**Unit 9: Statistical Thermodynamics: A Brief Overview**
*Thermodynamics describes the behavior of huge collections of
microscopic particles through the use of such averaged properties as
temperature, density, volume, entropy, energy, and so forth. In doing
so, thermodynamics avoids the extraordinarily difficult task of directly
computing the behavior of such collections of particles by using
Newton’s equations of motion. The “dynamics” of thermodynamics has to do
with changes in these averaged properties and not with the details of
the microscopic configuration of the materials in the system.*

*However, statistical mechanics is based on one fundamental assumption:
that all possible microscopic configurations that are consistent with
the observed averaged thermodynamic properties are equally likely to
occur. This is an application of a philosophical position that has
proven extremely useful in science: that we do not occupy a special
place in the universe. (This idea is often called the “mediocrity
principle” and may originally have been introduced by
16 ^{th}-century astronomer Nicolaus Copernicus.) If all possible
microscopic configurations are equally likely, then the physics we
measure are most likely to be those of the most common possible
configurations. This simple principle allows us to interpret the
thermodynamic properties of materials in terms of their component
particles and the interactions between them, which in turn leads to a
more fundamental understanding of why matter behaves the way it does.*

**Unit 9 Time Advisory**

Completing this unit should take approximately 10 hours.

☐ Subunit 9.1: 5 hours

☐ Subunit 9.2: 1 hour

☐ Subunit 9.3: 1 hour

☐ Subunit 9.4: 1 hour

☐ Subunit 9.5: 1 hour

☐ Subunit 9.6: 1 hour

**Unit9 Learning Outcomes**

Upon successful completion of this unit, you should be able to:
- estimate the probability of a molecule occupying any given state;
- define the *Boltzmann probability distribution*;
- differentiate microcanonical, canonical, and grand canonical
ensembles;
- define the term *microstate* and then relate the entropy to the
number of microstates of a system;
- state the differences between the molecular partition function (q)
and the canonical partition function (Q), and use them to determine
thermally accessible states for the system;
- calculate thermodynamic properties by using the partition function;
and
- calculate equilibrium constants by using the partition function.

**9.1 Introduction to Statistical Mechanics**
- **Lecture: Massachusetts Institute of Technology OpenCourseWare: Dr.
Moungi Bawendi and Dr. Keith Nelson’s “Lecture 24: Introduction to
Statistical Mechanics”**
Link: Massachusetts Institute of Technology OpenCourseWare: Dr.
Moungi Bawendi and Dr. Keith Nelson’s “Lecture 24: Introduction to
Statistical
Mechanics”

```
Also available in:
[iTunesU](http://ocw.mit.edu/courses/chemistry/5-60-thermodynamics-kinetics-spring-2008/video-lectures/lecture-24-introduction-to-statistical-mechanics/)
[MP4](http://ocw.mit.edu/courses/chemistry/5-60-thermodynamics-kinetics-spring-2008/video-lectures/lecture-24-introduction-to-statistical-mechanics/)
Instructions: Watch the video (approximately 52 minutes long),
which begins with a review of some topics covered in Unit 8 of this
course and then moves into a consideration of statistical mechanics.
Pay particular attention to how you can calculate the probability of
finding a molecule in a particular energy state. If we know the
distribution of all the molecules of a system in its various states,
then we can arrive at the macroscopic thermodynamic functions for
that system. You can find the lecture notes for this video here
(PDF).
Watching this lecture should take approximately 1 hour.
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-NonCommercial-ShareAlike 3.0 United States
license](http://creativecommons.org/licenses/by-nc-sa/3.0/us/).
```

**Reading: The University of Arizona: Professor W.R. Salzman’s**Link: The University of Arizona: Professor W.R. Salzman’s*“Dynamic Textbook” of Physical Chemistry*: “Notes on Statistical Thermodynamics – Partition Functions” and “Other Useful Information and Some Simple Models”*“Dynamic Textbook” of Physical Chemistry*: “Notes on Statistical Thermodynamics – Partition Functions”and “Other Useful Information and Some Simple Models” (HTML)Instructions: Study the material presented on both webpages. This reading essentially covers everything you need to know in order to meet the learning objectives set out for this unit of the course. You will learn how partition functions are defined, how they may be evaluated, and how they can be used to calculate the various thermodynamic functions of state. Focus on gaining a firm mastery of the material found in these readings, as an understanding of these topics will be crucial as you approach the remaining material in this unit of the course.

Reading this material should take approximately 4 hours.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.

**9.2 Partition Function (q): Large N Limit**
- **Lecture: Massachusetts Institute of Technology OpenCourseWare: Dr.
Moungi Bawendi and Dr. Keith Nelson’s “Lecture 25: Partition
Function (q)—Large N Limit”**
Link: Massachusetts Institute of Technology OpenCourseWare: Dr.
Moungi Bawendi and Dr. Keith Nelson’s “Lecture 25: Partition
Function (q)—Large N
Limit”

```
Also available in:
[iTunesU](http://ocw.mit.edu/courses/chemistry/5-60-thermodynamics-kinetics-spring-2008/video-lectures/lecture-25-partition-function-q-2014-large-n-limit/)
[MP4](http://ocw.mit.edu/courses/chemistry/5-60-thermodynamics-kinetics-spring-2008/video-lectures/lecture-25-partition-function-q-2014-large-n-limit/)
Instructions: Watch the video (approximately 51 minutes in length)
to learn about partition functions and how we use them to determine
what microstates are available to any given system. The lecture’s
examples of a perfect crystal, a monoatomic gas, and a polymer in
solution will help you visualize these abstract ideas in a more
concrete way. You can find the lecture notes for this video here
(PDF).
Watching this lecture should take approximately 1 hour.
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-NonCommercial-ShareAlike 3.0 United States
license](http://creativecommons.org/licenses/by-nc-sa/3.0/us/).
```

**9.3 Partition Function (Q): Many Particles**
*Note: Some of the material you need to know for this subunit is covered
by the reading assigned beneath Subunit 9.1, found above.*

**Lecture: Massachusetts Institute of Technology OpenCourseWare: Dr. Moungi Bawendi and Dr. Keith Nelson’s “Lecture 26: Partition Function (Q) — Many Particles”**Link: Massachusetts Institute of Technology OpenCourseWare: Dr. Moungi Bawendi and Dr. Keith Nelson’s “Lecture 26: Partition Function (Q) — Many Particles”Also available in:

Instructions: Watch the video (approximately 51 minutes in length) to learn how we can use the partition functions to calculate the thermodynamic functions for a system, thereby relating the microscopic and macroscopic properties of the system. You can find the lecture notes for this video here (PDF).

Watching this lecture should take approximately 1 hour.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States license.

**9.4 Statistical Mechanics and Discrete Energy Levels**
*Note: Some of the material you need to know for this subunit is covered
by the reading assigned beneath Subunit 9.1, found above.*

**Lecture: Massachusetts Institute of Technology OpenCourseWare: Dr. Moungi Bawendi and Dr. Keith Nelson’s“Lecture 27: Statistical Mechanics and Discrete Energy Levels”**Link: Massachusetts Institute of Technology OpenCourseWare: Dr. Moungi Bawendi and Dr. Keith Nelson’s “Lecture 27: Statistical Mechanics and Discrete Energy Levels”

Also available in:

Instructions: Watch the video (approximately 52 minutes in length) to learn about the statistical mechanical treatment of entropically driven processes. You will see how to calculate the entropy of mixing by using statistical mechanics. You can find the lecture notes for this video here (PDF).

Watching this lecture should take approximately 1 hour.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States license.

**9.5 Model Systems**
*Note: Some of the material you need to know for this subunit is covered
by the reading assigned beneath Subunit 9.1, found above.*

**Lecture: Massachusetts Institute of Technology OpenCourseWare: Dr. Moungi Bawendi and Dr. Keith Nelson’s“Lecture 28: Model Systems”**Link: Massachusetts Institute of Technology OpenCourseWare: Dr. Moungi Bawendi and Dr. Keith Nelson’s “Lecture 28: Model Systems”Also available in:

Instructions: Watch the video (approximately 51 minutes in length) to see an example of a statistical mechanical calculation carried out on a system in which the possible number of configurations (states) is nearly infinite. You can find the lecture notes for this video here (PDF).

Watching this lecture should take approximately 1 hour.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States license.

**9.6 Applications: Chemical and Phase Equilibria**
- **Lecture: Massachusetts Institute of Technology OpenCourseWare: Dr.
Moungi Bawendi and Dr. Keith Nelson’s “Lecture 29: Applications:
Chemical and Phase Equilibria”**
Link: Massachusetts Institute of Technology OpenCourseWare: Dr.
Moungi Bawendi and Dr. Keith Nelson’s “Lecture 29: Applications:
Chemical and Phase
Equilibria”

```
Also available in:
[iTunesU](http://ocw.mit.edu/courses/chemistry/5-60-thermodynamics-kinetics-spring-2008/video-lectures/lecture-29-applications-chemical-and-phase-equilibria/)
[MP4](http://ocw.mit.edu/courses/chemistry/5-60-thermodynamics-kinetics-spring-2008/video-lectures/lecture-29-applications-chemical-and-phase-equilibria/)
Instructions: Watch the video (approximately 52 minutes in length)
to learn how to use the partition function to determine K. You can
find the lecture notes for this video here (PDF).
Watching this lecture should take approximately 1 hour.
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-NonCommercial-ShareAlike 3.0 United States
license](http://creativecommons.org/licenses/by-nc-sa/3.0/us/).
```