MIDTERM EXAM, PHYSICS 5305, Spring, 2010, Dr. Charles W. Myles

Take Home Exam: Distributed Monday, March 8

DUE, CLASS TIME, MONDAY, MARCH 22!! NO EXCEPTIONS!

RULE: You may use almost any resources (library, internet, etc.) to solve these problems.

EXCEPTION: You MAY NOT COLLABORATE WITH ANY OTHER PERSON!

For questions/difficulties, consult with me, not with other students (whether or not they are in this class!), with people who had this course previously, with other faculty, with post-docs, or with anyone else I may have forgotten 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: Please read all of these before doing anything else!!!

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! They DO NOT mean to answer with only equations! Answers to such questions which are equations only with no explanation of what they mean will receive ZERO credit.

7.  NOTE!!! The setup (THE PHYSICS) of a problem will count more heavily in the grading than the detailed mathematics of working it out.

NOTE: I HAVE 16 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!!!! PROBLEM 1 IS REQUIRED! 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 ______

a.  Consider 8 true dice. (A “true” die is one that is a perfect, uniform density cube so that, on a given roll, the probability that any one face will be the upper face is  ≈ 0.16667). Calculate the probability that, when these 8 dice are thrown together, five and only five will land with four dots up. Calculate the probability that, when these 8 dice are thrown together, that at least one will land with four dots up.

b.  State the Fundamental Postulate of Equilibrium Statistical Mechanics. DISCUSS it’s physical meaning and some of it’s consequences.

c.  Write the 1st Law of Thermodynamics for an infinitesimal, quasi-static process in a system characterized by absolute temperature T & one external parameter x. EXPLAIN the physical meaning of every symbol. DISCUSS the physical meaning & the consequences of this law.

d.  Write the 2nd Law of Thermodynamics for the system of part a. EXPLAIN the physical meaning of every symbol. DISCUSS the physical meaning & the consequences of this law.

e.  Write the 3rd Law of Thermodynamics for the system in part a. EXPLAIN the physical meaning of every symbol. DISCUSS the physical meaning & the consequences of this law.

f.  DISCUSS the physical meaning of entropy.

NOTE!!!! WORK ANY 5 OUT OF PROBLEMS 2 through 7!

2.  The solution to Homework Problem 4 in Ch. 2, shows that for N spin  particles of magnetic moment μ in an external magnetic field H, the number of accessible states with energy in the range E to E + δE has the approximate form:

a.  Calculate the entropy in terms of E, H and N.

b.  The absolute temperature of the system is T. Calculate the internal energy E as a function of T, H, and N.

c.  In this system, the magnetic field H is an external parameter and the generalized force associated with H is the system magnetization M. Calculate M as a function of H, T, and N. The function M(H,T,N) is the equation of state for this system.

d.  Calculate the heat capacity CH,N at constant magnetic field H and constant particle number N for this system.

e.  Obtain an expression for the heat capacity CM,N at constant pressure M and constant particle number N for this system. (Hint: The procedure for this is analogous to the procedure outlined in Sect. 5.7 to obtain the heat capacity at constant pressure for a system of volume V).

3.  A solid contains N magnetic atoms of spin S. At high enough temperatures (T ® ∞), each spin is randomly oriented (so that it is equally likely to be in any one of it’s 2S + 1 possible states). This is called the paramagnetic state. At temperatures below a critical temperature T0, the interactions between the magnetic atoms causes them to all be lined up in the same direction. This is called the ferromagnetic state. A measurement of the heat capacity of this system below T0 shows that it has the form

C(T) = A[3(T/ T0)2 – 1], for  T0 < T < T0 (A is a constant)

= 0, otherwise.

For this system, calculate:

a.  An explicit expression for the constant A.

b.  The entropy S as a function of temperature T, S(T).

c.  The number of accessible states Ω(E) with energy in the range E to E + δE.

d.  The internal energy E as a function of temperature T, E(T). Assume that there are no relevant external parameters.

e.  When the system is in it’s ferromagnetic state, the temperature is increased from a value T1 to a higher value T2. Calculate the heat Q that is needed to raise the temperature from T1 to T2.

4.  In a quasi-static process, the volume of a certain gas changes from VA to VB (VB > VA, so it expands in this process). In the same process, the pressure decreases from PA to PB. It is known that the pressure P of the gas changes with it’s volume V as P = aV- where, a is a constant. Calculate 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 volume from VA to VB and the pressure changes from PA to PB).

a.  Process 1: This is a two step process. In the first step, the volume is expanded from VA to VB under isobaric (constant pressure) conditions. In the second step, the volume is constant and heat is extracted to reduce the pressure to PB.

b.  Process 2: The volume is increased from VA to VB and heat is supplied to cause the pressure to decrease linearly with volume from PA to PB.

c.  Process 3: The two steps of Process 1, described in part a are performed in the opposite order.

NOTE!!!! WORK ANY 5 OUT OF PROBLEMS 2 through 7!

5.  Consider a monatomic, NON-IDEAL gas with N particles confined to volume V which is in thermal equilibrium at temperature T. An approximate equation of state for this gas has the form:

P = kBT[(N/V) + B2(T)(N/V)2].

P is the mean pressure and B2(T) is called the “2nd Virial Coefficient”. In general, it depends on the

temperature T. Find expressions for:

a.  The number of accessible states Ω(E) with energy in the range E to E + δE.

b.  The entropy S.

c.  The mean energy E. That is, find the dependence of E on T and V.

d.  The heat capacity at constant volume, Cv.

e.  The heat capacity at constant pressure volume, Cp.

Note: For each part, assume that B2(T) is a known function & express your answers in general in terms of it.

6.  Work Problem 14 in Chapter 5 of the book by Reif.

7.  Work Problem 15 in Chapter 5 of the book by Reif.