| Day of week and Date | Lecture Content and Homework Assignment |
|---|---|
| Tue. Jan. 10, 2017 |
Lecture: An overview of elementary thermal physics: Chapter 1,
sections 1.1 - 1.6. [Temperature, thermal equilibrium, Ideal Gas Law,
nR=Nk, elementary kinetic theory, equipartition of energy. Heat, Work,
Internal energy (U) and the First Law: ΔU = ΔQ +
ΔW. Heat capacity at constant volume (CV), at constant pressure
(CP) and the relation between the two. Latent heat of
transformation, and Enthalpy (H).] Homework #1 (due Thu. Jan. 19, 2017): Problem 1.34, Problem 1.51. |
| Thu. Jan. 12, 2017 |
Lecture: Classical Thermodynamics vs Statistical Mechanics. The Laws
of Thermodynamics. Permutations. Combinations. The Stirling
approximation in the large N limit. The Einstein Solid. The number of
ways of distributing q energy quanta among N oscillators. Microstates
and Macrostates. Homework #1 (due Thu. Jan. 19, 2017): Problem 2.17. |
| Tue. Jan. 17, 2017 |
Lecture: The width of the multiplicity as a function of energy. The
entropy of an ideal gas: the Sackur-Tetrode equation. Discussion of
problem 2.26. Homework #2 (due Thu. Jan. 26, 2017): Problem 2.32. |
| Thu. Jan. 19, 2017 |
Lecture: The entropy of mixing and why the total entropy changes more
when we mix dissimilar gases. Definition of temperature. Finding the
internal energy U and the entropy S in terms of N, T, and V. Homework #2 (due Thu. Jan. 26, 2017): Problem 2.37, Problem *2.38. |
| Tue. Jan. 24, 2017 |
Lecture: The paramagnet: magnetization, internal energy, and entropy
as a function of temperature (rather, as a function of x =
μB/kT). Negative temperatures! The energy and magnetization in the
high-temperature limit. Some discussion of Problem 3.20. Homework #3 (due Thu. Feb. 2, 2017): Problem 3.20. |
| Thu. Jan. 26, 2017 |
Lecture: Justification of the various terms in the expression for dU:
the TdS term, the -PdV term, and additional kinds of work done:
μdN, etc. Defining temperature and pressure this way, and the
chemical potential. Some relations involving partial
derivatives. Relating P, T to partial derivatives of S. Equilibrium
conditions from maximizing S: equality of temperature, pressure, and
μ of two thermal systems in different kinds of contact.
The chemical potential: why it is not merely 3kT/2, and why it can
even be negative! Example of the Einstein solid with low numbers
of N, q. Discussion of problem 3.35. Homework #3 (due Thu. Feb. 2, 2017): Problem 3.36. |
| Tue. Jan. 31, 2017 | Lecture: What the Second Law of
Thermodynamics says about maximizing entropy subject to (a) energy
exchange only: heat flows from the higher temperature region to the
lower temperature region; (b) volume exchange only: the partition
moves from the higher pressure region towards the lower pressure
region; (c) number of molecules exchange only: molecules move from the
region of higher chemical potential to the region of lower chemical
potential. Two elements of heat engines: calculation of the work done along an isotherm (T fixed) and along an adiabat (no heat absorbed by the gas or released to the environment). [This is the area under a curve traversed reversibly on a PV diagram]. Homework #4 (due Thu. Feb. 9, 2017): Problem 3.39. |
| Thu. Feb. 2, 2017 | Lecture: The Carnot cycle
[a reversible cycle between two temperatures: TH and
TL]. Efficiency of the Carnot engine. The Carnot engine run
in reverse: the Carnot refrigerator and the Carnot heat pump, and
their coefficients of performance. Entropy as the new quantity which
is a state variable. Q and W are not state variables. But
ΔU = Q+W is [the change in] a state variable for
reversible paths! T-S diagrams. The Carnot theorem: why building a
better engine than the Carnot engine violates the II Law of
Thermodynamics. Homework #4 (due Thu. Feb. 9, 2017): Problem 4.1. |
| Tue. Feb. 7, 2017 |
Lecture: Review of Chapters 1-4. Homework #5 (due Thu. Feb. 16, 2017): No homework. |
| Thu. Feb. 9, 2017 |
In-class test #1 based on Chapters 1-4. Homework #5 (due Thu. Feb. 16, 2017): Problem 4.18. |
| Tue. Feb. 14, 2017 |
Lecture: Extending the concept of enthalpy to other "thermodynamic
potentials" such as G, F, etc. Extensive and Intensive variables. The
Gibbs Free Energy as "available work" and also as equivalent to
maximizing entropy in processes occuring under constant pressure and
temperature. Discussion of problems 5.1, 5.2, 5.3. Homework #6 (due Thu. Feb. 23, 2017): Problem 5.5. |
| Thu. Feb. 16, 2017 |
Lecture: Part I: Which thermodynamic potential to minimize: U, H, F, or G?
The answer depends on the conditions the system finds itself in:
constant T, P, or adiabatic and isochoric, etc. Part II: The natural variables of the four thermodynamic potentials. The thermodynamic identity (total derivative) for each of the four thermodynamic potentials. Maxwell relations from the equality of mixed second partial derivatives. The GPHSUVFT mnemonic. Homework #6 (due Thu. Feb. 23, 2017): Problem 5.23. [Note: part (d) of this problem can be handled in a variety of ways, so any reasonable approach will be sufficient.] |
| Tue. Feb. 21, 2017 |
Lecture: The Joule expansion and its surprises: (1) The gas
temperature T does not change despite the gas expanding, (2)
work is not done despite the integral of PdV being non-zero and
positive, and (3) we can calculate ΔS by integrating dS
along the isotherm despite the process being irreversible.
The Joule-Thomson expansion and cooling of gases by
expansion. The Joule-Thomson coefficient expressed as
μJT = [T(∂V/∂T)P - V] / CP.
The Joule-Thomson coefficient is zero for ideal gases, and therefore
there is a need for a better equation of state. Motivations for
("derivation" of) the van der Waals equation of state for gases:
[P + an2/V2][V - nb] = nRT. Homework #7 (due Thu. Mar. 2, 2017): Do a careful calculation of (∂V/∂T)P for a gas described by the van der Waals equation of state and show that μJT = [2a/RT - b] / CP. Keep in mind that the CP in the denominator of the first expression for μJT, the one in the lecture description above, is for n moles of gas, i.e., n times the heat capacity at constant pressure for one mole of gas. |
| Thu. Feb. 23, 2017 |
Lecture: Physics of the atmosphere: the variation of pressure and
temperature with height. Heat of condensation, and the ozone
layer. Phase transformations and phase diagrams: diagrams of phase
boundaries on a P-T plot. Slope of a boundary obtained using equality
of the Gibbs Free energy for the two phases: the Clausius-Clapeyron
equation. Applications: Trouton's rule. Diamond and graphite, and the
depth at which we should find diamonds. Homework #7 (due Thu. Mar. 2, 2017): Problem 5.32. |
| Tue. Feb. 28, 2017 |
Lecture: Derivation of the Boltzmann Factor in a couple of different
ways. Higher energy levels have lower probability unlike what
we said earlier about all microstates having equal probability. Why?
Because the system is in thermal equilibrium at temperature T with its
surroundings, and the reservoir has slightly lower energy and
therefore fewer microstates when the system is in a higher energy
state. The partition function. The central importance in statistical
mechanics of the Boltzmann Factor and the partition
function. Discussion of / solutions to problems 6.2, 6.5, 6.6, 6.7. Homework #8 (due Thu. Mar. 16, 2017): Problem 6.12. |
| Thu. Mar. 2, 2017 |
Lecture: The basics of cooling: via the Joule-Thomson effect, via
adiabatic demagnetization, and via laser cooling. A visit to the Webb
lab to see real dilution refrigerators (thanks to Prof. Crittenden). Homework #8 (due Thu. Mar. 16, 2017): No homework. |
| Tue. Mar. 14, 2017 |
Lecture: How to measure low temperatures, and how to get
the 3He back out of the 4He in a dilution
refrigerator. A careful calculation of
(∂V/∂T)P for a gas described by the van der
Waals equation of state to obtain a useful (non-zero) expression for
μJT = [T(∂V/∂T)P - V] / CP. Once again: why is the Boltzmann factor not just unity - weren't all states supposed to be equally likely? Calculating the average energy of an ensemble of identical quanta: the average energy is the total energy divided by the number of quanta and equals the probability of each state times its energy. Homework #9 (due Thu. Mar. 23, 2017): Problem 6.20. |
| Thu. Mar. 16, 2017 |
Lecture: The partition function of diatomic molecules when the two
atoms are different; for identical atoms the number of transitions is
typically half of that due to selection rules such as those for E1 (electric
dipole) transitions. Derivation of the equipartition theorem for energies that depend
quadratically on positions or momenta. Close to the minimum all
potentials have this quadratic dependence! Some interesting ways to do
the integral that arises. Problem 6.31. The number of states with
velocities in the interval [v, v + dv], where v is the magnitude of
the 3-dimensional vector velocity v. The Maxwell-Boltzmann
velocity distribution for gases. Homework #9 (due Thu. Mar. 23, 2017): Problem 6.32, parts (a) and (b). *Graduate students should do the whole problem. |
| Tue. Mar. 21, 2017 |
Lecture: Properties of the Maxwell distribution of speeds for an
ideal gas: the mean, rms, and most likely speed of molecules of the
gas derived from the distribution using properties of the Gamma
function. Review of first 3 sections of Chapters 5 and 6. Homework #10 (due Thu. Mar. 30, 2017): Problem 6.39. |
| Thu. Mar. 23, 2017 |
In-class test #2 based on all material covered up to and including
the lecture on 3/16. [Emphasis on material covered after test #1.] Homework #10 (due Thu. Mar. 30, 2017): |
| Tue. Mar. 28, 2017 |
Lecture: Discussion of results of test #2. Students are urged to
review partial derivatives, integration, and probability densities in
one and more dimensions. Memorize the value of Boltzmann's constant!
Partition functions again, for one particle. Partition functions for
many particles and how they partition into products of one-particle
partition functions.
Homework #11 (due Thu. Apr. 6, 2017): Problem 6.41. |
| Thu. Mar. 30, 2017 |
Lecture: A general expression for entropy, S, depending only on the
probabilities Pi of the system to be in the ith
state. Application of this expression to obtain the dependence of
entropy (S) and Helmholtz free energy (F) on the partition function Z.
Extending this approach we obtain all the thermodynamic properties of
the system [U, F, S, P, H, G, CV, ...] from the partition
function Z. Example: the ideal gas (to be continued).
Homework #11 (due Thu. Apr. 6, 2017): Problem 6.42. |
| Tue. Apr. 4, 2017 |
Lecture: Sums vs integrals: the fractional difference in the case of
the harmonic oscillator. The difference in the case of the
1-dimensional ideal gas partition function. The method of Lagrange
undetermined multipliers, and an application to deriving the
Boltzmann Factor from the expression for S in terms of probabilities.
Derivation of the thermodynamic properties of the ideal gas, i.e., of
U, F, S, P, H, G, CV, ... for the ideal gas, from the
partition function (translational degrees of freedom only).
Homework #12 (due Thu. Apr. 13, 2017): Problem 6.45. |
| Thu. Apr. 6, 2017 |
Lecture: Extending the microcanonical and canonical ensembles to the
Grand Canonical Ensemble (these are all gedanken ensembles) which give
rise to the extended Boltzmann Factor which includes the chemical
potential. A simple example: preferential bonding of the CO molecule
to hemoglobin (as opposed to oxygen). Indistinguishability and the
filling up of states at low temperatures forces us to include the
restrictions of quantum mechanics: indistinguishable particles are
truly indistinguishable, and the Pauli exclusion principle.
Switching from interpreting the Boltzmann
Factor as the "probability" for a particle to be in a certain state to
being the "probability" for a state to be occupied by one or more
particles. ("Probability" is in quotes here because we have to divide
by the grand partition function to get a probability). The occupancy
distribution for fermions (i.e., the Fermi-Dirac distribution) and
the occupancy distribution for bosons (i.e., the Bose-Einstein
distribution). Discussion of problem 7.8. Homework #12 (due Thu. Apr. 13, 2017): Problem 7.10. |
| Tue. Apr. 11, 2017 |
Lecture: Application of the Fermi-Dirac distribution to the electron "gas"
in a metal. The density of states as a function of quantum number n
and as a function of electron energy E. Calculation of the Fermi
energy: the energy of the highest occupied energy level at zero
temperature. The Fermi energy for electrons in a metal is in the 5 to
50 eV range. The electron gas is really "cold", i.e., room temperature
is rather low compared to the Fermi temperature
(EF/kB). The total energy U of the electron gas,
the pressure P, and the bulk modulus B. The latter two quantities are
rather large (almost a TeraPascal). Some discussion of why solid
objects do not go through each other, of metallic hydrogen, of the
collapse of stars into neutron stars, and of the latter into black
holes. Homework #13 (due Thu. Apr. 20, 2017): Problem 7.28. |
| Thu. Apr. 13, 2017 |
Lecture: The spectrum of light was discovered by Newton and subsequent
improvements due to better prisms led to better observations by
Fraunhofer who found missing frequencies. Kirchoff noticed the relation
between absorption and emission, and therefore attempted to create a perfect
emitter by creating a perfect absorber: a "blackbody" which is a hole
in a cavity. Subsequently he found the cavity could be made of any
material but the energy density of photons in it was the same: a
universal function of only the temperature T and frequency ν. Stefan discovered that the
total power radiated was proprtional to T4, a law that was
subsequently proved theoretically by his student Boltzmann and now
known as the Stefan-Boltzmann Law. Wien
studied the spectral distribution and found the correct high
frequency dependence, while the Rayleigh-Jeans expression (u ~
ν2kT) worked well
at low frequencies, but at high ν it led to the "ultraviolet
catastrophe". In 1900, in the first of two papers, Planck introduced
a law that extrapolated well between the two extremes. Today we
understand Planck's Law simply as a product of the density of states, the photon
energy for a mode of frequency ν, and the occupancy of the mode. Homework #13 (due Thu. Apr. 20, 2017): Problem 7.39. |
| Tue. Apr. 18, 2017 |
Lecture: Applications of the Planck (blackbody) spectrum: the total
energy, total number, and CV and the entropy per unit
volume. The energy flux, momentum flux, and pressure for
electromagnetic radiation in (a) a laser and (b) from a hole in a
cavity (i.e., for blackbody radiation). The energy balance of the
earth leads to a temperature of 300K for this planet. A discussion of
Einstein's A and B coefficients. Homework #14 (due Thu. Apr. 20, 2017): Problem 7.41, *7.46. |
| Thu. Apr. 20, 2017 |
Lecture: Review of some course material and discussion of final exam. Homework #14: No homework. |
| Thu. Apr. 27, 2017 9:00 AM - 11:30 AM | FINAL EXAM: Covers ALL material! |