Training Exam 2
F14 Phys 4510
Nov , 2014
__________
Your name
You can skip any three of the seven problems.
If you work more than four problems, I will grade all of them and count the
best grades toward the exam grade.
Bonus problem: See rules under ‘problem 8’.
Zero grade for illegible parts and for logic that is impossible to follow.
Problems for which more than one answer is offered will receive reduced
credit, even if one answer is completely correct. To avoid this problem,
cross out redundant answer parts.
In general, show your work for full credit.
Problem 1 Thermodynamic identities
Schroeder chapter 5
a) Derive the formula for isentropic compressibilities of a material
n/b The isothermal and adiabatic compressibilities are defined as
1 𝜕𝑉
1 𝜕𝑉
𝜅 𝑇 = − ( ) 𝑎𝑛𝑑 𝜅𝑆 = − ( )
𝑉 𝜕𝑝 𝑇
𝑉 𝜕𝑝 𝑆
b) Check for an ideal gas that the formula is true.
Problem 2
Consider a gas with a van der Waals equation of state.
a) Use Gibbs’ Free Energy to calculate the shape the function G(P/PC) and relate it
to an isotherm in the temperature range where a phase transition between gas
and liquid will occur.
b) Discuss the function and the isotherm.
Problem 3 Short essay question (~ 1 full page single spaced):
Discuss Negative Temperatures.
Problem 4
Schroeder chapter 3 and 5
System: Two-State Paramagnetic Gas with one species of molecules.
a) Derive the thermodynamic identity for G and express entropy, volume, and
chemical potential for G based on suitable partial derivatives.
b) Draw S/kB as function of U/B for this gas. Discuss the graph
c) Draw either C/NkB or M/N as function of kBT/B. Discuss your chosen
graph.
Hint: You may be able to discuss qualitative features of a graph even when
you cannot come up with the correct drawing.
Hint: in a P-V-T system (i.e. not our system), the enthalpy of a system is its
energy plus the mechanical work needed to make room for it; the Helmholtz
Free Energy is the total energy needed to create a system, minus the heat
one can get for free from an environment at temperature T; the Gibbs Free
Energy is the system’s energy minus the heat term that’s in FHH plus the
work term that’s in H.
Problem 5
Schroeder chapter 6
Consider a classical ‘degree of freedom’ that is linear rather than quadratic: 𝐸 =
𝑐 ∙ |𝑞|, for some positive constant c. An example would be the kinetic energy of a
highly relativistic particle in one dimension.
a) Calculate the partition sum.
b) Calculate the average energy from the partition sum.
Problem 6
The entropy of an ideal monatomic gas that lives in a two-dimensional universe is
𝑆
𝑘𝐵
1
2𝜋𝑚𝑈𝐴 𝑁
) ).
ℎ2
= ln ((𝑁!)2 (
a) Simplify the expression for entropy.
b) Determine temperature and pressure of this gas (in 2-d pressure is defined as force per unit
length). Simplify your results as much as possible and explain whether the results make sense.
Problem 7 Carnot Engines
a) Show the cycle of the Carnot Engine in the p-V, S-T, and p-T diagrams.
b) Explain the Third Law in the context of the S-T di-gram.
c) Compare the efficiency of the Carnot Engine with a regular heat engine for
the case when W= 220[J], TH = 600 [K], TC= 300[K], QH= 2180[J], QC= 1960[J].
Problem 8
?????????
Bonus Problem
Need to take your mind off the exam for a minute? Check these jokes out:
For those who want some proof that physicists are human, the
proof is in the idiocy of all the different units which they use for
measuring energy.
-- Richard P. Feynman, _ The Character of Physical Law_
A mathematician may say anything he pleases, but a physicist
must be at least partially sane.
-- J. Willard Gibbs, quoted _The Scientific Monthly_,
December, 1944
Murray Gell-Mann:
Niels Bohr brainwashed a whole generation of physicists into
believing that the problem [of the interpretation of quantum mechanics]
had been solved fifty years ago.
@R: Acceptance speech Noble Price (1976)
Scientific discovery may not be better than sex, but the satisfaction lasts
longer -- Stephen Hawking (BBC News, January 16, 2002)
"Heavier-than-air flying machines are impossible."
Lord Kelvin, president, Royal Society, 1895.
X-rays will prove to be a hoax.
Lord Kelvin, while president of the Royal Society
Physics is not a religion. If it were, we'd have a much easier time
raising money. - Leon Lederman
"Professor Goddard does not know the relation between action and reaction
and the need to have something better than a vacuum against which to react.
He seems to lack the basic knowledge ladled out daily in high schools."
-1921 New York Times editorial about Robert Goddard's revolutionary
rocket work.
"Correction: It is now definitely established that a rocket can function
in a vacuum. The 'Times' regrets the error." NY times, July 1969.
... that, in a few years, all great physical constants will have been
approximately estimated, and that the only occupation which will be left to
men of science will be to carry these measurements to another place of
decimals.
-- James Clerk Maxwell (1831-1879) [Scottish physicist]
Scientific Papers 2, 244, October 1871.
Max Planck (1858 - 1947):
If anybody says he can think about quantum problems without getting giddy,
that only shows he has not understood the first thing about them.
On a paper submitted by a physicist colleague:
"This isn't right. This isn't even wrong."
-- Wolfgang Pauli, Austrian physicist (1900-1958)
All science is either physics or stamp collecting.
-- Ernest Rutherford, New Zealand physicist (1871-1937)
Winner Nobel prize chemistry!! (1908)
Source given is JB Birks "Rutherford at Manchester," 1962.
"It was as if you fired a 15-inch shell at a sheet of tissue paper and its
came back to hit you."
-- Ernest Rutherford, about the scattering of alpha particles from gold
foil, which resulted in the discovery of the atom nucleus.
If you want to be a physicist, you must do three things
mathematics, second, study more mathematics, and third,
-- Arnold Sommerfeld (German Physicist, 1868-1951) in
Paul Kirkpatrick, in "The physicsts" (1978) by Daniel
-do
an
J.
First, study
the same.
interview with
Kevles.
It was absolutely marvelous working for Pauli. You could ask him anything.
There was no worry that he would think a particular question was stupid,
since he thought all questions were stupid.
-- Victor Frederick Weisskopf
What Einstein said wasn't all that stupid.
-- Wolfgang Pauli as a student, after hearing Einstein, 20 years his
senior give a lecture.
Physics is becoming so unbelievably complex that it is taking longer and
longer to train a physicist. It is taking so long, in fact, to train a
physicist to the place where he understands the nature of physical problems
that he is already too old to solve them.
-- Eugene Wigner
Feynman played many jokes on colleagues. In one case he found the combination to a locked filing
cabinet by trying the numbers a physicist would use (it proved to be 27–18–28 after the base of natural
logarithms, e = 2.71828...), and found that the three filing cabinets where a colleague kept a set of
atomic bomb research notes all had the same combination. He left a series of notes as a prank, which
initially spooked his colleague, Frederic de Hoffman, into thinking a spy or saboteur had gained access to
atomic bomb secrets.
Feynman’s students competed keenly for his attention; he was once awakened when a student solved a
problem and dropped it in his mailbox; glimpsing the student sneaking across his lawn, he could not go
back to sleep, and he read the student's solution. The next morning his breakfast was interrupted by
another triumphant student, but Feynman informed him that he was too late.
The Feynman Problem Solving Algorithm:
1) Write down the problem.
2) Think very hard.
3) Write down the solution.
A student recognizes Einstein in a train and asks: Excuse me, professor, but does New York stop by this
train?
"Physics is like sex: sure, it may give some practical results, but that's not why we do it." R. Feynman