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MIDTERM EXAM, PHYSICS 5305, Spring, 2007, Dr. Charles W. MylesTake Home Exam: Distributed, Wednesday, March 21

DUE, IN MY OFFICE OR MAILBOX, 5PM, WED., MARCH 28.

NO EXCEPTIONS!EXAMRULE: You are allowed to use almostany resources (library books, etc.) to solve these

problems. EXCEPTION: YouMAY NOT COLLABORATE WITH ANY OTHERPERSONin solving them! If you have questions or difficulties with the problems, you

may consult with me, but notwith fellow students (whether or not they are in this class!),with other faculty, or with post-docs. You are bound by the TTU Code of Student

Conduct not to violate this rule! Anyone caught violating this rule 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 yourgrade!!

1. PLEASE write on ONE SIDEof 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 wordsEXPLAINandDISCUSSmean 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!!!! YOUMUSTANSWER PROBLEM1! WORK ANY 5 OUT OF THE OTHER 6

PROBLEMS! So, you must answer 6 problems in total. Problem 1 is worth 10 points. Each ofthe others is equally weighted and 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 _______________________________

Signature1. THIS QUESTION IS REQUIRED !!

a. A coin is tossed 400 times. Find the probability of obtaining 215 heads. Use the Gaussianapproximation.b. Write the 1st Law of Thermodynamics for an infinitesimal, quasi-static process in asystem characterized by absolute temperature T and one external parameterx.EXPLAINthephysical meaning of every symbol you write andDISCUSSthe physical meaning of the lawitself.

c. Write the 2nd

Law of Thermodynamics for the system of part a.EXPLAINthe physicalmeaning of every symbol you write andDISCUSSthe physical meaning of the law itself.d. Write the 3rd Law of Thermodynamics for system in part a.EXPLAINthe physicalmeaning of every symbol you write andDISCUSSthe physical meaning of the law itself.e. DISCUSSthe physical meaning of entropy.f. State the Fundamental Postulate of Statistical Mechanics.DISCUSSits physical

meaning and some of its consequences.

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NOTE!!!! WORK ANY 5 OUT OF PROBLEMS 2 through 7!2. The number of accessible states for a system ofN particles confined to volume V and with

energy in the range E to E + E is given by (E,V,N) = AE(7N/3)exp[BNV(3/5)E] where A &B are constants.

a. Calculate the entropy in terms ofE,V and N.

b. The absolute temperature of the system is T. Calculate the internal energy E as afunction ofT,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 particlenumberN for this system. Express this as a function ofN & V only.e. Calculate the volume coefficient of expansion and the isothermal compressibilityfor this system. (See Reif, Ch. 5)f. Calculate the heat capacity CP,N at constant pressure P and constant particlenumberN for this system. Express this as a function ofN & Vonly.

3. Consider a classical simple harmonic oscillator in one dimension. The total energy is written as

E = (p2/2m) + ()kx2. p is the particle momentum, x is its position, kis the spring constant.

a. For fixed energy E, sketch the allowed region in classical phase space (the x-p plane)that is accessible to the oscillator for all energies less than or equal to E. If the energyis in the very small range E to E + E, sketch (on the same figure you just sketched)the allowed region of phase space that it accessible to the oscillator.

b. Fork = 2 N/m, m = 2 kg, and amplitude xm = 3 m, calculate the number of allowed

states for this oscillator for all energies less than or equal to E. Take the area of thecell in phase space to be Plancks constant h. Put inNUMBERS! I want aNUMERICAL VALUE!

c. Suppose the position x of the oscillator is changed a small amount to x + x, while the

momentum p is unchanged. Find an expression for the corresponding change in energyE, in this case. Forx = 0.01 m,NUMERICALLYcalculate E for the oscillator of

part b. Use this result toNUMERICALLYcalculate the number of accessible statesin the range E to E + E, for this same oscillator.

d. Using the quantum mechanical expression for the energy E of the same simple

harmonic oscillator in part b, estimate the quantum numbern. Compare this result tothat of part b.

4. A substance has the following properties (A,B,V0,T0, and a are constants):

i) At constant temperature T0, the work done by it on expansion from volume V0to

volume V is W0 = 5A(T0)3ln(V/V0)

ii) The entropy is given by S = B(V0/V)2(T/T0)

a

a. Calculate the change in Helmholtz free energy F = F(T,V) F(T0,V0) in taking thesystem from temperature T0 and volume V0 to temperature T and V.

b. Find the equation of state.c. Calculate the work done on expansion from volume V0 to volume V at arbitrary

temperature T.

NOTE!!!! WORK ANY 5 OUT OF PROBLEMS 2 through 7!5. Consider the chemical reaction A + BC. The molar heat capacities (in units of [J/(mole-K)]) at

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constant pressure of substances A,B, and C are the following functions of absolutetemperature T by: CA = 5T

, CB = 8T, CC = 15T

. This reaction is carried out at constanttemperature T = 300K.

a. Calculate the entropy change per mole, S, of substance Cproduced. Does the signof your answer violate the 2nd Law of Thermodynamics? If not, why not? EXPLAIN

briefly using complete, grammatically correct English sentences.b. Calculate the heat released per mole, Q, of substance C produced.c. Assuming that the mechanical work, W, done in this reaction is zero, calculate the

total change in internal energy per mole Eproduced in this reaction.6. Consider a system of total charge q in a static electric field, which is characterized by an

electrostatic potential . The system is in thermal equilibrium at absolute temperature T.Consider an infinitesimal, quasi-static process in which the electrostatic potential is changedby d. Recall that, in such a situation, the mechanical work done is - qd. For this system,use the following definitions (E is the internal energy, S is the entropy): Enthalpy: H = E - q,Helmholtz Free Energy: F = E TS, Gibbs Free Energy: G = E TS - q.In solving thisproblemPLEASE be VERY CAREFUL in your partial derivative notation about whichvariables are held constant when the derivative is taken.

a. Write, in differential form, involving the differential dE, the combined 1st & 2nd Laws

of Thermodynamics for this system, assuming that the entropy S and the electrostaticpotential U are independent variable.

b. Use the properties of differentials and the results of part a to express T and q as partial

derivatives ofE.c. Use the properties of partial derivatives and the results of part b to relate an

appropriate partial derivative ofq to a partial derivative ofT, hence deriving one ofMaxwells relations for this system.

d. Repeat parts a,b,c, except begin by writing the combined 1st and 2nd Laws in terms of

a differential ofF, assuming T and are independent variables and expressing S and

q as appropriate partial derivatives ofF.e. Repeat parts a,b,c, except begin by writing the combined 1st and 2nd Laws in terms of

a differential ofH, assuming S and q are independent variables and expressing T and as appropriate partial derivatives ofH.

f. Repeat parts a,b,c, except begin by writing the combined 1st and 2nd Laws in terms of

a differential ofG, assuming T and q are independent variables and expressing S and as appropriate partial derivatives ofG.

NOTE!!!! WORK ANY 5 OUT OF PROBLEMS 2 through 7!7. Consider a mole of steam under high pressure, so that the Van der Waals equation is the

proper equation of state. Van der Waals constants for steam: a = 4.0 (liter)2atm/(mole)2b = 0.5 liter/mole. It is initially at pressure Pi = 50 atm and has volume Vi = 0.5 liter. It is then

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expanded to twice that volume.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 fand gONLY, assume that the ideal gas law holds, but that = cp/cv = 1.96.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.