PHY252S:
Thermal Physics

Marked assignments will be available from Geoff starting tomorrow. He will also post on his door a list of term marks. Of course, the list will be anonymous, with the marks identified only by student number. Please check your numbers and report any discrepancy to Geoff.

I will be available Monday and Tuesday to answer your questions. I do recommend, however, that you e-mail me first so as to be sure to find me in my office.

The final exam date is Wednesday April 30, from 14:00-17:00, in room BN2S. For general exam schedule details, see the FA&S web schedule . A scientific non-programmable calculator and a standard letter format (or else, A4) aid sheet are allowed. This aid sheet must be handwritten and may be written on both sides of the paper (size-reduction using a photocopier or image reduction on a computer is NOT permitted!). There is no restriction on what you may write on the aid sheet. I will also provide a single-sided sheet of expressions with the exam paper, but it will contain no information whatsoever on the expressions supplied. Everything you need to solve the questions will be there, but not necessarily in ready-to-use form. For instance, you may have to combine two of the expressions to obtain the desired one. I would suggest, therefore, that you devote some care to the preparation of your aid-sheet. What will be provided with the exam is only a back-up to that aid-sheet. There will be five equally-weighted questions (no choice). The material covers the whole course, but it goes without saying that there will be more questions on chapters 5, 6, or 7 since you have not been tested on them. There will be no question requiring the use of the grand partition function (Gibbs sum). Most will be exercises (numerical or algebraic), but you can expect to be asked for explanations.
Here is last year's final exam .

The final grade F for the course will be a 60%-40% 'flip-flop' split of your term grade T and your final exam grade E- with the weighting in favour of the better grade. The term grade T will be computed as 60% homeworks H and 40% midterm M. Overall, then, we have the formula:
F = 0.6*max(T,E)+0.4*min(T,E) where T=0.4*M+0.6*H.
Note: all homeworks have the same weight, i.e. each is worth 15% of your term grade T.

• Mon 06 Jan : Degeneracy; Spin system I [KK pp1-18]
• Wed 08 Jan : Spin system II; Average value, Fundamental assumption of thermal physics [KK pp18-26]
• Fri 10 Jan : Ensembles; Two systems in thermal contact and Maximization of entropy at equilibrium, Definition of temperature [KK pp27-41] (tutorial on Wed 15 Jan)
• Mon 13 Jan : Entropy, Entropy always increases for large N, Laws of Thermodynamics; Reservoir and Boltzmann factor [KK pp41-61]
• Mon 20 Jan Partition function; Reversibility; Pressure, Differential form of First Law, Helmholtz free energy F [KK pp62-69]
• Wed 22 Jan Maxwell relation, F minimized at constant T,V; Two-state system; Ideal gas I, quantum concentration [KK pp69-76]
• Mon 27 Jan Ideal gas II, ideal gas law, Sackur-Tetrode equation; Photon gas I [KK pp76-78 and 89-94]
• Wed 29 Jan Photon gas II, Stefan-Boltzmann law, Planck radiation law, spectral density, Wien displacement law, entropy [KK pp94-98]
• Mon 03 Feb Photon gas III, blackbody; Phonons, Debye temperature and Debye T³ law [KK pp 99-109]
• Wed 05 Feb Chemical potential, Equilibrium condition in grand canonical ensemble, Physical meaning of chemical potential; Isothermal atmosphere, Scale height [KK pp 118-126]
• Mon 10 Feb Differential form of First Law including chemical potential; Lead-acid battery
• Wed 12 Feb Reservoir with particle and energy exchange; Gibbs factor; Grand partition function; Average particle number; Average energy
• Mon 24 Feb: pre-midterm review lecture
• Wed 26 Feb: pre-midterm question-and-answer session
• Fri 28 Feb: MIDTERM!!
• Mon 03 Mar Spin-statistics; fermions and Pauli exclusion principle, bosons; Fermi-Dirac distribution function
• Fri 07 Mar Bose-Einstein distribution function, large and small temperature behaviour; classical limit and connection to ideal gas; changing chemical potential
  (tutorial on Wed 05 Mar)
• Mon 10 Mar Equipartition theorem for ideal gas; rotational, vibrational and spin degrees of freedom; encoding internal degrees of freedom in the partition function; heat capacity at constant pressure; isothermal and isentropic processes; expansion into a vacuum.
• Wed 12 Mar (TUTORIAL - midterm solutions)
• Fri 14 Mar (PDF, 229K) (FUN BLACK HOLE / STRING LECTURE)
• Mon 17 Mar General properties of a quantum Fermi gas; Fermi distribution; gound-state of a 3-dim Fermi gas (Fermi energy, average energy per fermion).
• Wed 19 Mar Degenerate Fermi gas: application to astrophysics (white dwarf, neutron star); density of states of a Fermi gas.
• Mon 24 Mar Calculation of integrals over energy at non-zero temperature; energy and heat capacity of a near-degenerate fermion gas at non-zero temperature.
• Wed 26 Mar Temperature dependence of the chemical potential of a near-degenerate fermion gas; application to conduction electrons in a solid; low-temperature limit of the heat capacity of a solid conductor.
• Mon 31 Mar Density of states, density of occupied orbitals of a Bose-Einstein gas; occupancy of the lowest (ground-state) orbital; Einstein condensation temperature; Bose-Einstein condensation.
• Wed 02 Apr Bosons as composite systems of fermions; statistical fluctuations of the number distributions; superfluid helium-4.
• Mon 07 Apr Connection between entropy and the Second Law of thermodynamics; exchange of ordered (work) and disordered (heat) energy.
• Wed 09 Apr Review of post-midterm material.

Note: please ignore the numbers in the top right hand corners of the jpg scans of my notes. (They're just a personal bookkeeping device.)

P.S.: Boltzmann's grave really does have the famous formula S=k_Bln(W) on the headstone! See image (0.5MB!) ...

Assessment

Regarding homework . You will have two weeks to do each homework assignment. Homeworks will be due in-class at 11:10am (the beginning of class) as follows:

• HW#1, posted Wed 15 Jan, due Wed 29 Jan [Solutions #1 , posted Mon 03 Feb]
• HW#2, posted Wed 05 Feb , due Wed 26 Feb [Solutions #2 , posted Wed 26 Feb]
• HW#3, posted Wed 05 Mar , due Wed 19 Mar [Solutions #3, posted Mon 24 Mar]
• HW#4, posted Wed 26 Mar (note change of question 5!), due Wed 09 Apr [Solutions #4, posted Mon 14 Apr]

All homeworks must be done individually, and late homeworks (accepted only until solutions posted) will incur a 25% penalty. For details, see Homework Policies .

The midterm was held as a one-hour in-class test on Friday 28th February. For the midterm, a scientific calculator and an aid sheet were allowed. This aid sheet had to be handwritten and on only ONE side of the paper (size-reduction using a machine was NOT permitted!). Here are last year's midterm and solutions (students got 90 minutes to do it, in the end...)

For important university deadlines and dates, see the Faculty of Arts and Science Calendar . For a handy summary of date information for PHY252S, see this course calendar .

General Information

Material(from the course catalogue): "This is a core physics course for the Major and Specialist Programmes. This course is designed to explain macroscopic interactions using statistical concepts. The course will discuss the dynamical basis of temperature, entropy, chemical potential and other equilibrium thermodynamic quantities. The statistical methods will be illustrated by examples in which quantum statistics is essential in understanding the macroscopic behaviour. Topics covered will be: The quantum statistical basis of macroscopic systems; definition of entropy in terms of the number of accessible states of a many-particle system leading to simple expressions for absolute temperature, the canonical distribution, and the laws of thermodynamics; specific effects of quantum statistics at high densities and low temperatures."

The text is Kittel and Kroemer, Thermal Physics . Copies are on sale in the UofT Bookstore. You might find enlightening the 2000-1 Prof's opinions on the text and references for this course.

Raw mark statistics