Callen Thermodynamics Homework Solutions

Physics 846 - Statistical Physics I - Fall 2003

Current reading assignment

Please read sections 4.A, 4.B, 4.C, and the introduction to section 4.D of the textbook. When you are done, fill out the questionnaire. The deadline for this assignment is Thursday 11/13 at 3:59am, i.e., you would be well served to finish it by Monday evening. You will not be graded on which answers you give but you are required to fill out the form.

Contact Information

General Information

Classes:Tuesdays and Thursdays, 10:30am-12:18pm
Location:Caldwell Lab 0277
Web page:
Makeup class:The first class on 9/25 had to be rescheduled since I was out of town. We will meet on Tuesday 10/07, 9:30am-10:18am in Smith lab 4079. We will do the same on Tuesday 10/21 also in SMith Lab 4079.
First class:Tuesday, September 30
Midterm exam:Thursday, October 30, 10:30am-12:18pm
No class:Tuesday, November 11 (Veteran's day)
No class:Thursday, November 27 (Thanksgiving)
Last class:Thursday, December 4
Final exam:Thursday, December 11, 9:30am-11:18am
Credit hours:4
Prerequisites:622 (undergraduate statistical physics II) and 664 (undergraduate theoretical mechanics) or equivalent

Problem Sets

Here, problem sets and their solutions will be posted as the class progresses.

Lecture notes

Here, you can find the lecture notes as the class progresses. Beware, these are scanned from my hand-written notes and thus rather large files.

  • 09/29 - introduction and state variables
  • 10/02 - equations of state, first and second law
  • 10/07 - entropy, third law, fundamental equation, thermodynamic potentials
  • 10/09 - thermodynamic potentials (handout)
  • 10/14 - thermodynamic potentials example, local equilibrium conditions
  • 10/16 - response functions, local stability conditions
  • 10/21 - Le Chatelier's principle, Gibbs paradoxon, first chapter summary
  • 10/21 - phase transition examples, Gibb's phase rule
  • 10/28 - classification of phase transitions, Clausius-Clapeyron equation
  • 11/04 - van-der-Waals gas, superconductors
  • 11/13 - helium, magnetic systems, chapter II summary, probability, stochastic variables
  • 11/18 - characteristic function, jointly distributed variables, central limit theorem, law of large numbers
  • 11/25 - Maxwell velocity distribution of an ideal gas, microscopic definition of entropy
  • 12/02 - microcanonical ensemble for a discrete system
  • 12/04 - microcanonical ensemble for continuous system, chapter IV summary


The grades were distributed as follows:



This class and the Statistical Physics II (847) class in the winter quarter will give an overview over thermodynamics and statistical physics. At the end of the winter quarter we will have explored the phenomenology of systems with a large number of particles and what methods exist to deal with this large number of degrees of freedom. We will be able to make very general statements on any such system as well as to analyze specific models.

The fall quarter will cover the following topics:

  • thermodynamic variables
  • laws of thermodynamics
  • thermodynamic potentials
  • thermodynamic stability
  • Maxwell relations
  • classification of phase transitions
  • Clausius-Clapeyron equation
  • microscopic definition of entropy
  • microcanonical ensemble


Linda Reichl, A Modern Course in Statistical Physics, ISBN 0-471-59520-9

Don't be afraid about the size of the book. We will cover only selected chapters in class. Specifically, we will concentrate on the main sections of chapters 2, 3, 4, and 7. At times, we will also deviate a little bit from the textbook. However, the chapters and sections which we will not be able to cover make for a good reference on thermodynamics and statistical physics.

Other books

If you do not like the textbook or if you are just interested in reading things from a different point of view, you may want to look at the following textbooks. A long list of statistical mechanics graduate textbooks is also available at

Herbert B. Callen, Thermodynamics and an introduction to thermostatistics, ISBN 0-471-86256-8
Very thorough coverage of thermodynamics.
Walter Greiner, Horst Stocker, and Ludwig Neise, Thermodynamics and statistical mechanics, ISBN 0-387-94299-8
Good overview of both thermodynamics and statistical physics.
Kerson Huang, Statistical mechanics, ISBN 0-471-81518-7
Very short on thermodynamics but a good book on statistical physics.
Leo P. Kadanoff, Statistical physics: statics, dynamics, and renormalization, ISBN 9-810-23758-8
Very advanced book.


The grade for this class is determined by a weighted average of the weekly problem sets, the midterm exam and the final exam. There will be no curve grading and I hope that everybody will get an A. The final grades will be determined according to the following scheme:


Problem Sets (20%)

Problem sets will be handed out every Thursday in class starting on September 25 (although I will be out of town on 9/25, there will be copies of the first (shorter) problem set in the classroom on this day.) The problem sets will also be available on the course web page. The problem sets are due the following Thursday in class. Exceptions to this rule are set 5 that is due on Tuesday, November 4, and set 9 that is due on Tuesday, December 2. Students who cannot attend class are requested to give their solutions to a fellow student or deposit them in my or the grader's mailbox in the physics business office on the same day. If you have a good reason for not being able to complete your homework by the due date please contact me for approval of an extension and exclusively put it into my mailbox by the approved extended deadline. In the interest of protecting the grader from extra work due to homework trickeling in late unapproved late homework will not be accepted.

The solutions of the problem sets will in general be discussed during the second half of each Thursday's class, i.e., 11:30am-12:18pm. However, there will be a few exceptions to this rule:

  • The fifth problem set will be discussed in the first half of the class on Thursday, November 6
  • The ninth problem set will be discussed in the second half of the class on Tuesday, December 2
Every student will have to present the solution to one problem on the board during the quarter. The problems will be assigned to the students in advance and the students about to present a problem are required to meet with me prior to the problem session to prepare their presentation.

You are strongly encouraged to discuss about the problem sets (and any other aspect of the class) with your fellow students. Group solutions to problem sets by two people working together are accepted. However, the solutions have to be hand-written by both students, either alternating weekly or within one week's solutions. Typed solutions will not be accepted.

Midterm Exam (40%)

The midterm exam will take place on Thursday, October 30, 10:30am-12:18am during class. Students which have any problem with the date of the exam are asked to contact me as soon as possible. It will contain several problems which will be related to the problem sets up to the one due on October 23 (set number 4) and to the lectures up to the class on October 28. There will be one problem asking for conceptual understanding and knowledge of basic definitions. It will be a closed book exam.

Final Exam (40%)

The final exam will take place on Thursday, December 11, 9:30am-11:18am. Students which have any problem with the date of the exam are asked to contact me as soon as possible.

The exam will contain several problems which will have a close resemblance to the problem sets of the full quarter with a strong emphasis on the second half of the quarter. There will be one problem asking for conceptual understanding and knowledge of basic definitions. It will be a closed book exam.

Reading assignments (0% but required)

From time to time reading assignments of sections of the textbook will be given. These reading assignments will be announced on top of the course web page as well as in class. On days on which reading assignments are discussed in class you have to answer some questions about the reading on the web by 3:59am the day of class. Your answers will not be graded; the only requirement is that you do take the questionnaires. However, since the answers to these questions will help focus the discussion during the class to those issues you had difficulties with during your reading, it is in your owm interest to answer these questions as honestly as possible. It is also not implausible that some of these questions might reappear on the exams. In order to get a passing grade you are not allowed more than two missed reading assignment questionnaires. If you missed more than two reading assignments during the quarter, you will have to schedule an oral examination on the missed assignments with me before obtaining a passing grade.

Attendance (0%)

Attendance during the lectures is not required. However, the problem sets will be closely related to the contents of the class and it is every students responsibility to keep up with announcements, etc., made in class if not present.

Special needs

Students with any special needs are asked to inform me at their earliest convenience.

[OSU physics][College of Mathematical and Physical Sciences][The Ohio State University]

Instructor: Roland E. Allen

Room M213 Mitchell Institute Building (Fall, 2017), [phone removed by department as economy measure]

Office hours: Monday, Tuesday, Wednesday 3-4 p.m.

The class meets each Tuesday and Thursday at 11:10 a.m. in MPHY 109. Each Tuesday at 4:00 p.m. we will have an optional homework presentation meeting, also in MPHY 109.

Textbook: Statistical Mechanics, 3rd Edition, by R. K. Pathria and P. D. Beale.
You may want to bring your copy of the textbook to class for easier note-taking.


Homework                               40%
2 “midterm” exams                 40%
final exam (comprehensive)    20%

Homework is due at the beginning of class each Tuesday.
Homework late by < 48 hours, 1/2 credit. Homework late by > 48 hours, no credit.

The “midterm” exams and final exam will be in the evening at 7:30 - 10:30 p.m. in Room MPHY 213 on the designated dates: October 5, November 16, December 8.

Some topics in astrophysics will be amplified compared to the textbook's treatments, with short and simple treatments of neutron stars and white dwarfs (including the Chandrasekhar limit), stellar pulsation, the Saha ionization equation, and reactions in cosmology. Topics not in the textbook include (i) review of some techniques in thermodynamics (from Herbert B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd Edition ), (ii) basics of the BCS theory of superconductivity (Linda E. Reichl, A Modern Course in Statistical Physics, 4th edition, pp. 224-231), and (iii) the Rayleigh-Bénard instability as an example of a nonequilibrium phase transition (Reichl, pp. 378-385).

HOMEWORK [P means Pathria & Beale (the textbook), M means Statistical Mechanics by D. A. McQuarrie, and G means States of Matter by D.L. Goodstein -- each assignment should be regarded as tentative rather than final until two weeks before it is due.

August 31: M1-58, 1-59

September 5: P 1.9; M 1-28, 1-30, 1-29, 1-35

September 12: P 1.7, 1.16; M 1-24, 1-61, 1-63

September 19: P 2.1, M 1-51, Callen 3.4-11, Callen 3.4-12, Kubo 1.36

September 26: P 3.6, P 3.7, P 3.15, P 3.16; M 2-17

October 3: P 3.19, 3.20, 3.21 (a) classical only plus 3.22, 10.19, 3.25

October 10: M 7-32, M 8-15, M 8-16, P 3.35 (a), P 3.43

October 17: P 4.1, P 4.7, P 3.29 1st part, P 3.29 2nd part, M 10-4

October 24: P 8.4 (a), P 8.6, P 8.8, P 8.13, M 10-19

October 31: P 8.4 (b) and (c), P7.5, P 7.13 first paragraph, P 7.23, M 10-27

November 7: P 6.1 first part, P 6.7, P 6.10, P 7.25, P 7.26

November 14: P 9.8, P 11.1 (a), P 7.33, P 7.34, P 7.35

November 21: P 12.1, P 11.8 (counts as 2 problems), G 5.12, G 6.3 (b) & 6.4 (counts as 1 problem)

For extra credit (maximum credit - 2 homework sets): 10-minute talks on Tuesday, December 5, at 7:00 p.m., in our classroom MPHY 109.

Some of the many possible very broad topics for talks: sample-607-topics.pdf .

Phys607Exam1-2017.pdf , Phys607Exam1-2017solution.pdf

Phys607Exam2-2017.pdf , Phys607Exam2-2017solution.pdf

Phys607Final-2017.pdf , Phys607Final-2017solution.pdf

Exam1-2017-607.pdf , Exam1-2017-607-solution.pdf

Exam2-2017-607.pdf , Exam2-2017-607-solution.pdf

Final-Exam-2017-607.pdf , Final-Exam-2017-607-solution.pdf




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