University of Minnesota
School of Physics & Astronomy

Phys 2201.100

Introductory Thermodynamics and Statistical Physics

Exam Solutions
modified 20-Oct-2017 at 5:50PM by Paul Crowell
Quiz 1 Solutions | Download posted 20-Oct-2017 at 5:50PM

Week 7
modified 16-Oct-2017 at 1:37AM by Paul Crowell

We will continue working through the consequences of our assumption that the entropy of two systems will be maximized in equilibrium. In the first case, the volume and particle number are fixed, and the two systems can exchange energy. We will then deal with the cases in which the volumes and particle number can change.

Note that Professor Crowell will be away Tuesday - Thursday and will not be holding office hours this week.

By Monday, October 16: 3.1
By Tuesday, October 17: pp. 98 - 99 and 103 - 104. You will need this for discussion section.
By Wednesday, October 18: 3.2
By Friday, October 20: rest of 3.3 and 3.4 through p. 111

Due, Tuesday, October 24
3.19, 3.20, 3.22, 3.25(d-e)

More problems will be posted later.

Most of 3.19 will be done in discussion section on October 17.

For 3.20, you will need to invert the hyperbolic tangent to complete the last part. This is not difficult! If you write out the hyperbolic tangent in the form tanh(x) = (e2x - 1) / (e2x + 1)- , you can invert the expression tanh(x) = 0.99- , with a couple lines of algebra. Alternatively, both Mathematica and Matlab have inverse hyperbolic functions (ArcTanh in Mathematica, atanh in Matlab ).

Although calculating the energy and magnetization of the system as a fraction of their maximum values is relatively easy, the entropy takes a little more work. There are two options, but my intention is for you to do the following: Use your answer for M / Mmax- to determine the fraction of the moments that are up or down. Then you can calculate the entropy using an expression you will have derived in discussion section.

It would be possible to use the result of Problem 3.23 to compute the entropy in 3.20, but I do not want you to waste time deriving the expression in 3.23. There is no physics in that....

Week 6
modified 12-Oct-2017 at 5:55PM by Paul Crowell

Homework problems:

Due Tuesday, October 17
2.37, 2.40, 2.42, 3.1, 3.3, 3.5, 3.6

Due Friday, October 20
3.7, 3.10, 3.12, 3.13, 3.14, 3.16

More problems will be added later.

NOTE that Quiz 1 covers material through Schroeder 2.4, but NOT Section 1.7.
This week I will hold an office hour on Tuesday from 2.

Quiz 1 is Wednesday, October 11. Please read the announcement under Week 5
Last name A - Sc: Tate B20
Last name Sp-Z: Tate 110

On Monday, October 9, I will address any question posted on the course Moodle site (under the Week of October 9). Please post your questions before 6 PM Sunday evening.

By Friday, October 13: Schroeder 3.1

Problems (due the week of October 16) will be posted later.

Discussion Problems and Solutions
modified 18-Oct-2017 at 5:12PM by Paul Crowell
Week 7 Discussion | Download posted 18-Oct-2017 at 5:12PM
Week 6 Discussion | Download posted 11-Oct-2017 at 12:10AM
Week 5 Discussion | Download posted 3-Oct-2017 at 7:50PM
Week 4 Discussion | Download posted 26-Sep-2017 at 4:24PM
Week 3 Discussion | Download posted 19-Sep-2017 at 2:41PM
Week 2 Discussion | Download posted 12-Sep-2017 at 5:22PM

Homework Solutions
modified 12-Oct-2017 at 6:02PM by Paul Crowell

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Lecture Slides
modified 16-Oct-2017 at 12:43AM by Paul Crowell

I will post my lecture slides about once a week. These are not intended to be complete lecture notes. They are simply the visual aids I use in lecture. In some cases, I will add material, particularly when (as happened on Friday) I am not satisfied with an explanation I did in class.

Week of October 9 | Download posted 16-Oct-2017 at 12:43AM
Week of October 2 | Download posted 9-Oct-2017 at 7:31PM
Week of September 25 | Download posted 1-Oct-2017 at 11:05PM
Week of September 18 | Download posted 22-Sep-2017 at 6:15PM
Week of September 11 | Download posted 16-Sep-2017 at 4:08PM
Week of September 4 | Download posted 9-Sep-2017 at 6:40PM

modified 3-Oct-2017 at 12:33AM by Paul Crowell

I am looking for a good MatLab tutorial and will post it when I get a response.

Someone made a comment today about an error when using Matlab's nchoosek function. For sufficiently large values of n and k, it will give a warning stating that the answer may not have full double precision accuracy (16 digits). You can ignore this, as 15 digits is more than enough. There is nothing wrong with the function. It will definitely overflow when the arguments get too large, but for values less than 200 it is perfectly correct.

However, we will soon be dealing with numbers so large that this will all be moot, and the only choice for computing the factorials of genuinely large numbers (e.g. 10^23) will be Stirling's approximation.

Two scripts are below. One can be adapted for making plots of functions. The second implements Schroeder2.10. Note that actually doing the math is easy. Making the table requested with labels takes a few extra steps.

Matlab script for problems like Schroeder2.10 | Download posted 3-Oct-2017 at 12:33AM
Matlab script for plotting a function | Download posted 3-Oct-2017 at 12:33AM

Week 5
modified 30-Sep-2017 at 4:44PM by Paul Crowell


By Monday, 10/2: 2.5 through the top of p. 72
By Wednesday, 10/4: remainder of 2.5 and 2.6 through p. 76
By Friday, 10/6: remainder of 2.6


No problems due Tuesday (exam on Wednesday)

Due Friday, October 13th
2.32, 2.34, 2.35, 2.36

Quiz 1

Quiz 1 will be held on Wednesday, October 11 at the usual class time. Please go to the correct room:
Last name A - Sc: Tate B20
Last name Sp-Z: Tate 110
The quiz will cover material through Section 2.4, with most of the questions on Chapter 1. I am posting last year's quiz as an example. This year's will be similar in format. My exams consist of problems (not multiple choice). Some may be very short, and others longer. There will be 4 - 5 total. I will post a sheet with numbers and formulae that will be provided. The quiz is otherwise "closed book." You will need a simple scientific calculator.

Data, constants, and formulae to be provided for Quiz | Download posted 30-Sep-2017 at 4:44PM
Quiz 1 from 2016 | Download posted 30-Sep-2017 at 4:38PM

Week 4
modified 29-Sep-2017 at 11:12AM by Paul Crowell


By Monday, September 25: 2.1, 2.2, and 2.4 through the top of p. 63
By Wednesday, September 27, 2.3
By Friday, September 29, 2.4 (pp. 63 - 67)

Problems to be done by Tuesday, October 3rd in discussion section
2.17, 2.19, 2.21, 2.23

Problems to be done by Friday, October 6th in lecture
2.26, 2.27, 2.28, 2.29, 2.30

Professor Crowell has a research program review Wed. and Thurs. Chris Conklin will lecture on Wed. Professor Crowell will hold office hours on Thursday 5:00 - 6:00, but he will not be able to hold the earlier office hour.

Important announcement about problem sets
modified 23-Sep-2017 at 2:15PM by Paul Crowell

Starting the week of September 25th, all problem sets must be handed in IN CLASS on the day due. Email submissions will not be accepted. Late homework will not be accepted.

Assignments due on Tuesday can be handed in in ANY discussion section. If for some reason, you have to miss your discussion section, you can hand the assignment in at another discussion section. You can give your problem set to someone else to hand in if necessary.

Assignments due on M, W, or F must be handed in during lecture.

Exceptions to this policy will be made only for university-approved absences arranged in advance.

Access to MatLab
posted 22-Sep-2017 at 6:36PM by Paul Crowell

The Physics department recommends that all majors learn to use MatLab. Over the next few years, all of the advanced laboratory courses will "support" MatLab, meaning that it will become the default application for graphing, data analysis, etc.

For this class, any program that does basic mathematics and graphics will suffice, but it is a good time to start using MatLab if you are not already doing so.

1. CSE students can download MatLab for free:

2. If you are not a CSE student, you CAN obtain access to MatLab by opening a CSE computer account:
You will be asked to enter a sponsor and a reason for requesting a CSE computer account
I am the sponsor: Paul Crowell
My username is crowell
The reason for requesting an account should be "Enrolled in PHYS2201."

Once you have a CSE computer account, you can download MatLab using the same link for CSE students.

Hints on Homework
modified 23-Sep-2017 at 1:58PM by Paul Crowell

1. All problem sets must be stapled.

2. When answering numerical problems, you must use an appropriate number of significant figures.

3. For any problem (numerical or not), work out the problem symbolically. If the problem is numerical, then substitute the numbers at the end.

4. Do not hand in a draft! When you have figured out a problem, prepare a final version that you hand in. Depending on your style, this may mean that you have to prepare a "clean copy" of your solutions. A few cross-outs are fine, but the grader should be able to read your solution easily.

5. I saw some good examples of carefully explaining the background of a problem, assumptions being made, and the sources of any information that was used. This is critical. In many cases, however, steps were skipped, or important information was not provided. Remember that the grader must be able to follow your reasoning in order to provide credit for a solution.

Week 3
modified 22-Sep-2017 at 7:00PM by Paul Crowell


By Monday, September 18: 1.7 (Note, I will discuss only very briefly in class, probably on Wednesday)
By Wednesday, September 20: finish 2.1 and read pp. 60 - 63 (top) from section 2.4. Also if you have a Math textbook that discusses simple combinatorics and statistics, it will be helpful to review counting problems and the binomial distribution.
By Friday, September 22: 2.2
By Monday, September 25, 2.3 and the rest of 2.4

Homework (Problems in bold to be handed in):

Due Tuesday, September 26 in discussion section:
2.1, 2,2, 2.3, 2.15, 2.16. Also hand in Additional Problem #2 (below)

Due Friday, September 29 in lecture:
2.5, 2.6, 2.9, 2.10, 2.11

For Problem 2.3,use a software package of your choice to make the plot in part (g). You do not have to use Stirling's approximation for this problem. The numbers are small enough that Excel or any other program can handle them.

For Problems 2.9, 2.10, and 2.11 you will also need to use a software package of your choice.

Additional Problems for Week 3 | Download posted 16-Sep-2017 at 4:36PM

Week 2
modified 16-Sep-2017 at 4:19PM by Paul Crowell


By Monday, September 11: 1.4 and 1.5
By Wednesday, September 13: 1.6 through p. 32
By Friday, September 15: rest of 1.6
By Monday, September 18, 1.7

Problems (those to be handed in are in bold)
Tuesday September 19: 1.41, 1.44, 1.46, 1.47, 1.48

Friday, September 23: 1,49, 1.50, 1.53, 1.54, 1.55

Regarding 1.50(f). Of course the sun is not made of methane, but the combustion of methane can be considered a "typical" chemical reaction. Given your answer to this problem, is it possible for the sun to be powered by ANY chemical reaction?

Office Hours
modified 7-Sep-2017 at 12:54PM by Paul Crowell

Office Hours for Paul Crowell (in PAN 222)
Thursdays: 2:00 - 3:15 and 4:00 - 5:00

Office Hours for Chris Conklin (in PAN 222)
Tuesday, 3:00 - 4:00

Office Hours for Bo Xiong (in PAN 222)
Monday, 2:00 - 3:00

PAN 222 is the room next door to Professor Crowell's office. If he is not in 222, you will find him in 220.

Week 1 (Assignment)
modified 4-Sep-2017 at 11:56AM by Paul Crowell


Complete by Friday, September 8: Schroeder, 1.1 - 1.4
Complete by Monday, September 11: 1.5 and 1.6 through p. 32


(Problems in bold to be handed in):
Due on Tuesday, Sept. 12 in discussion section: 1.7, 1.12, 1.15, 1.16, 1.21, 1.22(a-d), 1.24, 1.25
due on Friday, Sept. 15 in lecture: 1.27, 1.28, 1.29, 1.31, 1.32, 1.34, 1.37, 1.38, 1.40

Important Note

You will notice that many of the problems assigned this semester (for example, 1.15) will require you to look up information elsewhere OR to make reasonable estimates of certain quantities. Your written solution should state clearly the source of any information used to solve the problem and/or your rationale in making the estimates required.

First class meeting
modified 31-Aug-2017 at 7:35PM by Paul Crowell

We will meet for the first time on Wednesday, September 6th at 09:05 in Tate B20. Discussion sections will NOT meet on Tuesday, September 5. They will meet for the first time on September 12th.