A schedule of content and assignments set will appear below.
Week 1:
Notes:
I will give an overview of coming attractions and a reminder of various
things you probably remember from MP203 (or a different more introductory
thermal physics course you may have taken). In particular, we will go
over the
laws of thermodynamics and introduce some of the physical
systems we will be looking at in this course, such as gases, solids and
magnets. Notably we introduced the
classical ideal gas, the Einstein solid and the two-state paramagnet,
looking at microsopic and macroscopic variables. We also reviewed the
equipartition theorem for the thermal energy of a classical system.
Reading
There is a
summary of material
usually
covered in
MP203.
Please read this. It refers to various parts of the book by
Schroeder which are also useful to read. Please read at least chapter
1 this week.
Exercises:
Recommended exercises: 1.12, 1.17, 1.18, 1.21, 1.22, 1.24, 1.26, 1.34,
1.44 and 1.55 from Schroeder.
These exercises don't have to be handed in.
There will be special attention for 1.22 and 1.55 in the first tutorial
Week 2:
Notes: We used the equipartition theorem to
get information on the specific heats of monatomic and diatomic gases and
of solids. We introduced the first law of thermodynamics, relating
internal energy, heat and work. In particular, we considered work done
through a change of the volume of the system. Several special types of
processes were singled out (adiabatic, isochoric isothermal, isobaric,
cyclic). We introduced the second law of thermodynamics and defined the
entropy though the multiplicity of macrostates. We calculated the entropy
of the two-state paramagnet and the Einstein solid, gave a reminder of
Stirling's approximation, used that to find maximal entropy states.
Reading
Much of this material can be found in chapter 1 of Schroeder
The material which involves the entropy is mostly covered in chapters 2
and 3 of Schroeder.
Please read these. Much of this is still
summarized in my summary of material
usually covered in MP203.
Assignment:
Assignment 1 consists of problem 2.1 from this sheet and
Problems 2.18 and 2.22 from Schroeder HINT: Problem 2.1 on the sheet should be a lot easier if you
read paragraphs 4.1 and 4.3 in Schroeder first
Other recommended exercises (not for handing in):
Another recommended exercise appears on the assignment sheet.
Further recommended questions are the ones from last week if you didn't
do them already.
Week 3:
Notes:
We look at the entropy of classical systems, particularly the
classical ideal gas. The treatment of the ideal gas involves coarse graining of the
classical phase space (the space of configurations of momenta and
positions of all particles of the gas). We introduced two elements from
quantum mechanics into the description of the gas. First of all the
natural volume of the coarse graining cell is inferred from the
uncertainty principle (introducing Planck's constant into the
description). Secondly, if the particles of the gas are indistinguishable,
we consider any classical states which are related by a permutation of the
positions and momenta of the particles as the same state. This introduces
a factor of (1/N!) into the multiplicity of the gas, where N is the
number of particles.
By considering systems in thermal and mechanical equilibrium, we argued
that the derivatives of the entropy by energy and volume can be expressed
in terms of temperature and pressure, for any system.
Reading: Still Chapters 2 and 3 of Schroeder
Exercises: Please do exercises 2.26, 2.32 and 2.34 in
Schroeder.
These do not have to be handed in.
Week 4:
Notes:
There are no lectures in week 4, but I highly recommend reading ahead a
little, because I will set an assignment in the tutorial at the start of
week 5 (you will have until after study week to do it). See week 5 for
what is on the programme.
Week 5:
Notes:
Exploring the relation between entropy, energy and temperature, leads to
the thermodynamic relation.
We consider the thermodynamic relation in some detail and explore
some of its consequences. Notably we consider the relation between
entropy change in a system and absorption/release of heat. In
heating/cooling processes, we find that the entropy change can be found by
integrating the quotient of the heat capacity by the absolute
thermodynamic temperature. In processes at contant temperature (for
example phase transitions of pure substances like melting or evaporation)
the entropy change is just the (latent) heat absorbed divided by the
temperature. We introduce the third law of thermodynamics (the entropy
goes to zero as the temperature goes to zero) and we show that it implies
that the heat capacity (or specific heat) of any system should vanish at
zero temperature. We then consider how a knowledge of the entropy as a
function of energy, volume, number of particles etc. allows us, with the
help of the thermodynamic relation, to find energy, volume and
number of particles as a function of temperature, pressure and chemical
potential. As an example we calculate the energy of the two-state
paramagnet as a function of temperature.
Reading: This week's material is covered more or less one to
one in Chapter 3 of Schroeder.
Assignment
Assignment 2 consists of problem 2.2 from this sheet and
Problem 3.25 from Schroeder
Problems 3.33 and 3.34 from Schroeder are also recommended, but not
for handing in.
This is due on Monday, November 3rd (at the start of the
first tutorial after the study week).
Week 6:
Notes:
We look more carefully at the two-state paramagnet, deriving Curie's
law, and checking that the heat capacity goes to zero at zero temperature
(as implied by the third law of thermodynamics). We also discuss
magnetic cooling. We then introduce the thermodynamic potentials,
notably the (Helmholtz) Free energy F=U-TS, the Gibbs Free eergy
G=U-TS+pV and the enthalpy H=U=pV and we discussed how these can function
as potentials for work and/or heat in special situations (constant
temperature and/or pressure). We also derive the Maxwell relations.
Reading: We cover some loose ends that are still in Chapter 3
of Schroeder (for example Curie's law). A lot of information about
heat engines and refrigerators is in chapter 4. I recommend reading
through the whole chapter, even though some of it was not covered in the
letures so far. Efficiency of heat engines is in
section 4.1 and magnetic cooling is treated in section 4.4. Chapter 5
deals with thermodynamic potentials.
Exercises
Please do the exercises mentioned on this sheet.
You don't have to hand these in. The exercises include Ex. 1.46 and 5.14
in Schroeder's book.
Week 7:
Notes: We revisited the Maxwell relations and
applied one of them to derive the Clausius-Clapeyron equation which
describes the shape of the phase boundaries in the phase diagram (in this
case the (p,T) diagram) of a simple substance (this was mostly in the
tutorial).
We then introduced the canonical and
grand canoncial ensembles which describe the probabilities of microstates
of a system when it is in contact with a reservoir of energy at fixed
temperature, and for the grand canonical case also of particles, at fixed
chemical potential. (We introduced Boltzmann and Gibbs factors, partition
functions, thermal averages etc.). We rederived some results on
paramagnets using the canonical ensemble as an application. We start
looking at the partition functions of systems made up of non-interacting
subsystems. We also looked at the canonical formalism for
systems with continuous degrees of freedom, notably classical systems. We
showed that this gives a way to derive the equipartition theorem.
Reading:
I recommend a full read of Sections 5.1,5.2 and 5.3
of the book.
The rest of chapter 5 is pretty interesting too, but I probably
won't have time to cover it.
The material on canonical and grand canonical ensembles that we discussed
is treated in Sections
6.1,6.2, 6.6 and 7.1.
Hopefully we will eventually cover all of chapters 6 and 7.
Assignment
Assignment 3 consists of problems from Problems 6.17, 6.18 and 6.22 from
Schroeder
Since the lectures are a little behind the tutorials, you may hand this in
until the start of the lecture on Wednesday, November 20 (rather than
the tutorial on Monday).
Week 8:
Notes:
We derive the relation between the canonical partition function and the
free energy (F=-kT log(Z)).
We calculate the partition function of the classical ideal gas and find
the average energy and derive the ideal gas law.
We also started to consider the quantum mechanical ideal gas at
low density, or high tempperature.
Reading: Chapter 6, especially 6.3, 6.4, 6.5 and 6.6 and read into 6.7
if you can.
Assignment: I recommend 6.32, 6.44 and 6.45 from
Schroeder but you don't have to hand these in.
Week 9:
Notes:
We revisit the ideal gas, now treated quantum mechanically as a system
of identical non-interacting particles in a box, either with fixed or with
periodic boundary conditions. We looked at the limit where the gas is dilute
enough so that it does not matter whether the particles are bosons or
fermions. We calculate the partition function in terms of the "quantum
volume" and reproduce the results for the classical ideal
monatomic and diatomic gases - this will also give us another chance to
reflect on the range of validity of the equipartition theorem.
We next remind ourselves of the fact that all particles fall in two
classes: bosons and fermions, and we review
their basic properties and differences. We make a beginning with the
treatment of ideal gases of bosons and fermions.
Reading:
Sections 6.2 (the part on torations of diatomic molecules), 6.4 and
especially 6.7 and 7.2 in Schroeder
Exercises
If you have not done them yet, look at 6.44 and 6.45 from
Schroeder. I also recommend 6.48 and 6.50 and
7.10 and 7.15. You don't have to hand these in.
Next week's assignment will be 7.22, 7.23 and 7.28. This is a fair amount
of work, so it's worthhaving a look already!
Week 10:
Notes:
We derived the Bose Einstein and Fermi Dirac distribution functions
for the average occupation numbers of single particle states in bosonic
and fermionic systems. We also introduced the density of states. We
applied these tools to calculate properties of a Fermi gas at zero
temperature.
Reading:
Sections 7.2 and 7.3 in Schroeder
Assignment: Assignment 5 consists of problems 7.22,7.23 and
7.28 from Schroeder. This assignment is due on Wednesday
December 11.
Week 11:
Notes:
We give a treatment of the Sommerfeld expansion for the energy
and the relation between number of particles and chemical potential in
an electron gas at low (but nonzero) temperatures.
We then derived the Planck distribution of thermal radiation and we very
briefly consider the Debije model of a solid.
Reading:
Sections 7.3, 7.4 and 7.5 in Schroeder
Week 12:
No lectures in week 12 (we already had 24). An extra (revision) tutorial
will be
arranged for January - watch this space.