## Course Information

### Course Objectives

Two years ago, PHYS*2240 introduced you to the central role that entropy and the second law play in thermodynamics. The entropy concept originates from our lack of precise knowledge about which of the many, many microscopic states a system is actually in, despite our imposition of constraints on the system at a macroscopic level. This statistical perspective on entropy motivated the second law. We explored the consequences of the second law for equilibrium, for phase transitions, and for various applications and measurable quantities. PHYS*4240 continues the discussion begun two years ago.

We will follow the topic selection and structure in the textbook Thermal Physics by Daniel V. Schroeder. PHYS*4240 begins with a brief review of thermodynamics. This covers Chapters 1-3 and selected topics from Chapter 5. We will generalize the thermodynamic formalism from the isolated systems we considered in PHYS*2240, which have constant energy, volume, and particle number, to systems with other constraints, such as fixed temperature, fixed pressure, and/or fixed chemical potential. Free energies and extremum principles, which are ultimately connected to the entropy and the second law, are key concepts in these situations. The overall structure, and universal nature, of thermodynamics will be emphasized.

The central portion of the course (Chapter 6) develops convenient statistical methods for calculating free energies and other thermal-average properties of materials by counting ensembles of appropriately-weighted microstates. Systems we will consider include the ideal gas, the van der Waals fluid, the paramagnet, and the Einstein solid.

The final part of the course is devoted to the statistical mechanics of systems that are described by the laws of quantum mechanics (Chapter 7). We will resolve several issues with the classical treatment of the ideal gas that are related to the counting of microstates when the particles are indistinguishable. We will discuss the distinct statistics of fermions and of bosons. Topics include the quantum ideal gas, the heat capacity of the free electron gas in metals, blackbody radiation and the photon gas, Bose-Einstein condensation, and finally the treatment of quantized lattice vibrations (phonons) of a solid.

You will refine your analytical and problem-solving skills through regular written assignments.

### Class Schedule and Location

Day | Time | Location |
---|---|---|

Monday, Wednesday, and Friday | 11:30 am - 12:20 pm | MCKN 305 |

First Lecture: Friday, September 7th

Last Lecture: Friday, November 30th

The course runs for 12 weeks (36 lectures); there is no lecture on Thanksgiving (Monday, October 8th). Friday, November 30th is a Thanksgiving make-up lecture.

### Instruction

Lecturer | Office | Extension | |
---|---|---|---|

Rob Wickham | MacNaughton 448 | 53704 | rwickham@uoguelph.ca [1] |

#### Office Hours

Monday, 2:00 pm - 3:00 pm and Tuesday, 1:00 pm - 3:00 pm. Assignments will be due on (alternate) Wednesdays. Please send me an email if you can't find me and wish to schedule a meeting.

### Course Materials

#### Course Website

CourseLink: Login via https://courselink.uoguelph.ca/ [2]

#### Required Textbook

An Introduction to Thermal Physics, by D. V. Schroeder (Addison Wesley Longman, 2000).

#### Other, optional resources

A typed set of related notes by Eric Poisson which can be found on his web site (see faculty link on the departmental web site).

#### Some of the Classic References

- F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill,1965,QC 175.R43).
- F. Mandl, Statistical Physics, Second Edition (Wiley,1988,QC174.8.M27).
- D.L. Goodstein, States of Matter (Prentice Hall, 1975; Dover, 1985, QC 173.3.G66).
- K. Huang, Statistical Mechanics, Second Edition (Wiley,1987,QC174.8.H83).
- C. Kittel and H. Kroemer, Thermal Physics, Second Edition (Freeman, 1980, QC 311.5.K52).
- L.D. Landau and E.M. Lifshitz, Statistical Physics, Third Edition, Part 1 (Pergamon, 1980, QC 175.L32).
- R.K. Pathria, Statistical Mechanics (Pergamon,1972,QC175.P35).

At this stage of your education, you should be consulting more than one text to enhance your learning and understanding of the material. No particular book is perfect in all respects and scientists regularly refer to several books and papers to understand a concept.

### Evaluation

Assessment | Weight | Due Date |
---|---|---|

Assignments (5) | 30% | Sept. 26, Oct. 10, Oct. 24, Nov. 14, Nov. 28 |

Midterm Test | 30% | Monday, October 29th, 7-9 pm, place TBD |

Final Exam | 40% | December 7h, 7:00 - 9:00 pm, place TBD |

A medical certificate is required if the exam is missed.

Assignments are due at the beginning of class; late assignments will receive a grade of zero.

Physics is not done in a vacuum. (OK, sometimes it is...) Students may discuss assignments amongst themselves but their written solutions must not be shared with anyone (this would be an example of plagiarism).

Plagiarism is the act of appropriating the ``...composition of another, or parts or passages of his [or her] writings, or the ideas or language of the same, and passing them off as the product of one's own mind...'' (Black's Law Dictionary). A student found to have plagiarized will receive zero for the work concerned. Collaborators shown to be culpable will be subject to the same penalties.

### Course Outline

I. Thermodynamics: Review and unfinished business [Chapters 1 to 3, 5.1, 5.2]

1. Equation of state, state variables, constraints, equilibrium

2. van der Waals equation of state, limiting cases, isothermal compressibility, phase coexistence, thermal, mechanical and diffusive equilibrium

3. First law, energy, heat, work, quasi-static processes, adiabatic processes

4. Isolated systems (U, V, N fixed) and the second law, entropy, example: van der Waals model, properties of entropy

5. Structure of thermodynamics: derivatives of entropy are equations of state, second derivatives are response functions, examples: equipartition, thermodynamic identity

6. Equivalent representations of thermodynamics, free energies, geometrical interpretation of the Legendre transform, extremum principles [5.1, 5.2]

7. Derivative relations and thermodynamic identities arising from free energies, Maxwell relations

8. Examples involving Maxwell relations: heat capacities and compressibilities

II. Calculating thermodynamic potentials: ensembles and examples (mainly Ch. 6)

9. Macrostates and microstates, multiplicity of the two-state paramagnet, fundamental assumption

10. Classical entropy of the ideal gas in the microcanonical ensemble [Chapter 2]

11. Two-state paramagnet in thermal contact with a reservoir, the Boltzmann factor [6.1]

12. General theory for systems in contact with a reservoir, partition function, averages in the canonical ensemble [6.2]

13. Examples, paramagnet, Einstein solid

--Thanksgiving--

14. Partition function and Helmholtz free energy, composite systems

15. Classical partition function for the ideal gas, and thermodynamics [8.1]

16. Equipartition of energy among degrees of freedom [1.3, 6.3]

17. Energy fluctuations in the canonical ensemble

18. Weakly interacting gases [8.1]

19. Identical particles, indistinguishability, and mixing

20. Issues with the classical model for ideal gas thermodynamics

21. Partition function for a quantum particle in a 1D box, 3D case, N non-interacting particles, internal degrees of freedom, heat capacity, rotation of diatomic molecules.

Midterm evening of Monday, October 29th (following lecture 22)

22. Chemical potential of an ideal gas, diffusive equilibrium in an external field: isothermal atmosphere, mobile magnetic particles

23. The Gibbs factor, grand partition function, averages in the grand canonical ensemble [7.1]

24. Example: adsorption of oxygen in the blood, particle number fluctuations

40th class day Friday, November 2nd

III. Quantum statistical mechanics (Chapter 7)

25. Fermions and bosons, microstates of N ideal, indistinguishable quantum particles, occupation numbers, quantum statistics, quantum volume

26. Fermi-Dirac and Bose-Einstein distribution functions, classical limit

27. Degenerate ideal Fermi gas, ground state of a Fermi gas (T=0)

28. Thermodynamic properties of the ground state of a Fermi gas

29. Non-zero temperatures, density of states, Sommerfeld expansion

30. Heat capacity of a degenerate ideal Fermi gas, electrons in metals

31. Chemical potential of Bose gas, ground and excited state occupancies

32. Bose-Einstein condensation, examples: liquid 4He, dilute gas

33. Gas of photons in thermal equilibrium, Planck distribution

34. Planck spectrum for blackbody radiation, thermodynamics

35. Debye theory of lattice vibrations in solids: phonons

36. Phonon thermodynamics, phonon contribution to the heat capacity of a solid

## Course Policies

### Course Evaluation

The Department of Physics requires student assessment of all courses taught by the Department. These assessments provide essential feedback to faculty on their teaching by identifying both strengths and possible areas of improvement. In addition, annual student assessment of teaching provides part of the information used by the Department’s Tenure and Promotion Committee in evaluating the faculty member's contribution in the area of teaching.

The Department's teaching evaluation questionnaire invites student response both through numerically quantifiable data, and written student comments. In conformity with University of Guelph Faculty Policy, the Department’s Tenure and Promotions

Committee only considers comments signed by students (choosing "I agree" in question 14). Your instructor will see all signed and unsigned comments after final grades are submitted. Written student comments may also be used in support of a nomination for internal and external teaching awards.

Note: No information will be passed on to the instructor until after the final grades have been submitted.

### E-mail Communication

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### When You Cannot Meet a Course Requirement

When you find yourself unable to meet an in-course requirement because of illness or compassionate reasons, please email the course instructor to make arrangements.

### Drop Date

At Guelph, the last date to drop one-semester courses, without academic penalty, is Friday, November 2nd. For regulations and procedures for Dropping Courses, see the Undergraduate Calendar. [3]

### Copies of out-of-class assignments

Keep paper and/or other reliable back-up copies of all out-of-class assignments: you may be asked to resubmit work at any time.

### Accessibility

The University of Guelph is committed to creating a barrier-free environment. Providing services for students is a shared responsibility among students, faculty and administrators. This relationship is based on respect of individual rights, the dignity of the individual and the University community's shared commitment to an open and supportive learning environment. Students requiring service or accommodation, whether due to an identified, ongoing disability or a short-term disability, should contact Student Accessibility Services (SAS) as soon as possible.

For more information, contact SAS at 519-824-4120 ext. 56208 or visit the SAS website [4].

### Academic Misconduct

The University of Guelph is committed to upholding the highest standards of academic integrity and it is the responsibility of all members of the University community – faculty, staff, and students – to be aware of what constitutes academic misconduct and to do as much as possible to prevent academic offences from occurring. University of Guelph students have the responsibility of abiding by the University's policy on academic misconduct regardless of their location of study; faculty, staff and students have the responsibility of supporting an environment that discourages misconduct. Students need to remain aware that instructors have access to and the right to use electronic and other means of detection.

Please note: Whether or not a student intended to commit academic misconduct is not relevant for a finding of guilt. Hurried or careless submission of assignments does not excuse students from responsibility for verifying the academic integrity of their work before submitting it. Students who are in any doubt as to whether an action on their part could be construed as an academic offence should consult with a faculty member or faculty advisor.

The Academic Misconduct Policy is detailed in the Undergraduate Calendar. [5]

### Recording of Materials

Presentations which are made in relation to course work—including lectures—cannot be recorded or copied without the permission of the presenter, whether the instructor, a classmate or guest lecturer. Material recorded with permission is restricted to use for that course unless further permission is granted.

### Resources

The Academic Calendars [6] are the source of information about the University of Guelph’s procedures, policies and regulations which apply to undergraduate, graduate and diploma programs.