FINAL EXAM, PHYSICS 5305, Spring, 2011, Dr. Charles W. Myles
In Class Exam: May 7, 2011
1. PLEASE write on one side of the paper only!! It wastes paper, but it makes my grading easier!
2. PLEASE don’t write on the exam sheets, there is no room! If you don’t have paper, I’ll give you
some.
3. PLEASE show ALL work, writing down at least the essential steps in the problem solution. Partial
credit will be liberal, provided that essential work is shown. Organized work, in a logical, easy to
follow order will receive more credit than disorganized work. Problems for which just answers are
shown, without the work being shown, will receive ZERO credit!
4. The setup (PHYSICS) of a problem counts more than the mathematics of working it out.
5. PLEASE write neatly. Before handing in the solutions, PLEASE: a) put problem solutions in
numerical order, b) number the pages & put them in order, & c) clearly mark your answers. If I can’t
read or find your answer, you can't expect me to give it the credit it deserves.
6. NOTE!! The words “DISCUSS” & “EXPLAIN” below mean to write English sentences in the
answer. They DON’T mean to answer using only symbols. Answers to such questions containing only
symbols without explanation of what they mean will get ZERO CREDIT!!! It would also be nice if
graduate physics students would try to write complete, grammatically correct English sentences!
NOTE: I HAVE 17 EXAMS TO GRADE!!! PLEASE HELP ME
GRADE THEM EFFICIENTLY BY FOLLOWING THESE
SIMPLE INSTRUCTIONS!!! FAILURE TO FOLLOW THEM
MAY RESULT IN A LOWER GRADE!!
THANK YOU!!
NOTE!! YOU MUST ANSWER ALL PARTS OF QUESTION 1! ANSWER ANY 3 OUT OF 2,3,4,5,
6!
So, answer ALL PARTS of 4 questions. Each is equally weighted & worth 25 points for 100 points
total.
If you want to & if there is time, work on a 4th problem from 2,3,4,5, 6 will count up to
10 BONUS POINTS!! YOU MUST TELL ME WHICH ONE YOU WANT TO COUNT FOR THIS BONUS!!
1. ALL PARTS OF THIS QUESTION ARE REQUIRED!!
a. Briefly DISCUSS, using WORDS, with as few mathematical symbols as possible, the PHYSICAL
MEANINGS of the following terms: 1. Microcanonical Ensemble, 2. Canonical Ensemble, 3. Grand
Canonical Ensemble, 4. Entropy, 5. Fermi Energy, 6. Pauli Exclusion Principle, 7. Classical Statistics, 8.
Quantum Statistics, 9. Equipartition Theorem. 10. Partition Function.
b. Briefly DISCUSS, using WORDS, NOT symbols, the fundamental differences between Fermions &
Bosons & how these differences lead to the fundamentally very different Fermi-Dirac & Bose-Einstein
Statistics. (That is, what are the basic, intrinsic properties that distinguish Fermions & Bosons?) In this
discussion, be sure to mention the many particle wavefunctions for both kinds of systems & include the
qualitative differences expected between the many particle ground states of the Fermi-Dirac & BoseEinstein systems.
c. In class, we discussed two different models to calculate the lattice vibrational contribution to the heat
capacity at constant volume, Cv, of a solid. In Ch. 7, we first discussed the Einstein Model. Then (for a
couple of lectures) we went forward to Ch. 10 to discuss the Debye model. Briefly DISCUSS, using
WORDS, NOT symbols, the major differences between the Einstein & Debye Models. Which model gives
a theoretical temperature dependence for Cv at low temperature which agrees with experiment?
NOTE!!!! WORK ANY 3 OUT OF PROBLEMS 2 through 6!
Work on a 4 problem from 2,3,4,5, 6 will count up to 10 BONUS POINTS! (Please say which one!!)
2. At absolute temperature T, the magnitude of the tension force F of a stretched plastic rod is related to
it’s length L by the expression F = aT2(L – L0). a is a positive constant. L0 is the rod’s unstretched
length. When L = L0, the temperature dependence of the heat capacity of the rod at constant length is
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CL(T,L0) = bT, where b is a constant. Assume that L is the only external parameter and
consider an infinitesimal, quasi-static process in which the length L changes by an amount dL. In what
follows, T0 is the temperature at length L0, E = Internal Energy and S = Entropy.
PROBLEM 2 CONTINUES ON THE NEXT PAGE!
PROBLEM 2 CONTINUED!!!
a.
b.
c.
d.
e.
Write, in differential form, using the differentials dE & dS, the combined 1st & 2nd Laws of
Thermodynamics for this system. Assuming that T and L are independent variables, derive a Maxwell
Relation for this system relating a partial derivative of S to a partial derivative of F.
Using the results of part a along with the function F given above, find an expression for the partial
derivative (S/L)T which involves T, a, L, and L0.
Assume that the entropy S(T0,L0) at L = L0 and T = T0 is known. Using the heat capacity CL(T,L0) at
L0 given above, find an expression for the entropy S(T,L0) for any T at length L0.
Using the results of parts b and c, find an expression for the entropy S(T,L) at all temperatures and
lengths.
Assume that, at an initial temperature Ti and initial length Li, the rod is stretched quasi-statically to a
given length Lf (Lf > Li). Calculate the final temperature Tf at the end of this process. Is Tf greater than
or smaller than Ti?
NOTE!!!! WORK ANY 3 OUT OF PROBLEMS 2 through 6!
Work on a 4 problem from 2,3,4,5, 6 will count up to 10 BONUS POINTS! (Please say which one!!)
3. Consider N identical, non-interacting magnetic atoms at thermal equilibrium at temperature T. The
system is in a static external magnetic field H in the z direction. Each atom has a magnetic moment μ.
Use CLASSICAL statistical mechanics to find CLASSICAL expressions for the thermodynamic
properties asked for in the following. Hint: You need to use the classical energy of a magnetic moment
μ in a magnetic field H: E = -μH cosθ. θ is the angle between μ & H. In what follows, the symbol E
denotes the mean energy <E> and the symbol M denotes the mean z component of magnetization
<Mz>.
th
a.
b.
c.
Calculate the partition function Z for this system.
Calculate the average (mean) z component of magnetic moment <μz>  μz, the average (mean) z
component of magnetization M and the mean energy E for this system.
Calculate the entropy S for this system.
For parts d, e, f, assume that the system undergoes an infinitesimal, quasi-static process in which the
external magnetic field is changed by dH. The mechanical work done by this process is dW = -MdH.
d.
e.
For this process, write, in differential form, involving the differential dE, the combined 1st & 2nd Laws
of Thermodynamics for this system, assume that the entropy S and the static magnetic field H are
independent variables. Use this result along with the properties of differentials to express T and M as
appropriate partial derivatives of E.
Use the properties of partial derivatives and the results of part d to relate an appropriate partial
derivative of M to a partial derivative of T, hence deriving one of Maxwell’s relations for this system.
NOTE!!!! WORK ANY 3 OUT OF PROBLEMS 2 through 6!
Work on a 4 problem from 2,3,4,5, 6 will count up to 10 BONUS POINTS! (Please say which one!!)
4. A classical monatomic, NON-IDEAL gas with N particles, in thermal equilibrium at temperature T, is
confined to volume V. In Ch. 10 of Reif, discussed in class, it is shown that the natural logarithm of
the partition function Z for this system has the form: ln(Z) = (3/2)N ln(A/β) + ln(ZU) – Nln(N) + N,
with A = (2πm/h2), β = (kBT)-1. ln(ZU) has the approximate form: ln(ZU) = N ln(V) - (N2/V)B2(T).
B2(T) is called the “2nd Virial Coefficient”. It is an integral which depends on the form of the
interaction potential between two particles. Ch. 10 also discusses that this Z gives an approximate
equation of state for this gas of the form P = kBT[n + B2(T)n2]. n = (N/V) is the number density.
There are other important properties of a gas besides it’s equation of state. For example:
th
a.
Derive expression for the mean energy E of this gas.
b.
c.
d.
Derive an expression for the heat capacity at constant volume, Cv of this gas.
Derive an expression for the entropy S of this gas.
Derive an expression for the chemical μ potential of this gas.
Note: For each part, assume that B2(T) is a known function & express your answers in general in terms
of it and it’s temperature derivatives.
NOTE!!!! WORK ANY 3 OUT OF PROBLEMS 2 through 6!
Work on a 4th problem from 2,3,4,5, 6 will count up to 10 BONUS POINTS! (Please say which one!!)
5. Consider a quantum mechanical ideal gas of N identical, structureless particles in thermal equilibrium
at temperature T and confined to volume V. Let the container be a cubic box of side L so that V = L3.
The quantized energies of the single particle quantum states are of the usual form for a particle in a
box:
εr = (ħ2π2)[(nx)2 + (ny)2 + (nz)2])/[2mL2]. nx, ny, nz, are integers and m is the particle mass.
Start with the canonical ensemble partition function Z for this gas. In Ch. 9 of Reif’s book, it is shown
that the natural logarithm of Z has the form (μ is the chemical potential):
ln(ZBE) = - μβN - ∑r ln{1 – exp[β(μ - εr)} for a Bose-Einstein gas.
ln(ZFD) = - μβN + ∑r ln{1 + exp[β(μ - εr)} for a Fermi-Dirac gas.
a.
Use the relations between ln(Z), the mean energy E, and the mean pressure P to show that the equation
of state of this gas is PV = (⅔)E, independent of whether the gas is composed Fermions or Bosons. (It
won’t hold if the Bosons are photons, which have a different equation of state.) You might also need
to use the general relation between the chemical potential μ and total particle number N. (Note: This
result should convince you that the classical “Ideal Gas Law” is not valid for quantum mechanical
gases!)
NOTE: In parts b and c, I want NUMBERS for Pf, & Tf, not just formal results with mathematical
symbols!
b. Consider an adiabatic, quasi-static expansion of this gas from an initial volume Vi = V to a final volume
Vf = 10V. If the initial pressure is Pi = 1 atm = 105 N/m2, calculate the final pressure Pf of the gas in
this process. This result should ALSO be independent of whether the gas is composed Fermions or
Bosons. In this calculation, neglect the interaction of the gas with the container walls.
c. Now, pecialize to the case of a Fermi-Dirac gas. For Fermions, it is shown in Ch. 9 of Reif’s
book that the mean energy at low temperatures T depends on T as E = E0 + AT2, where E0
and A are constants. In the process described in part a, if the initial temperature was Ti = 10K
and the initial volume was Vi = 1.0 m3, calculate the final temperature Tf . To obtain a
NUMBER for Tf, let the constant A = 3,000 Joules/K. (Note: The constant E0 should not be
needed!)
NOTE!!!! WORK ANY 3 OUT OF PROBLEMS 2 through 5!
Work on a 4th problem from 2,3,4,5, 6 will count up to 10 BONUS POINTS! (Please say which one!!)
6. A gas of N non-interacting hydrogen (H2) molecules is in thermal equilibrium at temperature T.
a. Assume that, for calculating vibrational properties, each H2 molecule can be treated as a
quantum mechanical simple harmonic oscillator with natural frequency ω. Find an expression
for the vibrational partition function Zvib of this gas.
b. Assume that, for calculating rotational properties, an H2 molecule can be treated as a quantum
mechanical rigid rotator. Thus, the quantized rotational energy states have energies of the form
EJ = J(J+1)(ħ)2/I where J is the rotational quantum number and I is the moment of inertia, for
which you may assume a classical “dumbbell” model. Recall from quantum mechanics that, in
addition to the quantum number J, each rotational energy state is also characterized by a
quantum number m, which can have any of the 2J +1 values m = -J, -(J - 1), -(J - 2),…,….(J
- 2), (J - 1), J. So, each rotational energy EJ is (2J + 1)-fold degenerate. Of course, this
degeneracy must be accounted for when the partition function is calculated. Write a formal
expression (“formal expression” means leave it as a sum or an integral which can’t easily be
evaluated in closed form) for the rotational partition function Zrot of this gas. Evaluate it in the
high temperature limit. What does the phrase “high temperature limit” mean here?
c. Still in the high temperature limit, calculate the total mean energy, including translational,
vibrational, and rotational parts.
d. Calculate the specific heat at low temperatures, assuming that the temperature is still high
enough that the N H2 molecules remain in a gaseous form.