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MIDTERM EXAM, PHYSICS 5305, Spring, 2008, Dr. Charles W. Myles
Take Home Exam: Distributed, Wednesday, March 5
DUE, 5PM, FRI., MARCH 14!! NO EXCEPTIONS!
Bring it to my office or put it in my mailbox.
(I prefer it in my mailbox! Put it in a sealed envelope!)
RULE: You may to use almost any resources (library, internet, etc.) to solve these problems.
EXCEPTION: You MAY NOT COLLABORATE WITH ANY OTHER PERSON! If
you have questions/difficulties, consult with me, not with other students (whether or not they are in this
class!), with people who had this course last year, with other faculty, with post-docs, or with anyone else I
may have forgotten to list here. You are bound by the TTU Code of Student Conduct not to violate
this! Anyone caught violating this will, at a minimum , receive an “F” on this exam!
INSTRUCTIONS: Read all of these before doing anything else!!! Failure to follow them may lower your
grade!!
1. PLEASE write on ONE SIDE of the paper only!! This wastes paper, but it makes my grading easier!
2. PLEASE do not write on the exam sheets, there will not be room! Use other paper!!
3. PLEASE show all of your work, writing down at least the essential steps in the solution of a problem.
Partial credit will be liberal, provided that the essential work is shown. Organized work, in a logical,
easy to follow order will receive more credit than disorganized work.
4. PLEASE put the problems in order and the pages in order within a problem before turning in this
exam!
5. PLEASE clearly mark your final answers and write neatly. If I cannot read or find your answer, you
can't expect me to give it the credit it deserves and you are apt to lose credit.
6. NOTE: The words EXPLAIN and DISCUSS mean to do this briefly, using complete,
grammatically correct English sentences!
7. NOTE!!! The setup (THE PHYSICS) of a problem will count more heavily in the grading than the
detailed mathematics of working it out.
PLEASE FOLLOW THESE SIMPLE DIRECTIONS!!!! THANK YOU!!!
NOTE!!!! YOU MUST ANSWER PROBLEM 1! WORK ANY 5 OUT OF THE
OTHER 6 PROBLEMS! So, you must answer 6 problems total. Problem 1 is worth 10
points. Each of the others is equally weighted & worth 18 points. So, 100 is the maximum points possible.
Please sign this statement and turn it in with your exam:
I have neither given nor received help on this exam Signature
______________________________
1. REQUIRED QUESTION!!
a. A fair coin is tossed 500 times. Use the Gaussian approximation to find the
probability of obtaining 290 heads.
b. Write the 1st Law of Thermodynamics for an infinitesimal, quasi-static process in a
system characterized by absolute temperature T and one external parameter x. EXPLAIN the
physical meaning of every symbol you write. DISCUSS the physical meaning of this law.
c. Write the 2nd Law of Thermodynamics for the system of part a. EXPLAIN the physical
meaning of every symbol you write and DISCUSS the physical meaning of this law.
d. Write the 3rd Law of Thermodynamics for system in part a. EXPLAIN the physical
meaning of every symbol you write and DISCUSS the physical meaning of this law.
e. DISCUSS the physical meaning of entropy.
f. State the Fundamental Postulate of Statistical Mechanics. DISCUSS it’s physical
meaning and some of it’s consequences.
NOTE!!!! WORK ANY 5 OUT OF PROBLEMS 2 through 7!
2. Consider a classical plane pendulum. The mass m, is suspended from the ceiling by a massless wire of
length ℓ. It oscillates in a plane with no friction. The total mechanical energy is
E = [(L2)/(2mℓ2)] + mgℓ(1 – cosθ). L is the angular momentum about the
suspension point. θ is the oscillation angle. θ = 0 is where the wire is vertical. See
figure 
In what follows, make the small θ approximation, so that (1 – cosθ) may be
approximated as (½)θ2.
a. For fixed energy E, sketch the allowed region in the classical phase space for this pendulum
(the θ-L plane) that is accessible to the pendulum for all energies less than or equal to E. If the
energy is in the very small range E to E + δE, sketch (on the same figure you just sketched) the
allowed region of phase space that it accessible to the pendulum.
b. For length ℓ = 1.5 m, mass m = 2 kg, and maximum angle θm = 10º, calculate the number of
allowed states for this pendulum for all energies less than or equal to E. Take the area of the
cell in phase space to be Planck’s constant ħ. Put in NUMBERS! I want a NUMERICAL
VALUE!
c. In the small θ approximation, this pendulum behaves approximately as a simple harmonic
oscillator. So, the quantum mechanics of it is approximately the same as it is for the quantum
mechanical harmonic oscillator. Using the approximate quantum mechanical expression for
the energy E of the same pendulum as in part b, estimate the quantum number n. Compare this
result to the results of part b.
3. See figure. In a quasi-static process, a certain gas changes from
macrostate A to macrostate B (A  B). The mean pressure P of the gas
is found to change with it’s volume V as P = aV-(5/3). a is a constant.
Find the quasi-static work W done and the net heat Q absorbed by the
gas in each of the following processes. (Each process changes the gas from
A to B).
a. The system is expanded from it’s original volume to it’s final
volume under isobaric (constant pressure) conditions. The volume
is then kept constant and heat is extracted to reduce the pressure
to 106 dynes/cm. This process is labeled by a in the figure.
b. The volume is increased and heat is supplied to cause the pressure to decrease linearly with
volume. This process is labeled by b in the figure.
c. The two steps of the process described in part a are preformed in the opposite order. This
process is labeled by c in the figure.
4. Consider the chemical reaction A + B  C. The molar heat capacities (in units of [J/(mole-K)]) at
constant pressure of substances A,B, and C are the following functions of absolute temperature T: CA
= 3T2, CB = 5T½, CC = 10T. This reaction is carried out at constant temperature T = 300K.
a. Calculate the entropy change per mole, S, of substance C produced.
b. Does the sign of your answer violate the 2nd Law of Thermodynamics? If not, why not?
EXPLAIN briefly, using complete, grammatically correct English sentences.
c. Calculate the heat released per mole, Q, of substance C produced.
d. Assuming that the mechanical work, W, done in this reaction is zero, calculate the total
change in internal energy per mole E produced in this reaction.
NOTE!!!! WORK ANY 5 OUT OF PROBLEMS 2 through 7!
5. The number of accessible states for a certain system of N particles confined to volume V, with energy
in the range E to E + δE is Ω(E,V,N) = AE(5N/6)exp[BN3V(4/3)]. A & B are constants.
a. Calculate the entropy in terms of E,V and N.
b. The absolute temperature of the system is T. Calculate the internal energy E as a function of
T,V, and N.
c. Find the equation of state P(V,T,N) for this system, where P is the pressure.
d. Calculate the heat capacity CV,N at constant volume V and constant particle number N for this
system.
e. Calculate the volume coefficient of expansion α and the isothermal compressibility κ for this
system. (See Reif, Ch. 5)
f. Calculate the heat capacity CP,N at constant pressure P and constant particle number N for this
system.
6. Consider a system of magnetic moment μ in a static magnetic field B. The system is in thermal
equilibrium at absolute temperature T. Consider an infinitesimal, quasi-static process in which the
magnetic field is changed by dB. Recall that, in such a situation, the mechanical work done is - μdB.
For this system, use the following definitions (E is the internal energy, S is the entropy): Enthalpy: H = E
– μB. Helmholtz Free Energy: F = E – TS, Gibbs Free Energy: G = E – TS - μB. In solving this
problem PLEASE be VERY CAREFUL in your partial derivative notation about which variables are
held constant when the derivative is taken.
a.
b.
c.
d.
e.
f.
Write, in differential form, using the differential dE, the combined 1st & 2nd Laws of
Thermodynamics for this system, assuming that the entropy S & the magnetic field B are independent
variables.
Use the properties of differentials and the results of part a to express T and μ as partial derivatives of
E.
Use the properties of partial derivatives and the results of part b to relate an appropriate partial
derivative of μ to a partial derivative of T, hence deriving one of Maxwell’s relations for this system.
Repeat parts a,b,c, except begin by writing the combined 1st and 2nd Laws in terms of a differential of
F, assuming T & B are independent variables and expressing S and μ as appropriate partial derivatives
of F.
Repeat parts a,b,c, except begin by writing the combined 1st and 2nd Laws in terms of a differential of
H, assuming S and μ are independent variables and expressing T & B as appropriate partial derivatives
of H.
Repeat parts a,b,c, except begin by writing the combined 1st and 2nd Laws in terms of a differential of
G, assuming T and μ are independent variables and expressing S & B as appropriate partial derivatives
of G.
7. Consider a mole of steam under high pressure, so that the Van der Waals equation is the equation of
state. Van der Waals constants for steam: a = 6.0 (liter)2–atm/(mole)2, b = 0.8 liter/mole. It is initially
at pressure Pi = 40 atm and has volume Vi = 0.4 liter. It is then expanded to Vf = 0.8 liter.
a. Calculate the initial temperature of the steam.
b. If the expansion is isobaric (at constant pressure), calculate the final temperature.
c. Calculate the mechanical work (in Joules) done in the isobaric expansion of part b.
d. If the expansion is isothermal (at constant temperature), calculate the final pressure.
e. Calculate the mechanical work (in Joules) done in the isothermal expansion of part d.
For parts f and g ONLY, assume that the ideal gas law holds, but that γ = cp/cv = 2.15.
f. Calculate the final temperature and pressure if the expansion is adiabatic (at constant entropy).
g. Calculate the mechanical work (in Joules) done in the adiabatic expansion of part f.