University of Minnesota
School of Physics & Astronomy

Phys 2201.100

Introductory Thermodynamics and Statistical Physics

Final Exam
modified 17-Dec-2017 at 2:19AM by Paul Crowell

The final exam for PHYS 2201 will be held on Tuesday, December 19, from 1:30 - 4:30 PM in Tate B50.

A review session will be held on Saturday, December 16, 3:00 - 5:00 PM in Tate B20. Post problems that you would like to go over on the course Moodle page by noon on Saturday. Problems must be posted in advance.

Professor Crowell will have an office hour this Thursday from 2:00 - 5:00. He will be in his office Monday from 4:00 - 6:00.

The exam is cumulative.

I am posting the final exam from 2016 below. I have added a couple comments, particularly on the one problem that involved the type of derivation that I do not consider important.

I will post a formula sheet by early next week. Otherwise, the exam will be closed book. Although it will be similar in format to last year's exam, there may be more questions like the first problem of the second midterm (short answer, with no partial credit awarded).

I am posting below a summary of topics that I believe are important. If, after reading this, you have any questions along the lines of "What can be on the exam?", please ask them during Wednesday's class.

Summary of important topics | Download posted 11-Dec-2017 at 8:12PM
Formula Sheet for Final Exam | Download posted 17-Dec-2017 at 2:19AM
2016 Final Exam and Solutions | Download posted 6-Dec-2017 at 7:35PM

Homework Solutions
modified 13-Dec-2017 at 11:54AM by Paul Crowell

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Discussion Problems and Solutions
modified 13-Dec-2017 at 11:30AM by Paul Crowell
Week 15 Discussion | Download posted 13-Dec-2017 at 11:29AM
Week 14 Discussion | Download posted 10-Dec-2017 at 7:17PM
Week 13 Discussion | Download posted 28-Nov-2017 at 11:41PM
Week 12 Discussion | Download posted 21-Nov-2017 at 7:44PM
Week 11 Discussion | Download posted 15-Nov-2017 at 8:29PM
Week 10 Discussion | Download posted 8-Nov-2017 at 1:39PM
Week 9 Discussion | Download posted 1-Nov-2017 at 11:40AM
Week 8 Discussion | Download posted 25-Oct-2017 at 8:41PM
Week 7 Discussion | Download posted 25-Oct-2017 at 12:02AM
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

Lecture Slides
modified 13-Dec-2017 at 11:35AM 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 December 11 | Download posted 13-Dec-2017 at 11:35AM
Week of December 4 | Download posted 8-Dec-2017 at 5:51PM
Week of November 27 | Download posted 1-Dec-2017 at 7:51PM
Week of November 20 | Download posted 21-Nov-2017 at 7:52PM
Week of November 13 | Download posted 19-Nov-2017 at 10:12PM
Week of November 6 | Download posted 10-Nov-2017 at 11:07AM
Week of October 30 | Download posted 6-Nov-2017 at 10:51AM
Week of October 23 | Download posted 27-Oct-2017 at 7:44PM
Week of October 16 | Download posted 22-Oct-2017 at 10:10AM
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

Week 15
posted 9-Dec-2017 at 11:17AM by Paul Crowell

No new reading or problems. You should have read Chapter 5 up through p. 172, which represents the official "end" of the material covered by this class. In summary, the material which we have covered (and which therefore may appear on the final) is Chapter 1 (except 1.7), Chapter 2, Chapter 3 (keeping in mind that some of the material covered in Chapter 3, such as the paramagnet, is handled more efficiently using the canonical ensemble in Chapter 6), Chapter 4 through p. 132, Chapter 5 through p. 172, and Chapter 6. As a rule, the homework and discussion problems provide a guide for the types of problems that are likely to appear on the final exam.

I will attempt to schedule a review session, and we will discuss potential times in class on Monday. In the meantime, please post questions that you would like to review on the Moodle page.

Week 14
posted 1-Dec-2017 at 6:35PM by Paul Crowell

We will conclude by considering free energy, electrochemical work, and perhaps a little bit on phase transitions.

By Monday, December 4, 5.1 through p. 152 and 156 - 158
By Wednesday, December 6, pp. 152 - 156
By Friday, December 8, 5.2
By Monday, December 11, 5.3 through p. 172

Problems posted under Week 13

The link below is to a lecture I gave a year ago. You may want to listen to it:

Week 13
modified 29-Nov-2017 at 5:00PM by Paul Crowell

The lectures on Monday and Wednesday will be catching up with the reading

By Monday, November 27: 6.4 and 6.5
By Wednesday, November 29: 6.6 and 6.7
By Friday, December 1: 5.1 through p. 152

Problems Due Friday, December 8:
5.1, 5.2, 5.8, 5.9

Due Tuesday, December 12:
5.22, 5.29, 5.30

Exam Solutions
modified 21-Nov-2017 at 7:46PM by Paul Crowell

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Week 12
modified 29-Nov-2017 at 4:47PM by Paul Crowell

We will continue with statistical mechanics using the canonical ensemble.

By Monday, November 20: 6.4
By Wednesday, November 22: 6.5
By Monday, November 27, 6.6 and 6.7

Note that 6.5 assumes familiarity with the Helmholtz free energy, which is introduced in Chapter 5. You only need to know two things. First the definition of the Helmholtz free energy is F = U - TS . Second, you should read the argument on p. 161, which shows that for a system of fixed volume (and with a fixed number of particles) in contact with a reservoir at temperature T, the Helmholtz free energy is minimized in equilibrium. Make sure you understand Fig. 6.14.

Problems Due Tuesday, November 28
6.26, 6.31, 6.34, 6.36

Problems Due Tuesday, December 5
6.39, 6.41, 6.47, 6.48, 6.52
For 6.39(c), start by assuming that the moon once had an atmosphere with a composition and temperature similar to that of the Earth. For 6.47, you can apply the logic we used for rotational systems, in which we worried only about the ground state and the first excited state when calculating the Boltzmann factors at low temperature. Finally, for 6.48, you will need to calculate the rotational partition function, but do assume that you are in the high temperature limit. The rotational energy for O2 is given in Problem 6.24.

If you are bored over Thanksgiving, take a look at this (optional) problem. It is not mathematically difficult, but it introduces the concept of a phase transition. If you understand it, you are on the way to becoming a certified statistical mechanic.

DNA Problem | Download posted 18-Nov-2017 at 12:40PM

Week 11
modified 14-Nov-2017 at 1:35AM by Paul Crowell

Quiz 2 will be given on Wednesday, November 15

It will cover material through 4.3 (NOT 4.4, and nothing on steam engines)
There will not be any questions on chemical potential on this quiz, although there may some on the final exam (after we have done part of Chapter 5).

Last year's quiz and the formula sheet that will be provided are posted under Week 10.


Tate B20: A-M
Tate 110: N – Z

As a reminder, in Monday's class, I will do any relevant problem that is posted on the Moodle page by 7 PM Sunday.

I will hold my office hour this week on Tuesday, from 2:00 - 4:30. I will not hold regular office hours on Thursday, but I will probably be in my office from 2 - 3:15.


By Monday, November 13, Read 6.1
By Friday, November 17, Read 6.2 and 6.3
By Monday, November 20, Read 6.4

Problems (due Tuesday, November 21):
6.6, 6.11, 6.14, 6.16, 6.20 (b - e) 6.22 (parts b, c, and d)

For 6.20, you can use the result from part (a) without proof. It is just the standard expression for the sum of an infinite geometric series. For Problem 6.22, you can use without proof the result given in part (a). Use Excel, Matlab,or Mathematica to complete the graphs in 6.22(c). You do not need to do parts (e) and (f) of 6.22. Note, however, that the magnetization at high temperature obey's Curie's Law (M \propto 1/T ), just like the two-state paramagnet.

Week 10
modified 6-Nov-2017 at 11:23AM by Paul Crowell

I am away until Friday and will not be able to hold office hours this week. I will answer questions posted to the Moodle page, and I will hold an extra office hour next Tuesday (probably 2:00 - 4:00). Chris Conklin will do the lecture on Wednesday.

On Schroeder 3.37: As I noted in class, the "physical" solution of this problem goes as follows: If I have a volume of gas at z = 0 and then raise that same volume of gas by a height z, then the energy per molecule in that gas must have increased by mgz. Therefore the chemical potential at height z is just the chemical potential at z=0 PLUS mgz.

For the "rigorous" solution, see the file below.

See slides 15 - 18 of the "Week of October 30th" under Lecture Slides for a calculation of the Carnot efficiency without (I hope) any sign errors! I will go over this very quickly on Monday.

There is a very useful website for visualizing engines:

In particular, we will discuss the Stirling cycle, Otto cycle (used in the two-stroke engine), and Diesel cycle. The common four-stroke engine is also shown, but I will not discuss it in any detail.

Reading for Week 10:
By Monday, November 6: 4.1 and 4.3
By Wednesday, November 8: 4.2 and 4.4 (through p. 144)
We will not be discussing steam engines in any detail.
By Friday, November 10: 6.1 (through p. 225)

Problems due Friday, November 17:
6.1, 6.3, 6.5

Quiz 2 will be given on Wednesday, November 15

It will cover material through 4.3 (NOT 4.4, and nothing on steam engines)
There will not be any questions on chemical potential on this quiz, although there may some on the final exam (after we have done part of Chapter 5).

The second quiz from 2016 is posted below.

Formula Sheet for Quiz 2 | Download posted 4-Nov-2017 at 4:24PM
Quiz 2 from 2016 | Download posted 4-Nov-2017 at 4:21PM
Schroeder 3.37 | Download posted 3-Nov-2017 at 8:00PM

Week 9
modified 8-Nov-2017 at 1:42PM by Paul Crowell

We will complete our discussion of the consequences of maximizing entropy in equilibrium by relaxing the assumption that the total number of particles in a system is fixed. This will lead us to the concept of chemical potential. To sump up, systems in thermal equilibrium (exchanging energy) have the same temperature, those in mechanical equilibrium (exchanging volume) have the same pressure, and those in chemical equilibrium (exchanging particles) have the same chemical potential. A common example of chemical equilibrium is water in coexistence with water vapor. If the air above a pool of water is too dry, water will evaporate until the chemical potential of the water vapor (water in air) is the same as that of liquid vapor.

Note that this week Chris will be holding his office hour on Wednesday at 3 pm rather than on Tuesday.


By Monday, October 30: Schroeder 3.5 and 3.6
By Wednesday, November 1: 4.1
By Friday, November 3: 4.2 and 4.3

Problems Due Tuesday, November 7
4.1, 4.2, 4.3, 4.5, 4.6
The attached file contains some hints for 4.6, which does require some slightly messy algebra.

Due Friday, November 10
4.10, 4.13, 4.15, 4.18
For 4.13, you can ignore the last sentence ("Discsuss the implications, giving a numerical example..")

In addition, we will do several problems in class:
4.5, 4.7 - 4.9, 4.14, and the Diesel and Stirling engines. Take a look at these....

Hint for 4.6 | Download posted 30-Oct-2017 at 7:11PM

Note on Problem 4.3
modified 8-Nov-2017 at 1:41PM by Paul Crowell

In part (c), you are asked to compute the "rate at which water would evaporate" (due to the exhaust heat). There is an ambiguity here, which is whether you are to assume that the water is heated up to 100 C and then evaporates, OR whether it simply evaporates at the temperature of the environment. If the cooling towers are large enough, you can assume that the water evaporates at the same temperature as the river, and the latent heat in this case is the latent heat of evaporation of water at 298 K. This is perfectly well-defined, even though the water is not "boiling." You can look up the relevant number...

Week 8
modified 1-Nov-2017 at 4:03PM by Paul Crowell

Note that there will not be any problems due on Friday, October 27. Week 7 problems are due on Tuesday, October 24, and the next problem set (to be posted later) will be due on Tuesday, October 31.

This week we will generalize the concept of maximizing entropy in equilibrium to systems in which the volume and particle number can vary.

By Monday, October 23rd: 3.4 through p. 110
By Wednesday, October 25th: Rest of 3.4
By Friday, October 27th: 3.5 and 3.6

Homework: Due Tuesday, October 31
3.31, 3.32, 3.34,

Homework: Due Friday, November 3
3.36, 3.37, 3.38
Note: for Problem 3.36, start with the multiplicity function of the Einstein solid given in Problem 3.25.

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.

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.