistical determinism and why is it an important concept in this course?
d) What is the "orbital picture"? Describe its main features. Explain why was it necessary for solving certain problems in this course.