MIDTERM EXAM, PHYSICS 5305, Spring, 2009, Dr. Charles W. Myles
In Class Exam: Thursday, March 12, 2009
Note: I’ve changed my mind about having both an in class and a take home exam.
This in-class exam will be the only Mid-Term Exam we will have.
INSTRUCTIONS: Please read ALL of these before doing anything else!!!
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 11 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 QUESTION 1! ANSWER ANY 3 OF THE OTHERS!
So, answer 4 questions total. Each is equally weighted & worth 25 points for 100 points total.
1. REQUIRED QUESTION!!
a. 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.
b. 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.
c. Write the 3rd Law of Thermodynamics for the system in part a. EXPLAIN the physical meaning of every symbol you write and DISCUSS the physical meaning of this law.
d. DISCUSS the physical meaning of entropy.
e. State the Fundamental Postulate of Equilibrium Statistical Mechanics. DISCUSS it’s physical meaning and some of it’s consequences.
f. Consider a game in which 6 true dice are rolled. (A “true” die is one which is a perfect cube of uniform density so that, on a given roll, the probability that any one of the faces will end up as the upper face is (1/6) ≈ 0.16667). Calculate the probability that, from these 6 dice, a player will obtain one and only one ace. (An “ace” is a die with only one dot showing on the upper face.). Calculate the probability that, from these 6 dice, a player will obtain at least one ace.
NOTE!!!! WORK ANY 3 OUT OF PROBLEMS 2 through 6!
2. 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(7N∕8) exp[BN(5/8)V(2/3)]. 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 º V-1(¶V/¶T)P,N] for this system. Calculate the isothermal compressibility [κ º -V-1(¶V/(¶P)T,N] for this system.
f. Calculate the heat capacity CP,N at constant pressure P and constant particle number N for this system.
3. Consider a classical simple harmonic oscillator in one dimension. The total energy is written as
E = (p2/2m) + (½)kx2. p is the particle momentum, x is it’s position, k is the spring constant.
a. For fixed energy E, sketch the allowed region in classical phase space (in the p-x plane) that is accessible to the oscillator 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 oscillator.
b. For k = 3 N/m, m = 3 kg, and for oscillation amplitude xm = 4 m, calculate the number of allowed states for this oscillator for all energies less than or equal to E. Take the area of the cell in phase space to be Planck’s constant h (= 6.6 ´ 10-34 Joule – sec). Put in NUMBERS! I want a NUMERICAL VALUE! (Hint: The area of an ellipse is A = πab, where a is the semi-major axis of the ellipse and b is the semi-minor axis).
c. Using the quantum mechanical expression for the energy E of the same simple harmonic oscillator in part b, estimate the quantum number n. Compare this result to that of part b.
4. A system consists of a closed electrical circuit in thermal equilibrium with a heat reservoir at temperature T. The circuit carries a current I & is in a static external magnetic field which produces a magnetic flux F through the circuit. In this situation, the infinitesimal, quasi-static mechanical work done when the magnetic flux changes by dΦ is dW = IdΦ. Let E = internal energy S = entropy & use the definitions: Enthalpy: H = E + IΦ, Helmholtz Free Energy: F = E - TS, Gibbs Free Energy: G = E – TS + IΦ. In solving this, PLEASE be VERY CAREFUL in your partial derivative notation about which variables are held constant when the derivative is taken.
a. Write, in differential form, using the differential dE, the combined 1st & 2nd Laws of Thermodynamics for this system, assuming that S & F are independent variables. Use the properties of differentials and the result for dE to express T & I as partial derivatives of E.
b. Use the properties of partial derivatives and the results of part a to relate an appropriate partial derivative of I to an appropriate partial derivative of T, (deriving one of Maxwell’s relations for this system).
c. Repeat parts a & b, except begin by writing the combined 1st & 2nd Laws in terms of a differential of the H, assuming that S & I are independent variables and expressing T & F as appropriate partial derivatives of H.
d. Repeat parts a & b, except begin by writing the combined 1st & 2nd Laws in terms of a differential of F, assuming T & F are independent variables and expressing S I as appropriate partial derivatives of F.
e. Repeat parts a & b, except begin by writing the combined 1st & 2nd Laws in terms of a differential of G, assuming T & I are independent variables and expressing S & F as appropriate partial derivatives of G.
NOTE!!!! WORK ANY 3 OUT OF PROBLEMS 2 through 6!
5. 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 = 4T2, CB = 2T(3/2), CC = 25T. This reaction is carried out at constant temperature T = 300K.
a. Calculate the entropy change per mole, DS, 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, DQ, 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 DE produced in this reaction.
6. An ideal gas contains N non-interacting molecules. However, it is confined to a 2-dimensional box of area A in the x-y plane, instead of to the usual 3-dimensional volume V. So, it has 2N degrees of freedom. It is at thermal equilibrium at absolute temperature T.
a. Consider this gas for fixed internal energy E. By an analysis similar to that done in class and in Ch. 2 of our textbook, DERIVE the dependence of the number of accessible states W(E) on the internal energy E and on the area A. (Note: Even if you know the correct result without going through the derivation, write the essential steps of the derivation anyway. Do this in sufficient detail to convince me that you know what to do for this question).
b. Calculate dependence of the entropy S of this gas on the internal energy E and on the area A. Derive the relation between the absolute temperature T and the internal energy E for this gas.
c. Assume that the pressure P for this gas is defined similarly to the way it is defined for the 3-dimensional gas that P is the generalized force associated with the external parameter A. Derive the equation of state for this gas. (That is, derive the “ideal gas law” for 2 dimensions).
d. Consider an infinitesimal, quasi-static process in which the area A of confinement for this 2-dimensional gas changes by an amount dA. Write, in differential form, using the differential dE, the combined 1st & 2nd Laws of Thermodynamics for this gas, assuming that S & A are independent variables. Use the properties of differentials & the result for dE to express T & P as partial derivatives of E.
e. Use the properties of partial derivatives & the results of part d to relate an appropriate partial derivative of A to an appropriate partial derivative of T, (deriving a Maxwell’s relation for this 2-dimensional gas).