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Chapter 4

Partition Functions and Statistical Thermodynamics: Spreadsheet Activities To Promote Connections in Physical Chemistry Craig M. Teague* and Truman H. Jordan Department of Chemistry, Cornell College, 600 First Street SW, Mount Vernon, Iowa 52314, United States *E-mail: [email protected]

We have reimagined the traditional physical chemistry sequence to emphasize connections between topics. In this chapter, we focus on a key set of linking topics in physical chemistry: partition functions and statistical thermodynamics. We cover the Boltzmann distribution in the first course, then return to discuss partition functions at the beginning of the second course. We then work with statistical thermodynamics throughout the rest of the second course. Through a set of spreadsheet exercises in the second course, students gain facility with partition functions and statistical thermodynamics and how they interface with both quantum mechanics and classical thermodynamics. Starting from spectroscopic data, students calculate partition functions, thermal energies, entropies, temperature independent energies of reaction, and ultimately free energies of reaction in order to calculate thermodynamic quantities for observed reactions and predict whether or not a speculative reaction will occur. In this way, students keep using statistical thermodynamics while further aspects of quantum mechanics, classical thermodynamics, and kinetics are covered in class and in the laboratory. Student performance and response to the spreadsheet exercises are discussed, followed by possible implications for other physical chemistry faculty members.

© 2018 American Chemical Society Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

Background Introduction Undergraduate physical chemistry is typically divided into a two-course sequence where one course focuses mostly on classical thermodynamics and one course focuses mostly on quantum mechanics (1). Some schools teach the thermodynamics course first while others teach the quantum mechanics course first; in other schools students can take either course first. Kinetics, dynamics, and statistical thermodynamics are often fit into these two courses in a variety of ways: sometimes at the end of the thermodynamics course, sometimes at the end of the quantum mechanics course, sometimes in other places. Occasionally, a school is able to offer a third physical chemistry course focused on kinetics and dynamics, while a few schools have implemented other ways of covering the material of undergraduate physical chemistry (2–5). In our view, helping students see and use connections within the physical chemistry material is of critical importance (6, 7). While faculty members naturally see these connections because of their deep engagement with the material through teaching and research, it is often difficult for students to see connections since they are learning many of these topics for the first time (4). In particular, classical thermodynamics and quantum mechanics are large topics with several important aspects that take time to develop in the classroom and laboratory. There are so many details that students can lose sight of the forest while mucking about in the trees. As such, students sometimes miss the full power of physical chemistry concepts, how they fit together with each other, and how they can apply to a variety of situations and systems. To address this, we have reimagined the physical chemistry sequence to explicitly emphasize connections within the content. Briefly, this involves covering an introduction to all areas (8) in the first course: kinetics, dynamics, quantum mechanics, spectroscopy, statistical mechanics, and classical thermodynamics. Then, in the second course, all of these areas are covered in more depth and sophistication. By structuring the courses in this way, students see and work with topics again while also having the background to understand how the concepts within the topics relate to each other and how they can apply in a variety of situations. This chapter focuses on a key aspect of our approach: how we cover partition functions and statistical thermodynamics and how students interact with these topics, which we view as perhaps the most important way of connecting various other topics in physical chemistry. We briefly review the concept of partition functions and the topic of statistical thermodynamics, especially with an eye toward how they link other ideas together. We then discuss our course structure and pedagogy, especially with respect to partition functions and statistical thermodynamics. Next, we turn to describing the key set of assignments where students engage with partition functions and statistical thermodynamics over an extended period of time while we continue to cover other concepts in more depth in the classroom and laboratory. This set of assignments is a multipart spreadsheet exercise titled Use and Significance of the Partition Function, and we discuss the general structure and goals of the set before describing each part of the set. 50 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

We conclude the chapter with a discussion of student performance and response to the spreadsheet exercises as well as possible implications for other physical chemistry faculty members. We provide the student handout for the spreadsheet activities as an appendix.

Partition Functions and Statistical Thermodynamics (9) The concepts of the Boltzmann distribution and partition functions, along with statistical thermodynamics as a general topical area, are key linking ideas in physical chemistry (4, 10–14). For example, from quantum mechanics we develop the characteristics of different types of energy levels: translational, rotational, vibrational, and electronic. By using appropriate simplified models (e.g., particle on a line, particle on a sphere, etc.) we can discuss the spacing and degeneracy of these levels, and we can examine the differences in these energy levels through various forms of spectroscopy. A natural question then arises: How many of these levels are occupied? Or, put another way, what are the relative populations of these energy levels? The Boltzmann distribution and partition functions are the way to answer these questions. This question is quite important, for only occupied levels contribute to the overall internal energy and entropy of a system. We can use statistical thermodynamics to get to these quantities. Once we have information about the internal energy and entropy, the rest of classical thermodynamics can be addressed. Therefore, these ideas successfully link quantum mechanics and classical thermodynamics, which is the single most important connection students can make in undergraduate physical chemistry. These three key linking ideas, the Boltzmann distribution, partition functions, and statistical thermodynamics, relate to other areas of physical chemistry as well. The intensities of spectral lines relate to populations of energy levels. The Maxwell distribution of speeds can be thought of as one case of the Boltzmann distribution. Using mean energies in statistical thermodynamics, one can derive the internal energy and heat capacity of gases and recover the classical equipartition values. A full understanding of transition state theory relies on the partition function of the vibrational mode that causes the transition state to decompose into products, and transition state theory itself relates to understanding likely mechanisms for reactions through the enthalpy of activation and entropy of activation. Equations of state, including the ideal gas law, can be derived from statistical thermodynamics. The partition function relates to the chemical potential, which in turn relates to phase stability, phase transitions, reaction progress, and chemical equilibrium. These are just some of the examples of how students can see and use these three key linking ideas to understand more of the power and even beauty within physical chemistry. Despite these connections, we think partition functions and statistical thermodynamics often get shortchanged in the undergraduate curriculum (1). We believe there are multiple related reasons for this. First, classical thermodynamics, a powerful set of concepts itself, can be developed without reference to energy levels. Second, as mentioned above, classical thermodynamics and quantum mechanics are large topics and we could easily spend multiple courses just 51 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

covering these two topics. Third, some physical chemistry faculty members are unsure of when to cover partition functions and statistical thermodynamics because to understand them even partially it is best if one already knows something about both classical thermodynamics and the nature of energy levels through quantum mechanical models. However, we argue that the importance of partition functions and statistical thermodynamics, especially their ability to link together other concepts, means that we should work to overcome these barriers to giving these topics appropriate attention in the undergraduate curriculum. We also argue that a modern treatment of classical thermodynamics should involve at least some discussion of the molecular view of matter and energy; just because we can discuss classical thermodynamics without reference to energy levels does not mean we should (4, 10).

Course Structure and Pedagogy At Cornell College, we have a two-course sequence in physical chemistry where both courses are required for the chemistry major. These courses are mathematically rigorous; multivariable calculus is a prerequisite for Physical Chemistry I. This first course is an option for our biochemistry / molecular biology major, but few students in this major choose this option in part because of the mathematical prerequisite. We often have physics majors in our physical chemistry courses as well. We are able to offer Advanced Physical Chemistry every other year, which is an upper level option for our chemistry major. However, this chapter deals only with how we address partition functions and statistical thermodynamics in Physical Chemistry I and II. All of our physical chemistry courses have both classroom and laboratory components; there is no separate physical chemistry laboratory course. Our department is certified by the American Chemical Society’s Committee on Professional Training (15). In the classroom, we use student-centered pedagogy through Process Oriented Guided Inquiry Learning (POGIL). In a POGIL classroom, students work in groups to discover the course material themselves by working through carefully designed activities. POGIL has been described in detail elsewhere (16–18). We also use a supplementary lecture after every few POGIL activities in order to summarize the content the students have learned and to draw explicit connections between different topics. We use POGIL classroom activities that are either published or in the beta testing phase. Three separate collections of POGIL activities have content related to the Boltzmann distribution, partition functions, and statistical thermodynamics, and we use all three (19–21). We are involved with the POGIL-Physical Chemistry Laboratory (POGIL-PCL) project to write and implement POGIL laboratory activities, and we use these activities in the laboratory portions of our courses (22–24). In Physical Chemistry I, we begin by covering some aspects of chemical kinetics, the Maxwell distribution of speeds, and quantum mechanics. We then turn to the Boltzmann distribution to describe how energy levels are populated at thermal equilibrium. We do this through the POGIL activities 52 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

Spectroscopy.4: Population of Quantum States in Shepherd and Grushow (20) and/or ChemActivity 18: Thermal Energies in Moog, Spencer, and Farrell (19). In the supplemental lecture to follow these activities, we draw explicit connections back to the Maxwell distribution of speeds and a general treatment of how the Boltzmann distribution gives the populations of the energy levels students have learned so far. Students see the partition function for the first time through one of the Boltzmann distribution equations; we point this out and we indicate that we will use the partition function extensively in Physical Chemistry II. In the rest of Physical Chemistry I, we develop some of classical thermodynamics with an eye toward the molecular nature of matter and energy. We refer back to the Boltzmann distribution and other topics as we discuss classical thermodynamics. We begin Physical Chemistry II by covering the partition function. This is first done through a POGIL activity currently in the beta testing phase in Shepherd, Grushow, Garrett-Roe, and Moog (21). It is also possible to cover some of this through ChemActivity 19: Thermal Energies of Molecules in Moog, Spencer, and Farrell (19). We follow this with a lecture that connects back to the Boltzmann distribution from the first course and provides more detail about the partition functions and their connections to classical thermodynamics. In this lecture, we do not cover every aspect of these connections; rather, we indicate that much information can be obtained from working with the partition functions but do not go into detail. Students discover many of these details themselves through the spreadsheet activities. We then introduce these spreadsheet activities, including their general structure and goals as well as some of the aspects of the calculations necessary for Part I. In the rest of Physical Chemistry II, we cover other areas of physical chemistry in more detail: kinetics, dynamics, quantum mechanics, spectroscopy, and classical thermodynamics. As we do so, we constantly relate back to the Physical Chemistry I material and partition functions. For example, in chemical kinetics we are able to do a robust treatment of transition state theory early in the course because students already know about the partition function. As the students learn more about different types of energy levels from further development of quantum mechanics, we relate this to the partition function in class and through the spreadsheet activities. When students discover more depth within classical thermodynamics we spend time discussing how that information can be obtained or derived from the partition function and how the spreadsheet activities can be used to calculate these quantities using statistical thermodynamics.

Spreadsheet Activities Overview, Structure, and Goals The set of spreadsheet activities is titled Use and Significance of the Partition Function and consists of five parts as described below. The specific calculations within each part are different, yet the five parts build on each other. At the beginning of Physical Chemistry II, we give students a detailed handout that describes all five parts (see the Appendix to this chapter). This handout describes 53 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

what they are to calculate, it develops some of the equations they need to implement in their spreadsheet, and it gives some guidance regarding how the results should be presented. However, the handout does not discuss how to write spreadsheet formulas or other spreadsheet techniques. Each part is due separately, with due dates throughout most of the second course. For each part, students upload their spreadsheet and a short report to the learning management system for the course. We grade each part separately, providing comments on both the spreadsheet and the report. In many cases, students need to fix any spreadsheet errors because later parts of the set of activities depend on earlier parts. Our students come into physical chemistry with a fairly wide range of experience and comfort with spreadsheet work. We provide support and structure to this activity in multiple ways. First, though students do most of their spreadsheet work outside of class, we do devote some time in class for student spreadsheet work. This time is more structured for Part I, where we often project a spreadsheet at the front of the room and walk students through a few key spreadsheet techniques in the context of the calculations needed for Part I. The in-class time becomes less structured later in the course; it is usually open working time for the students. The total class time devoted to the entire set of activities is 1.5-2 h. Second, if students send us a draft of their spreadsheet calculations before the due date we offer a no-penalty check of their work. We do not go into detail about how to fix incorrect formulas; rather, we highlight cells that students need to check and send it back to them. Students can also ask us questions during the open working time in class or during office hours. Third, we provide a spreadsheet template for Parts I and II. This template lays out a structure for their calculations and provides some information about which equations to use, but it does not provide any formulas within the spreadsheet. See Figure 1 for part of the template for Part I. We find that this template eases student concerns as we begin the spreadsheet activities, especially because Part I is fairly involved. The template also makes it easier to grade; we simply use a key in the same format as the template. We do not provide a template for subsequent parts since students should be more comfortable with the nature of the assignment and how the results should be presented within the spreadsheet. We have several related goals for these spreadsheet activities. First, as noted above, we want students to continue to work with important linking concepts as we cover other physical chemistry material in more depth throughout the second course. The activities also provide a touchstone for us as we can point out how concepts work throughout the second course in the context of the ongoing activities. Second, the activities allow students to explore and use a direct link between the microscopic and the macroscopic. The activities provide students direct applications of what we cover in class: students use spectroscopic data about individual molecules to calculate properties of bulk substances and reactions. Specifically, we want students to demonstrate properties of partition functions and how they relate to entropies, Gibbs free energies, and thermodynamic parameters of reactions. Third, we want students to develop their spreadsheet skills. These transferrable skills will serve students well beyond our courses. Specifically, in this set of activities the following spreadsheet skills are emphasized: absolute and relative cell referencing, how to work with complicated formulas in a way that 54 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

makes them easy to troubleshoot, effective and efficient use of the spreadsheet to take advantage of the strengths of the program, construction of scatter plots, sheet copying and working with similar but not identical sheets, cell referencing between sheets, working within a spreadsheet workbook that has many individual sheets, and revising and updating sheets for future use in a different specific chemical application.

Part I: Partition Functions In this part, we ask students to calculate partition functions for two gases, N2 and I2, at a specified pressure and at multiple temperatures. Starting from given vibrational transitions and bond lengths along with other information they look up (e.g., atomic masses, fundamental constants, conversion factors), students calculate the translational, rotational, and vibrational partition functions at multiple temperatures. Along the way, they calculate other important molecular parameters such as reduced mass, moment of inertia, rotational constant, and thermal wavelength. Students discover the relative magnitudes of the three partition functions, how each of the three depends on temperature, and how the molecular differences between N2 and I2 affect the results.

Figure 1. Portion of the template provided to students for Part I of the spreadsheet activities. 55 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

As discussed above, we provide a student template for this part. This template has the following areas on the sheet: experimental data, fundamental constants, conversion factors, calculated values that do not involve temperature (subdivided into three parts, one each for calculated values that relate to the translational, rotational, and vibrational partition functions), calculated values that use temperature, and a summary table. Structuring the template in this way helps us achieve some of our goals related to spreadsheet skills: students recognize when to use absolute and relative cell referencing, they get a sense for how to effectively design a spreadsheet, and they gain some insight into how to construct complicated formulas in an efficient way. Throughout the template, we guide the students toward the appropriate units to use, but students must enter all data and spreadsheet formulas themselves and they must use the spreadsheet to do all unit conversions. The spreadsheet formulas required are reasonably complex, and students must keep track of multiple masses and unit pitfalls as they set up formulas. Part II: Thermal Energies Here we ask students to calculate the thermal energy of three gases, NO2, H2, and XeF2, at a variety of temperatures. Importantly, students do not calculate the full internal energy in this part; the internal energy includes both temperatue-dependent parts and temperature-independent parts (e.g., zero point energy) (25, 26). We use the term thermal energy to indicate the portion of the internal energy that is temperature dependent. Since we treat these as ideal gases, and because we neglect any electronic contribution to the thermal energy, the thermal energy is divided into three parts: translational, rotational, and vibrational. The translational and rotational thermal energies are calculated from their equipartition values, while the vibrational thermal energy is calculated from the mean energy of the vibrational mode. Students are given vibrational transitions for the three molecules. They discover, or rediscover, how the translational and rotational thermal energies vary with temperature and with molecular shape (linear or nonlinear). They also discover the relative magnitude of the vibrational thermal energy and its temperature dependence as well as the fact that each vibrational normal mode has its own associated thermal energy. Because of the way we have students carry out this part, Part II does not depend on Part I. We still provide a template with similar areas on the sheet, though the formulas the students need to enter into the spreadsheet are generally not as complex as in Part I. Students must take care in identifying the shape of the molecules and understanding how the shape affects the rotational thermal energy. Furthermore, students must deal with the degenerate vibrational mode of XeF2 in the correct way, and it is not necessarily obvious to students how to do this. Part III: Entropies In this part we ask students to calculate the total entropy for four different gases: Ar at one temperature, CO at one temperature, N2 at various temperatures, and I2 at various temperatures. In the handout, we provide the bond length and 56 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

vibrational transition for CO; similar data they need for N2 and I2 is given in Part I. Beginning with this part, the handout also explains and derives the formulas necessary to complete the spreadsheet activities because we have not covered these formulas in other ways in class. For Part III, the handout shows how the entropy relates to both the partition function and the thermal energy and it derives equations for translational, rotational, vibrational, and total entropy. Part III explicitly depends on both Parts I and II. For each chemical species, students need to set up both a partition function sheet and a thermal energy sheet before setting up a final sheet with cell referencing to the first two sheets to calculate the entropy. The number of sheets in the workbook really begins to grow at this point. Since Ar has only translational energy levels, this one is the simplest for students to calculate. For CO, students need to realize that they may need to change the spreadsheet formula they used for reduced mass, depending on how they set that formula up in Part I. They also need to recognize that the symmetry number of CO differs from what they used in Part I. For N2 and I2, students are familiar with these molecules because of Part I. Though we do suggest a format for the final table of entropies for a chemical species, we do not provide a template for Part III. The entropy calculations themselves are not that complex once students have the partition functions and thermal energies calculated. To provide a check for students, in the handout we list what the total entropies should be for Ar and CO at the temperature we specify. Part IV: Temperature-Independent Energies of Reaction With this part, students begin working with thermodynamic parameters of reactions. We ask students to work with three gas phase reactions, each consisting of a noble gas atom A reacting with F2 to give AF2. Two of these reactions, where A = Xe and A = Kr, are known to occur while the third, A = Ar, is speculative. The main goal of this part is for students to calculate the temperature-independent contribution to the heat of reaction, ΔrxnU(0), for these three reactions (i.e., the change in internal energy of the reactions at 0 K). Students also calculate an estimated value for the standard heat of reaction for the Ar reaction. We provide the experimental standard heats of formation for KrF2 and XeF2 and the molar bond enthalpy of F-F along with an estimated molar bond enthalpy for Ar-F. We also provide vibrational transitions for F2 and KrF2 along with estimated vibrational transitions for ArF2. Information for XeF2 is found in previous parts of the handout. Part IV does not depend explicitly on earlier parts of the activity, though many of the necessary calculations are similar to Part II. Students begin Part IV by estimating the standard heat of formation for ArF2 by using molar bond enthalpies and the number of bonds broken and formed, which is a straightforward calculation. The handout derives the equations necessary to calculate the temperature-independent contribution to the heat of reaction through a term that appears to be a thermal enthalpy. For each chemical species, this thermal enthalpy includes a translational term and a PV term, and for molecules it includes rotational and vibrational terms as well. In other words, thermal enthalpies are calculated like thermal energies except for the additional PV term. Students 57 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

need to calculate the thermal enthalpy for each chemical species in the reaction, then use these values along with the standard heat of reaction to calculate the temperature-independent contribution to the heat of reaction.

Part V: Free Energies of Reaction and Temperature Dependence of Free Energy In this culminating part, we ask students to calculate standard state free energies for the same three reactions as in Part IV. Furthermore, we ask them to try to find the temperature where the reactions switch from nonspontaneous to spontaneous. Finally, with respect to the speculative ArF2 reaction, we ask them to describe the prospects for making this compound through the standard state formation reaction. In this part, the only given information they need is bond lengths for F-F, Xe-F, Kr-F along with an estimated bond length for Ar-F. All other needed data is given in previous parts of the handout. For Part V, the handout develops the equations that relate the free energy to the partition function and temperature-independent contribution to the heat of reaction. Equations developed in Parts III and IV are needed to derive the equations in Part V. Part V builds on all the other parts of the set. Students must calculate partition functions and other quantities for all chemical species in the reactions, then put the quantities together to calculate free energies of reaction. This requires many separate sheets in the workbook. While the needed calculations are mostly familiar at this point, there are a few key wrinkles to which students must be attentive. First, as just noted there are many individual sheets to keep track of for this part. Second, students need to be careful about how they calculate reduced masses and moments of inertia for the triatomic products of the three reactions. This subtlety is easy to overlook. Third, because we ask students to examine the temperature dependence of the free energy of reaction and because many of the individual pieces to the calculation are temperature dependent, students must carefully design the spreadsheet to take this into account. When students work on this part of the calculations, we suggest they have a single master temperature cell they can adjust on the last sheet and then have temperature cells on all the other sheets reference this master cell. When students complete Part V, they have started with spectroscopic data from individual molecules and, through the partition functions, they have calculated whether or not a bulk reaction will occur and how the reactions are affected by temperature. They have successfully merged quantum mechanics with classical thermodynamics in a concrete and hands-on way, which is the single most important connection for them to make in undergraduate physical chemistry.

Student Performance and Response In this section, we describe student performance on two levels: with the spreadsheet activities and in the physical chemistry course sequence as a whole. We then discuss student response at the same two levels. 58 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

Based on their spreadsheet work and the associated brief reports, students generally do well with the spreadsheet activities. It is not uncommon for several students to get all calculations correct within a given part of the set. Furthermore, if a student’s calculations are in error s/he usually can fix the issues by the time s/he needs to use that part again in subsequent parts of the set. Part II seems to be the easiest for the students to complete correctly. If students understand conceptually what they are to do, Part IV tends to go well also. However, getting to this conceptual understanding is sometimes difficult for students; the concepts covered here are generally unfamiliar to the students. While students usually master spreadsheet techniques like absolute and relative cell referencing and working with multiple sheets (including cell referencing to other sheets), their spreadsheet design is sometimes hard to follow in Parts III, IV, and V. In other words, when they do not have a template, students will often get the correct answers but their spreadsheets are hard to follow. This can mean that grading is more challenging for these parts of the set. Despite this, in their brief reports students often draw out the important aspects of the calculations and discuss them appropriately. One key thing we look for in the brief reports is students’ ability to relate the calculated quantities back to differences in molecular properties. For example, we want them to discuss the differences in calculated quantities with respect to spacing of energy levels (e.g., rotational versus vibrational) and with respect to molecular parameters (e.g., molecular mass and bond length of N2 versus I2). Students usually do well in explaining connections like this. With respect to student performance in the physical chemistry sequence as a whole, we give the American Chemical Society Division of Chemical Education Examination Institute’s Physical Chemistry Comprehensive Exam as part of our final exam in Physical Chemistry II. Like all ACS Exams, this exam is nationally normed (27). Over the past seven years, the mean of our students’ scores (N = 45) is at the 76th percentile nationally and the median of our students’ scores is at the 82nd percentile nationally while our standard deviation, 7.65, is smaller than the standard deviation of the national data, 8.28. While we are pleased with these results, we cannot say how much of our students’ success is due to our course structure, pedagogy, and/or emphasis on recognizing and using connections within physical chemistry. However, with these results we are confident that we are not harming our students. Student response to the spreadsheet activities is generally positive, with a few students noting they were the most beneficial aspect of Physical Chemistry II. Many students note that their spreadsheet skills improved as a result of these activities. Students appreciate the templates for Parts I and II, and some ask for templates for the remaining parts. At present we do not plan to do this, but we may provide more information about what the final table of results should look like in order to provide a bit more structure and also hopefully make these parts a little easier to grade. Students usually report that they spent significant time on the activities outside of class, and some students comment on the sense of accomplishment they get when they successfully implement a complicated spreadsheet workbook. 59 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

We do not get many comments on the overall structure of the courses, which we attribute to the fact that the students have not experienced physical chemistry in any other way so they have no basis for comparison. However, the comments we do get are positive. The most common comment is that students like seeing aspects of the material in both courses, especially when it comes to quantum mechanics. One other student comment stands out: after completing the ACS final exam, one student questioned why a particular exam question was in one section of the exam (e.g., thermodynamics) when she thought it belonged in another section (e.g., dynamics). We hope this meant that the student was seeing the connections within the content and viewing physical chemistry as a coherent whole rather than a set of disconnected topics. This is what we want for all our students.

Implications and Conclusions Faculty members at other institutions are free to implement the spreadsheet activities we describe here. Although we encourage faculty members to carefully consider their pedagogy and course structure, we believe the activities can be implemented with any pedagogy (POGIL, other student-centered pedagogy, lecture-based) or mixture of pedagogies and in many course structures. For example, faculty members need not completely reorganize topics within the physical chemistry sequence as we have done. In a more traditional sequence where classical thermodynamics is covered in one course and quantum mechanics is covered in another course, we recommend placing partition functions and the start of these activities at the beginning of the second course. Indeed, this is what we used to do before we integrated all topics into both courses. Regardless of the main content of the first course, either quantum mechanics or classical thermodynamics, we think the beginning of the second course is the place to discuss partition functions and start these activities. At schools that have only one physical chemistry course required for the major, we think these activities can still be implemented. The details of this kind of implementation will depend on the topics covered and the sequence of those topics, but perhaps the activities could be started about halfway through this course. This should allow enough time for students to complete the set of activities. With all that said, we are confident that faculty members can use these spreadsheet activities or variations of them in new and creative ways we have not thought of yet. We look forward to hearing from others about this. In this chapter, we described how partition functions and statistical thermodynamics link together many other topics within physical chemistry. Most important is the connection between quantum mechanics and classical thermodynamics. We discussed our pedagogy and how we structure our courses to emphasize connections within the material, especially with respect to the Boltzmann distribution, partition functions, and statistical thermodynamics. We then turned to the description of a multipart spreadsheet exercise titled Use and Significance of the Partition Function, where we discussed the structure and goals of the set of activities before describing the details of each part. We discussed student performance and response, both to the spreadsheet activities and to 60 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

the physical chemistry sequence as a whole. We closed with some potential implications for other physical chemistry faculty members, and we look forward to hearing how others implement or modify the ideas we shared.

Acknowledgments Most importantly, we thank our students over the course of many years. It has been our pleasure to interact with you in the classroom and laboratory. Our departmental colleagues are incredibly supportive of our efforts described in this chapter and in other endeavors. In addition, discussions with physical chemistry faculty at other schools have been valuable and quite fruitful. Finally, we thank Cornell College for travel funds to attend meetings and workshops to discuss these ideas.

Appendix Below we provide the student handout for the set of spreadsheet activities. This handout, along with the student template for Parts I and II, is also available electronically by emailing the corresponding author: [email protected]. Note that this handout does not give all the equations students need; we ask them to refer to material we covered in class for the other equations. In addition, instructors may wish to ask students to use chemistry databases to look up the molecular data we provide in Parts III-V. While this would take students more time, it is a valuable skill for students to learn to use such databases effectively. If instructors choose to ask students to find the appropriate data, we do recommend providing the estimated Ar-F bond energy and bond length as well as the estimated vibrational frequencies for ArF2 in Parts IV and V. Use and Significance of the Partition Function This set of exercises will run throughout the course. It is divided into five parts. Each part has its own due date, although you are welcome to work ahead. I am happy to provide feedback on draft calculations as noted below. For all parts of these activities, we will follow the Brief Report Guidelines with specifics as noted below. Work in Microsoft Excel or a similar program that I can open in Excel. Part I Due via upload to Moodle by 12:30 pm on xx/xx In this part of the exercises you will calculate partition functions for two gaseous molecules, N2 and I2. Specifically, calculate qT, qR, and qV for 1 mol of both of these gases at a pressure of 1 bar and the following temperatures: 100 K, 200 K, 400 K, 800 K, and 1600 K. 61 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

Helpful information: For N2: = 2331 cm-1 and r = 1.094×10-8 cm For I2: = 213 cm-1 and r = 2.666×10-8 cm Bond length = average internuclear separation = r You may assume ideal gas behavior: PV = nrt Rotational constant: Moment of inertia: I = μr2 Reduced mass: where m1 = mass of one atom and m2 = mass of second atom Formulas we discussed in class would also be useful You must calculate everything explicitly in your spreadsheet. In other words, only fundamental constants and the data given above for the two molecules can be entered as numbers in the spreadsheet; use formulas to calculate everything else, even simple conversions. Make sure all your units cancel; q is always unitless. As you work in the spreadsheet jot down notes about your progress, issues you overcame or decisions you made, and any observations you have on the calculations. Use the spreadsheet template given; your final calculations will have this form:

In addition to collecting this data in table form, constructing plots is important as well. In one plot or set of plots, show how the partition function(s) depend on temperature. In a second plot or set of plots, show how the partition function(s) depend on the properties of the molecules. If you are unsure how to set up these plots, or if you want to discuss ideas as to how you might show important relationships in these plots, I am happy to discuss this. I will gladly check your final results so you can have some feedback on your calculations; if you would like me to do this see me during a designated work time or during office hours. Also due via upload at the due date is a Word file; see the separate guidelines for what to include in this file. Part II Due via upload by 12:30 pm on xx / xx 62 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

In this part you will calculate thermal energy for three gaseous molecules, NO2, H2, and XeF2. We use the term thermal energy to indicate the part of the internal energy that is due to the system’s temperature. Specifically, calculate U(T) - U(0) (we sometimes labeled this quantity E in class) for 1 mol of each of these gases at the following temperatures: 100 K, 300 K, 500 K, 700 K, and 1400 K. Because we are treating these as ideal gases, and because we are neglecting any electronic contribution to the thermal energy, the thermal energy has three parts: translational, rotational, and vibrational. Use the equipartition values for translation and rotation; for vibration, use the formula given below (remembering that each vibration is a separate normal mode of motion and therefore must be calculated separately). Remember, there is a zero point energy that also exists for a molecule; this assignment does not calculate zero point energy but rather the energy difference between the zpe and the energy at temperature T; i.e., we are calculating the thermal energy. Helpful information: For NO2: = 750 cm-1, 1323 cm-1, and 1616 cm-1 For H2: = 4395 cm-1 For XeF2: = 213 cm-1, 213 cm-1, 513 cm-1, and 557 cm-1 Translational thermal energy: Rotational thermal energy:

Vibrational thermal energy: where where is the vibrational transition (in cm-1) for one vibration (this formula is simply the mean energy of a vibrational mode). As before, calculate everything in the spreadsheet. As you work in the spreadsheet, jot down notes about your progress, issues you overcame or decisions you made, and any observations you have on the calculations. Use the spreadsheet template given; your final calculations will have this form:

63 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

In addition to collecting this data in table form, constructing plots is important as well. In one plot or set of plots, show how the thermal energies depend on temperature. In a second plot or set of plots, show how the thermal energies depend on the properties of the molecules. If you are unsure how to set up these plots, or if you want to discuss ideas as to how you might show important relationships in these plots, I am happy to discuss this. I will gladly check your final results so you can have some feedback on your calculations; if you would like me to do this see me during a designated work time or during office hours. As before, a Word file on Part II is due at the due date. If you like, you can put all your comments for all parts in a single electronic document, just use headings to separate the parts of your document. Part III Due via upload at 12:30 pm on xx / xx In this part you will calculate entropy for four gaseous species: Ar, CO, N2, and I2. You will need to make use of Parts I and II for this assignment; they should be incorporated into this assignment where necessary. Specifically, calculate: 1. The entropy of 1 mol of Ar at its boiling point of 87.3 K. It should be 129.2 J K-1. 2. The entropy of 1 mol CO at 298 K. Data you will need: r = 112.8 pm; = 2168 cm-1. The total entropy should be 197.5 J K-1. 3. Calculate the entropy of 1 mol N2 and 1 mol I2 at 100 K, 200 K, 400 K, 800 K, and 1600 K. Data you will need are given in Part I. Helpful information: The total entropy is given by

However, in each of these cases we have a gas, where the particles are indistinguishable. This means that

where N is the number of particles. Substituting and focusing on the second term in Equation 1, we find that

due to the properties of logarithms. We now use Stirling’s approximation (see your text for a derivation),

which gets more accurate the more particles there are (say, for example, in a mole). Therefore, substituting Equation 4 into Equation 3, we have 64 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

Since we are dealing with 1 mol in each case, N = NA which is Avogadro’s number. Furthermore,

so we can write

Returning to the first term in Equation 1, the total thermal energy, [U(T) - U(0)], is simply equal to the sum of the components of the thermal energy: translational, rotational, and vibrational (we will ignore any electronic contributions). We have also seen that when the energy is a sum the partition function will be a product. Therefore,

again ignoring any electronic contributions. Showing each of the internal energy components explicitly and inserting the product qTqRqV for q (and because of the properties of logarithms) we find that

Of course, we can write the total entropy as the sum of the individual entropies, and Equation 9 gives us a way to do that. Note that translational motion is truly indistinguishable which is why we left NA in the translational term below. In other words, rotations and vibrations are localized to one molecule so they are always distinguishable. This also means that the last term, R, is actually part of the translational contribution since it came from Stirling’s approximation (which had to be made because the particles were indistinguishable). Therefore, we find the following:

In Part I you calculated q values; in Part II you calculated [U(T) – U(0)] values. This assignment builds on these two parts; use these last four equations to calculate entropies. 65 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

You must calculate everything explicitly in your spreadsheet. In other words, only fundamental constants and the data given above for the molecules can be entered as numbers in the spreadsheet; use formulas to calculate everything else, even simple conversions. Make sure all your units work out. You should have a sheet for q, a sheet for U, and a sheet for S for each question above. There is no template for the S sheet, but the format of your spreadsheet should look something like previous templates. In the S sheet use cell referencing from the q and U sheets as appropriate (i.e., don’t just type in numbers that you found in a different calculation). The final calcs in the S sheet should look like the following:

What might make sense to plot for this part? Discuss your ideas with me, then construct appropriate plots. I will gladly check your final results so you can have some feedback on your calculations; if you would like me to do this see me during a designated work time or during office hours. As before, a Word file on Part III is due at the due date. Part IV Due via Moodle dropbox by 12:30 pm on xx / xx In the last two parts, we will be calculating thermodynamic parameters for reactions using the ideas and calculations we have developed in Parts I, II, and III. Recall that when dealing with thermodynamics of reactions, we define Δrxn as products – reactants. For example, ΔrxnH° = ∑prodΔfH° − ∑reacΔfH°. In Part IV, you will calculate the temperature-independent contribution to the heat of reaction. In other words, you will calculate the internal energy change of reactions at 0 K, ΔrxnU(0). In effect, this is the change in zero point energy for a reaction. This part deals with three reactions, two experimentally measured and one hypothetical:

66 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

Goal of Part IV: Calculate an estimated value for ΔrxnH°298 K for the formation of ArF2 and ΔrxnU°(0) for each of the three reactions. Helpful information Enthalpy of a F-F bond: 147,988 J mol-1 Estimated enthalpy of a Ar-F bond: 8000 J mol-1 Vibrational transitions: 919.0 cm-1 for F2; 449 cm-1, 232 cm-1, 232 cm-1, and 599 cm-1 for KrF2; estimated transitions at 390 cm-1, 200 cm-1, 200 cm-1 and 410 cm-1 for ArF2 Other information was given in Part II. First, note that an approximate value for the enthalpy change for a reaction is given by

where bb stands for bonds broken and bf stands for bonds formed. Second, note that the standard state internal energy U° of any chemical species is equal to the standard state zero point energy U°(0) plus the standard state thermal energy U°(T):

This equation can be rewritten:

and then substituted back into the first version of the equation:

to yield what appears to be a trivial result, that U° = U°. However, let us add P°V to each side of Equation 17:

and then make use of the definition of enthalpy:

by substituting the standard state version of Equation 19 into Equation 18 on both the left and the right:

From the last equation we can identify the standard state enthalpy of a chemical species as the sum of two terms, the zero point energy U°(0) and a term that looks like the thermal enthalpy: [H° – U°(0)]. This thermal enthalpy term is similar to the thermal energy term you calculated in Part II, with one exception. It will have translational, rotational, vibrational, and PV parts. The PV part is present because this is enthalpy and not internal energy. Use the ideal gas law to find the PV part for each chemical species. Of course, individual atoms have no rotational or vibrational thermal enthalpy. As usual, we will neglect the electronic 67 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

contribution to the thermal enthalpy for all atoms and molecules. Thermal enthalpies for each chemical species can be calculated in a similar fashion as was done in Part II if one remembers the additional PV part. The goal here is to calculate thermodynamics for reactions, and in Part IV the specific goal is to calculate ΔrxnU°(0). From Equation 20, we can obtain a standard state enthalpy change for a reaction:

Finally, we can rearrange Equation 21 to obtain the desired term ΔrxnU°(0):

In Equation 22, ΔrxnH° is an experimentally determined quantity or, in the case of ArF2, a calculated estimated quantity. The other term on the right hand side can be recognized as the change in thermal enthalpy for a reaction, which can be obtained from individual thermal enthalpies of chemical species within the reaction. As always, calculate everything explicitly in your spreadsheet. There is no spreadsheet template for this part. It would make the most sense to have a separate sheet for each thermal enthalpy calculation (just like you did in Part II except modified for the PV term) and then another sheet for each reaction that references other sheets. Remember, your ultimate goal is four quantities: one ΔrxnH° and three ΔrxnU°(0). As before, a Word file is due along with the spreadsheet. Part V Due via Moodle dropbox by 12:30 pm on xx / xx In this culmination of these activities, you will calculate standard state free energies of reactions and experiment with temperature dependence of spontaneity. In doing so, we will successfully merge classical thermodynamics with statistical mechanics and information from spectroscopy to arrive at predictions relevant to real chemical reactions. Consider the following reactions:

The first two reactions are known to take place while the last one is highly speculative. You found ΔrxnH°298 K for the third reaction in Part IV; this time you will calculate ΔrxnG°298 K for each reaction. In addition, you will try to find the temperature at which each reaction changes from nonspontaneous to spontaneous (i.e., when ΔrxnG° changes sign). You do not need to find the exact temperature; close is good enough. This temperature is an approximation, in part because we will use assume the standard enthalpy of reaction does not depend on 68 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

temperature; i.e., we will use the 298 K values for standard enthalpy of reaction in all cases. Assume all chemical species are ideal gases at all temperatures. Use your calculations to address this question: What are the prospects for making ArF2 by the above reaction? Helpful information Xe-F bond length: 2.00×10-10 m Kr-F bond length: 1.871×10-10 m Ar-F estimated bond length: 1.8×10-10 m F-F bond length: 1.418×10-10 m Other information is given in previous parts.

In Equation 26, Δrxnn is simply the change in number of moles of gas within the reaction; note that Δrxnn can be positive, negative, or zero depending on the particular reaction. From modifying Equation 17 in Part IV, we have the following equation:

and substituting this into Equation 26 yields:

To proceed further, we use an expression modified from Part III:

which becomes the following equation when applied to reactions:

where the summations are over product chemical species and reactant chemical species. If we substitute Equation 30 into Equation 28, we have:

and then we get some nice cancellation: 69 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

Part V builds on all the other parts. Use sheets developed for those other parts as appropriate. Note: you may need to modify your equations for a few items such as reduced mass and moment of inertia. Think carefully about the chemical species you are working with in Part V compared with earlier parts. Pay careful attention to cell referencing—especially for T. Since a goal of this part is to find the T where the reaction switches from spontaneous to nonspontaneous, you may want to enter a single T value in your table below and then have the all your other sheets reference this cell as appropriate. The results of your calculations should be presented in a neat, orderly fashion. One possible format is:

As always, a Word file is due with the spreadsheet. This time, feel free to comment on Part V as well as the entire set of spreadsheet activities.

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Fox, L. J.; Roehrig, G. H. Nationwide Survey of the Undergraduate Physical Chemistry Course. J. Chem. Educ. 2015, 92, 1456–1465. Harris, H. A., Fitting Physical Chemistry into a Crowded Curriculum: A Rigorous One-Semester Physical Chemistry Course with Laboratory. In Advances in Teaching Physical Chemistry; Ellison, M. D., Schoolcraft, T. A., Eds.; ACS Symposium Series; American Chemical Society: Washington, DC, 2007; Vol. 973, pp 298−307. LoBue, J. M.; Koehler, B. P. Teaching Physical Chemistry: Let’s Teach Kinetics First. In Advances in Teaching Physical Chemistry; Ellison, M. D., Schoolcraft, T. A., Eds.; ACS Symposium Series; American Chemical Society: Washington, DC, 2007; Vol. 973, pp 280−297. Cartier, S. F. An Integrated, Statistical Molecular Approach to the Physical Chemistry Curriculum. J. Chem. Educ. 2009, 86, 1397–1402. 70 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

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Miller, S. R. Rethinking Undergraduate Physical Chemistry Curricula. J. Chem. Educ. 2016, 93, 1536–1542. Mack, M. R.; Towns, M. H. Faculty Beliefs about the Purposes for Teaching Undergraduate Physical Chemistry Courses. Chem. Educ. Res. Pract. 2016, 17, 80–99. Committee on Professional Training, Undergraduate Professional Education in Chemistry: ACS Guidelines and Evaluation Procedures for Bachelor’s Degree Programs [Physical Chemistry Supplement], American Chemical Society: Washington, DC, 2008. Holme, T. A.; Reed, J. J.; Raker, J. R.; Murphy, K. L. The ACS Exams Institute Undergraduate Chemistry Anchoring Concepts Content Map IV: Physical Chemistry. J. Chem. Educ. 2018, 95, 238–241. In this section, we chose not to develop or restate the many equations relating to these topics because they can be found in many standard undergraduate physical chemistry textbooks. We do note, however, that some textbooks do a better job of explaining the connections between all these topics. We also note that some of the relationships between statistical and classical thermodynamics are given in the student handout found in the Appendix to this chapter. Finally, some of the other references to this chapter include the development of some of these equations as well. Novak, I. The Microscopic Statement of the Second Law of Thermodynamics. J. Chem. Educ. 2003, 80, 1428–1431. Kozliak, E. I. Introduction of Entropy via the Boltzmann Distribution in Undergraduate Physical Chemistry: A Molecular Approach. J. Chem. Educ. 2004, 81, 1595–1598. Castle, K. J. High-Resolution Vibration-Rotation Spectroscopy of CO2: Understanding the Boltzmann Distribution. J. Chem. Educ. 2007, 84, 459–461. Fetterolf, M. L. Enhanced Intensity Distribution Analysis of the RotationalVibrational Spectrum of HCl. J. Chem. Educ. 2007, 84, 1062–1066. Cartier, S. F. The Statistical Interpretation of Classical Thermodynamic Heating and Expansion Processes. J. Chem. Educ. 2011, 88, 1531–1537. American Chemical Society Committee on Professional Training. https:/ /www.acs.org/content/acs/en/about/governance/committees/training.html (accessed January 14, 2018). Moog, R. S., Spencer, J. N., Eds. Process-Oriented Guided Inquiry Learning; ACS Symposium Series; American Chemical Society: Washington, DC, 2008; Vol. 994. Spencer, J. N.; Moog, R. S. The Process Oriented Guided Inquiry Learning Approach to Teaching Physical Chemistry. In Advances in Teaching Physical Chemistry, Ellison, M. D., Schoolcraft, T. A., Eds.; ACS Symposium Series; American Chemical Society: Washington, DC, 2007; Vol. 973, pp 268−279. POGIL. https://pogil.org/ (accessed January 15, 2018). Moog, R. S.; Spencer, J. N.; Farrell, J. J. Physical Chemistry: A Guided Inquiry: Atoms, Molecules, and Spectroscopy; Houghton Mifflin: New York, NY, 2004. 71 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.

20. Shepherd, T. D.; Grushow, A. Quantum Chemistry & Spectroscopy: A Guided Inquiry; The POGIL Project: Lancaster, PA, 2014. 21. Shepherd, T. D.; Grushow, A.; Garrett-Roe, S.; Moog, R. Thermodynamics & Statistical Mechanics: A Guided Inquiry; The POGIL Project: Lancaster, PA, unpublished activities currently in the beta testing phase. 22. Hunnicutt, S. S.; Grushow, A.; Whitnell, R. Guided-Inquiry Experiments for Physical Chemistry: The POGIL-PCL Model. J. Chem. Educ. 2015, 92, 262–268. 23. Stegall, S. L.; Grushow, A.; Whitnell, R.; Hunnicutt, S. S. Evaluating the Effectiveness of POGIL-PCL Workshops. Chem. Educ. Res. Pract. 2016, 17, 407–416. 24. Hunnicutt, S. S.; Grushow, A.; Whitnell, R. How is the Freezing Point of a Binary Mixture of Liquids Related to the Composition? A Guided Inquiry Experiment. J. Chem. Educ. 2017, 94, 1983–1988. 25. McQuarrie, D. A.; Simon, J. D. Physical Chemistry: A Molecular Approach; University Science Books: Sausalito, CA, 1997, p 701. 26. Engel, T.; Reid, P. Phyiscal Chemistry, 3rd ed.; Pearson: Boston, 2013, pp 17−18. 27. ACS Division of Chemical Education Examinations Institute. http:// uwm.edu/acs-exams/ (accessed January 15, 2018).

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