Chapter 4
Modeling Reaction Energies and Exploring Noble Gas Chemistry in the Physical Chemistry Laboratory
Using Computational Methods To Teach Chemical Principles Downloaded from pubs.acs.org by UNIV OF ROCHESTER on 05/15/19. For personal use only.
James A. Phillips* Department of Chemistry, University of Wisconsin – Eau Claire, 105 Garfield Ave., Eau Claire, Wisconsin 54701, United States *E-mail:
[email protected].
A two-part computational chemistry module for a physical chemistry lab course is described. The first part is concerned with the characterization of argon hydrofluoride (HArF), and the primary intent is to assess its energy relative to Ar and HF. The charge distribution is also considered, and the Ar-H and ArF distances are compared to those expected for bonding and non-bonding interactions. The second part involves a student-driven exploration of other noble gas compounds. The goal here is to cultivate literature proficiency and creative thinking skills within an accessible intellectual context. Ultimately, this endeavor illustrates how chemical computations enable a completely safe, open-ended scholarly project that provides original and substantive material for the introduction and discussion sections of a lab report. The extent to which the skills cultivated via this module manifest students’ ability to design and execute a subsequent calorimetry project is also considered.
Overview and Context This chapter describes the design and implementation of a two-part computational lab module for a physical chemistry lab course, and the motivation for doing so. As a whole, this exercise emphasizes underlying chemical concepts and how quantum chemistry can illuminate these, as opposed to the underlying theory of the theoretical models themselves. Part 1 is concerned with modeling the formation of argon hydrofluoride (HArF), the only known compound containing the element argon. Part 2 is a student-driven exploration of the design and characterization of other noble gas compounds.
© 2019 American Chemical Society
Beyond providing some practical experience with computations, this module targets several specific learning outcomes, specifically: Connecting macroscopic thermochemical quantities (e.g., ΔrH°) to specific, microscopic contributions to the molecular energy (Emolec). ii) Learning to navigate chemical literature databases, finding relevant research articles, and critically reading them to extract key information and insight. iii) Developing chemical creativity; asking meaningful research questions, and/or generating sound hypotheses. iv) Collecting relevant data, and using it to address hypotheses, and making meaningful, effective comparisons between chemical systems. i)
Since its initial inception in the early 2000’s, I have used this module in three different courses, though the goal has always been to address the same general objectives, despite a slightly different context in each case. In addition, though both computational methods and accessible computing speed have changed dramatically over the ensuing time period, the first part of module has changed but very little, and while the results obtained by the students are now much more consistent with those in the original research manuscript (1, 2), this has had no real effect on the educational impact. Originally, I taught this as the first lab in my physical chemistry I (thermodynamics and kinetics) course, with a goal of initiating and/or and reinforcing a conversation about microscopic molecular energies, in addition to providing students with the tools to theoretically model calorimetry results later in the term. Falling short of any rigorous treatment of statistical mechanics in my thermodynamics course, I still attempt to provide a microscopic rationale for various concepts (e.g., heat capacities of ideal gases, ΔvapS values, etc.), which mandates some understanding of how individual molecules store energy and interact. One advantage of approaching this issue in the lab setting is to provide a hands-on exercise that enables two-way interactions between student and instructor. Students’ ability to correctly answer questions related to outcome (i) stated above provides at least some indication that this module has been effective in achieving that stated learning goal. In addition, the fact that they can effectively apply these skills to calorimetry projects later in the term indicates that the practical experience gives them some confidence in using these theoretical tools, even if most students need some guidance in extending them to slightly more complicated reactions. I have also used this module as a first lab in the physical chemistry II (quantum) course, and for the last decade, after a consolidation of our physical chemistry and instrumental analysis labs, we have used it as one of the initial modules in our “physical-analysis laboratory” course, which runs in parallel with physical chemistry II lecture. Students in this course are in one of three cohorts: i) taking quantum chemistry in parallel to the integrated lab, ii) enrolled in our ACS certified biochemistry major which does not require physical chemistry II (as of yet), or iii) took the physical chemistry lecture sequence the previous year. The latter cohort is very much the minority of this group, and as such, the student population has little experience with quantum mechanics prior to this lab module. For the first cohort, this exercise seems to facilitate the transition between physical chemistry I and II (macroscopic to microscopic). For the latter, it is an effective overview of concepts that stream through both semesters of the yearlong sequence. It is worth noting that neither the inhomogeneity of the student population nor limited experience with quantum chemistry has ever significantly inhibited the implementation of this module.
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Methods Though there are numerous options for software platforms, and the choice is strictly a matter of taste, I have exclusively used various versions of Spartan in the classroom (details below). The advantages are ease of use, visual quality, and I have found that the menu-driven/dialog box structure of Spartan is rather effective in terms of highlighting the essential elements of a quantum-chemical computation for first-time users. The molecules are built (i.e., starting nuclear coordinates specified), by simply by clicking the mouse on a “builder” palette, a process much like using a chemistry desktop publishing application. In turn, the calculation is set up and executed using a series of drop-down menus in a single dialog box. There is no coding involved to run these jobs in Spartan (though some calculation parameters can be customized using optional keyword inputs). On the other hand, there are numerous software alternatives that are free or significantly less expensive, and many of colleagues with more experience in computational chemistry prefer these. The results presented herein come from two specific sources. The B3LYP (3) results obtained with the 6-31G* (aka 6-31G(d)) (4, 5) were obtained using Spartan student version 5.0.1 (6), with the default calculation parameters (i.e., no modifying keywords were used). These results, which replicate those initially generated by students, are emphasized in Part 1, and the atomic charges are Mulliken charges (see below) (3). Interestingly, the B3LYP/6-31G* results are incredibly (and fortuitously) accurate, can be obtained with the reasonably inexpensive student version of Spartan, and the jobs completed in less than one minute on an older desktop PC. (Some specific clocktime data: A B3LYP/6-31G* optimization of HArF took 41 seconds to complete, a subsequent B3LYP/6-311+G** (3–5) optimization of that geometry took 51 seconds.) The remainder of the results, obtained from methods including HF, B3LYP, MP2, and M06 (7), with the 6-31G(d), 6311+G(2df,2pd), aug-cc-pVTZ, and aug-cc-pV5Z basis sets (3–5) were obtained using Gaussian09 (G09) version B.0.1 (8), with coordinates and input files generated using GaussView version 5.0. Beyond exploring method performance (Table 2), which may be of interest to potential adopters of this module, a higher level of theory and greater control over the calculations were sought for the compounds described in Part 2, as to ensure reliability of these “published” results. If one is familiar with coding Gaussian files, running G09 is arguably a more convenient means of changing the parameters that dictate how the calculations are executed. Accordingly, the G09 results here were obtained using more stringent calculation parameters; an ultrafine integration grid (“int=ultrafine”) and tight geometry convergence criteria (“opt=tight”). Such settings are often desirable for quasistable systems, though omitting these settings in Spartan calculations has not led to any issues for students. Charges displayed with these latter results are from a Natural Population Analysis (NPA) (3, 9). Also note that because of the Gaussian calculation settings differ, the duplicate (B3LYP/631G(d)) results for HArF do exhibit some trivial differences from those obtained via Spartan (Table 1 vs. Table 2).
Part 1: Modeling the Formation of Argon Hydrofluoride (HArF) The first part of this module is taught in an instructional computer lab, in a mode best described as a “workshop”. In this setting, I deliver a few 10 to 15-minute mini-lectures, often interspersed, on various topics including: i) A description of the original experiment in which HArF was produced, identified, and characterized (if not during the preceding lab meeting), ii) an overview of the software program, and iii) a short overview of quantum chemical models and their execution. For the prelab assignment students are instructed to: i) Acquire the primary HArF reference article (1), ii) read it, iii) 35
read some background material on molecular energies (e.g., Atkins Chapter 0) (10), and iv) provide a Lewis structure and VSEPR geometry for HArF. HArF was originally produced in a low-temperature (20 K) argon matrix that had been seeded with HF and irradiated with deep UV photons (130-150 nm) (1, 2). However, we ignore any effects due to solvation by argon on the molecular energies, and model the reaction energy (path independent, solvation notwithstanding) as if it were a simple, direct gas-phase reaction, viz.
The theoretical framework used to compile the various contributions to the molecular energies, and subsequently, the reaction energy at (ΔrE), is described immediately henceforth. This is critical information for students, and is included in the lab manual text and my pre-lab lecture for this project, so it is described in detail here. Note also that the standard state symbol (°) is deliberately excluded; these values are formally not reflective of a standard condition. In compiling these energies, the rotational and translational energies are from some user-specified temperature, but the vibrational energy is from absolute zero (see below). In the reaction above, ΔrE is simply the difference in molecular energy (Emolec) between the products reactants, i.e.
For HArF and HF, the molecular energy (Emolec) is the sum of electronic, vibrational, rotational, and translational terms, viz.
For argon, which lacks vibrational and rotational degrees of freedom, equation 3 reduces to a sum of electronic and translational terms, viz.
The electronic energy (Eelec) is precisely the quantity that the quantum-chemical computations provide, and it dominates all other terms in equations 3 and 4. Similarly, the vibrational energy (Evib) is also obtained from the computations. Though not a primary output, frequencies can be determined from the electronic energies; the force constants are essentially the second derivatives of the electronic energy with respect to each normal coordinate. In this treatment, the total Evib is taken to be the total zero-point energy, equal to half the sum of the vibrational frequencies (ν), i.e.,
Argon, being monatomic, lacks any vibrational degrees of freedom, while HArF and HF, being linear molecules, have 3N-5 vibrational modes (4 and 1, respectively). Non-linear molecules have 3N-6 vibrational modes, and this can be relevant for the second part of this module. For the rotational (Erot) translational (Etrans) energies, we make a classical approximation, and use the equipartition theorem, and thus presume that the average energy of each degree of freedom for a given atom or molecule is equal to ½ RT (in molar energy units). Each reactant and product in 36
equation 1 has three translational degrees of freedom, making the translational energies equal at any given temperature (about 3.7 kJ/mol at 298 K), viz.
For rotational energies, it is essential to properly account for the number of rotational modes. Argon, again being monatomic, lacks any rotational degrees of freedom. For HArF and HF, being linear molecules, each has two rotational degrees of freedom, and thus their rotational energies are equal at any given temperature (approximately 2.5 kJ/mol at 298 K), viz.
Polyatomic molecules have three rotational degrees of freedom, and this underlies the difference in vibrational modes for linear and non-linear molecules, and again, this may warrant consideration in the second part of the module. For example, a bent triatomic has three rotational degrees of freedom and a single bending vibrational mode, whereas a linear triatomic has only two rotational degrees of freedom, and there are two degenerate bending vibrations. The conversion of ΔrE reaction to ΔrH reaction is given by,
in which Δn is the difference in moles of gas between products reactants (Δn = –1, for reaction 1). Alternatively, though I have never done so in my course, one could obtain the standard molar enthalpy values (H°m) of the reactants and products at 298 K, as determined by the values of the molecular partition functions. This approach more accurately accounts for the translational, rotational, and especially the vibrational energy (via contributions from thermally-accessible excited states), and could be critical when dealing with low-frequency vibrations. Moreover, the molar enthalpies are provided directly by most software packages when a frequency calculation is requested, including in Gaussian and Spartan. But in this process, the students do not deal directly with the physical origin of these quantities (unless they were to work through the partition functions as part of the exercise), and there may be issues with some software platforms as far as requesting a frequency calculation on Ar. Gaussian does provide thermodynamic data for Ar in this manner, however. Regardless, with the molar enthalpy data in hand, ΔrH° is simply the difference between the standard molar enthalpy values of the products and reactants, viz.
It is worth noting that the difference between the Hm values for HArF obtained via the method presented above and that obtained via the partition functions (and provided by Spartan) is only about 0.001 kJ/mol. Table 1 displays a set of sample thermochemical results for reaction 1 from B3LYP/6-31G* calculations (3–5). As noted above, these results take oat most a few minutes of clock time (in total) to execute, and as such, I most often use this model chemistry in the classroom for delivering Part 1. The table is constructed in a way that illustrates the various specific energetic contributions to ΔrE and note that Eelec dominates all other contributions. A sample student spreadsheet, essentially identical to that provided in our lab manual, is included as Figure 1. There are three sections: The top is simply a place to paste in raw computational results, the middle section is for planning the formula entries and unit conversions for use in the final reaction energy table, which comprises the 37
lower section (and mimics Table 1 to some extent). The final table computes ΔrE for each specific type of energy as well as the total, and the rotational and translational energies are designed to update for a different reaction temperature. The electronic and vibrational energies are not temperature dependent (within our treatment). As such, this module also provides an opportunity to teach intermediate spreadsheet skills, which are essential throughout our physical analysis lab course.
Figure 1. A sample spreadsheet for organizing student thermochemical calculations.
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Table 1. Energy Data for Ar, HF, HArF, and Reaction 1a B3LYP/6-31G*
Eelec
Evib
Erotb
Etransb
Ar
-1384995.67
–
–
3.72
-263653.04
23.84
2.48
3.72
(-100.42017)
(1993.3)
-1648071.93
20.90
2.48
3.72
(-627.717625)
(1746.9)
576.8
-2.9
0.0
-3.7
ΔrE d =
570.1
ΔrH e =
567.6
(-527.5171419) HF HArF ΔrEtypec
a kJ/mol
unless otherwise noted; additional Eelec and Evib values noted in parentheses. Eelec values in parentheses are in units of Hartrees or au. Evib values in parentheses are cm-1. The individual HArF frequencies are 463, 709 (2 modes), and 1639 cm-1. b Obtained assuming average energy per mode is ½ RT, with T = 298K. c Reaction energy difference for each type of energy, respectively by column. d Total reaction energy as obtained via the sum of the individual contributions. e Reaction enthalpy as determined from ΔEtot and equation 8.
Table 2 presents a comparison of various model chemistries (methods and basis sets) and the intent here is strictly to aid instructors in model selection; this is not intended as a student exercise. Balancing resources and accuracy is always a key consideration, and in this instance, the B3LYP/ 6-31G(d) results are among the most accurate relative to the high-level CCSD(T)/aug-cc-pV5Z results from the original manuscript (1). However, this is due to a fortuitous cancellation of errors; the discrepancy for B3LYP increases with larger basis sets which suggests a cancellation of model and basis set errors with smaller basis sets. Overall, each method highlighted in Table 2 (except HF) is reasonably accurate, and overall, I have only allocated minimal attention to the issue of model performance in teaching this module. Often, I will have students explore basis sets and the corresponding impact on clock-time. This does provide a first-hand illustration of the variation principle, and also a practical lesson in balancing time and (presumed) accuracy. In the follow-up assignment for Part 1, students are tasked with re-performing these calculations at a higher level of theory, usually M06/6-311+G** (3–5, 7), which is available via the full version of Spartan. I also note that I originally ran this module with the HF/3-21G level of theory (3), and though these results were quite inaccurate, there was little or no pedagogical impact. Note that the HF results in Table 2, even with larger basis sets, are by far the least accurate. One thing to note is the effectiveness of the density functional (DFT) methods (B3LYP (3) and M06 (7)), which are comparable to HF in terms of computational time (roughly twice the clock time), but include the effects of electron correlation, which greatly improves the energy predictions. The MP2 method involves an explicit correction to HF to account for electron correlation, and was used widely prior to the development of accurate DFT methods, but is much more demanding in terms of computational time and resources (6). It is also less accurate with the smallest (6-31G(d)) basis set in Table 2.
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Table 2. Model Chemistry Comparison for ΔrE (kJ/mol) method
basis set
ΔrEelecb
ΔrE
ΔΔrEelecc
% diff.
HF
6-31G(d)
732.3
731.0
166.3
29.4
6-311+G(2df,2pd)
655.9
653.6
89.9
15.9
6-31G(d)
577.7
571.3
11.7
2.1
6-311+G(2df,2pd)
532.5
527.7
-33.5
5.9
aug-cc-pVTZ
529.5
525.0
-36.5
6.4
6-31G(d)
648.4
640.1
82.4
14.6
6-311+G(2df,2pd)
578.1
574.8
12.1
2.1
aug-cc-pVTZ
560.6
557.7
-5.4
0.9
6-31G(d)
603.2
596.6
37.2
6.6
6-311+G(2df,2pd)
547.4
574.8
-18.6
3.3
aug-cc-pVTZ
546.4
542.3
-19.6
3.5
aug-cc-pV5Z
544.3
540.3
-21.7
3.8
aug-cc-pV5Z
566.0
–
–
–
B3LYP
MP2
M06
CCSD(T)a a Ref.
(1).
bΔ
rE value in terms of only electronic energies; difference in Eelec between product (HArF) and
reactancs (Ar, HF) in equation 1. (1).
c Difference
between the ΔrEelec value and the CCSD(T) value from Ref.
Another element of the follow-up assignment is to answer some discussion questions related to the original manuscript, focusing on the energetic issues and comparing our results wherever possible to those reported in that paper (1). These questions are also included in Table 3, for use with the module. The remaining part of the follow-up assignment is to make a high-quality graphical figure with a descriptive caption, which includes bond lengths, calculated atomic charges, and ΔrE. Proper precision for these data are, 0.1 Å for bond distances, 0.1° for angles, and 0.1 kJ/mol (or kcal/mol). A depiction of the B3LYP/6-31G* structure parameters for HArF, based on an image generated by Spartan, is displayed in Figure 2, with a Lewis structure, per the prelab assignment. One way that this module could be extended is in the extent to which the bonding in HArF is characterized. Alas, I have pursued only a few rather simplistic considerations, but I do take time to highlight the visualization capabilities that Spartan offers, such as maps of the “electron density” and “electrostatic potential” surfaces. Such images are now commonplace in textbooks, but that latter does shed light on one issue discussed in the original manuscript; the extent of ionic bond character in HArF, or stated somewhat differently, the weight of an HAr+/F– resonance structure (HAr+ is isoelectronic with HCl). This consideration leads to a brief discussion of computed atomic charges, and the difficulty of even defining them and thus somehow assigning electrons to a given atom, in light of the complexity of the computer-generated electron distribution. In spite of these potentially complicating issues, I do have students examine computed charges and include values in their figures; non-zero values indicate that the noble gas is taking part in actual bonding interactions. Mulliken charges are displayed in Figure 2; these tend to be reasonable when small basis sets are used (3). Because the results in the next section involve the much larger aug-cc-pVTZ basis set, values therein result from Natural Population Analyses (9). NPA charges are more stable with respect to basis set 40
size, but tend to be larger in magnitude overall than those computed from other models (3). Indeed, they are larger for HArF in Figure 2 (below). Though I tend not to engage in a detailed discussion of charge models with my students, the text by Cramer (3) does offer a thorough and accessible treatment of this topic. Another noteworthy point is that in practice, it is advisable not to indulge in a precise, quantitative interpretation of calculated atomic charges, but rather, trends in charges obtained from identical model chemistries for a series of analogous compounds are physically meaningful (3).
Figure 2. A Lewis structure and B3LYP/6-31G* geometry parameters for HArF, with Mulliken atomic charges. Rcov and Rvdw are distances predicted by summing covalent and van der Waals radii, respectively. One final characterization issue is how we establish that the adjacent atoms are truly bonded to the noble gas, beyond noting those shifts in atomic charges discussed immediately above. To this end, I typically have students consider predictions of bonding or non-bonding distances from sums of covalent (11) and van der Waals radii (12), respectively. In the case of HArF, the Ar-F distance (1.974 Å via B3LYP/6-31G*) is much closer to the sum of the covalent radii (1.68 Å) than to the sum of the van der Waals radii (3.35 Å). The case is more compelling for the Ar-H distance (1.426 Å), for which the sum of the covalent and van der Waals radii are 1.34 Å and 2.98 Å respectively. This is surely an old-fashioned approach perhaps, but nonetheless, I believe senior students should understand the application of bond radii, and internalize some knowledge of typical bond distances. In addition, I also note that these calculations yielded all real frequencies (i.e., no imaginary values); this indicates a true equilibrium structure for HArF. Regardless, a possible extension of this module would be to further characterize the bonding in HArF by a more sophisticated analysis such as Atoms in Molecules (AIM) (13) or Natural Bond Orbitals (NBO) (14). To summarize Part 1 of this module, students will: i)
For prelab: Obtain and read the main reference article and background material, plus draw the Lewis structure and VSEPR geometry for HArF. ii) In lab: Perform computations, build a thermochemical analysis spreadsheet, and undertake some consideration of the bond distances and atomic charges. iii) Follow-up assignment: Re-do the computational analysis using a higher level theory, answer discussion questions, and generate a high-quality figure of HArF with a figure caption that provides a consideration of the bond distances and charge distribution. 41
Table 3. Discussion Questions for Part 1 1) Briefly explain/rationalize the origin of the specific contributions to Erot and Etrans for each species in the net reaction (HF, Ar, and HArF). (Equations are essential.) 2) Our electronic structure calculations are much less sophisticated than those discussed in the HArF paper. Compare your calculated ΔrE to theirs (both B3LYP & M06, converting between units as necessary). What is the % difference in the reaction energy (relative to their highest quality result)? Which of our methods is more accurate (relative to theirs)? 3) What “type” of energy dominates the reaction energy (i.e., what is the biggest term in Equations (3) and (4))? Does the changing the temperature change ΔrE much? Why or why not? 4) Mechanistic Considerations: Formation of HArF. a) What is the source of energy that “drives” the reaction in the experiment (as conducted and described in the paper)? b) What range of energies (in molar energy units) does this source provide? c) Compare it to ΔrE. It is enough to complete the reaction? d) HF bond rupture (upon photo-excitation) is initial step in the formation of HArF. i) How much energy is required to break an HF bond? (Look up the Bond E) ii) How does that compare to the photon energy? (i.e., from your answers to “a” and “b” iii) What happens to the excess energy, i.e., that which is left over after breaking H-F with hν?
Part 2: Exploring New Noble Gas Compounds In Part 2 of this module, students characterize a noble gas compound of their own design, subject to a few guiding constraints. The goal here is to encourage students to ask appropriate and effective “chemical questions”, and thereby facilitate the growth of their creative thinking skills. One challenge associated with these intentions is that quite often, the context for a solid research question takes years of experience to fully comprehend. But in this case, the issue is quite accessible to nearly all chemistry students because it simply challenges well-established dogma: The Group 18 elements are notoriously unreactive, so is it possible to form bonds to them? In fact, there are have been numerous studies of noble gas compounds similar to HArF, with theory often preceding experiment (15–18), and these efforts continue at present (19, 20). A key issue from an educational standpoint is how the computational approach facilitates an open-ended inquiry, for one because it is simply easier to build molecules on a computer screen than to actually synthesize them, and in addition, the safety advantage is noteworthy. One simply could not safely supervise 24 students working with an array of such notoriously unstable compounds. With computational chemistry, however, there are no safety hazards whatsoever, and students can explore a wide range of bizarre and unstable compounds with essentially zero risk. Part 2 starts with a literature workshop that we hold in the library computer lab, and it is taught primarily by the science librarian. The broader goals are to facilitate the use and understanding of various literature databases and also help students grasp the broader scope of the chemical literature (e.g., What are the various types of research articles? Who publishes research journals? What is the scope or audience for a given journal?). More to the point of this project, the specific intent is to cultivate the skills needed to find all the pertinent information available on a noble gas compound of their choosing. Ultimately, the collection of relevant articles they find guides their design, and establishes an intellectual context for their report introduction. The students’ noble gas compounds are subject to a few design constraints. The point is not only to direct their thinking, but also to produce a set of results that are more-or-less comparable 42
across the entire class to illuminate key trends, but also because we award a trophy to the student that designs the highest-energy stable compound. These constraints level the playing field for this contest by establishing a common energetic benchmark. They are as follows: i) The compound must be neutral, ii) The formula can only contain one noble gas atom, and iii) One must be able to form the compound via a reference reaction (akin to equation 1 above) that involves only one equivalent of some stable compound. Also, it is essential that the compound exhibit bona fide chemical bonds to noble gas atoms, though this condition is not always easy to establish completely. As above, we have made this assessment by comparing distances to (sums of) covalent van der Waal’s radii, requiring that the frequencies are real, and noting non-zero values for calculated charges on noble gas atoms. Quite often, the interatomic distances involving the noble gas atom in a “stable” compound are somewhat longer that those predicted from covalent radii. Conversely, they are usually a great deal shorter than those predicted by van der Waals radii, and the truly unstable compounds tend to fragment completely during the geometry optimization (i.e., the noble gas atom gets pushed out to a location several angstroms from the other atoms). So, even on a purely structural basis, the distinction is reasonably clear when the results are viewed through the proper filter – if the distances are significantly shorter than the predicted non-bonded distances it is considered a valid compound for the project. In addition, vibrational frequencies should be real for true minimum-energy structures, but I have on occasion relaxed this requirement. Again, one possible extension of this lab would be to undertake a more a sophisticated analysis of the bonding to make this case. Despite of some degree of variability in the approach that students take to this challenge, the comparisons which most easily facilitate a discussion of noble gas stability tends to follow a simple bond-insertion mechanism, which is essentially a generalization of reaction 1 above, i.e.,
Often students follow this path, and many make a simple substitution relative to HArF (e.g., Kr for Ar, Cl for F, etc.), not only because it is a relatively simple extension, but also it sets up a direct, single-variable comparison to HArF. Other students that pursue more creative options realize at some point that to make a valid comparison, they need to consider a reference compound that differs from theirs by only one parameter (e.g., the noble gas atom or one of the X/Y groups). As such, I often I encourage them to coordinate with a fellow classmate, or just examine a second compound (or a series) so that they can do so, but only after I challenge them as to how they will construct a comparison based on the compound they chose. Students that deviate from the mechanism entirely (e.g., inserting a noble gas into a crown ether, or bonding it with a Lewis acid) sometimes have difficulty making effective comparisons between systems without a great deal of extra effort. Some degree of patience on the part of the instructor is critical, as is a willingness to let students pursue their ideas and encounter roadblocks and unexpected results. The first assignment for Part 2, following the literature session and some time to do their analysis, is a mock group meeting in which each student gives a brief, two- or three-slide presentation on their compound. They are specifically instructed to: i) introduce their compound and the hypothesis or question that led to it, ii) display its structural properties, charge distribution, and its energy value relative to the noble gas and the stable X-Y compound, and iii) make the case using bond radii that there are bona fide bonds to the noble gas atom. Almost always, students assemble a set of results from which stability trends can be deduced during the group discussion that ensues, at least among those 43
that fit the mechanism implied by reaction 10. The assessment of “stability” is based upon ΔrE for equation 10, as it applies to the various compounds. A few apparent trends in these compounds are: 1. Stability Increases with the Size of a Noble Gas Atom. Figure 3 displays structural properties and ΔrEelec values for H-Ar-F, as well as H-Ne-F, and HKr-F. These were obtained via M06/aug-cc-pVTZ, and B3LYP/6-31G(d) energies are also included for comparison. It is worth noting that there are no significant structural differences between the M06/aug-cc-pVTZ and B3LYP/6-31G(d) structures for any of the compounds discussed herein, and all exhibit only real vibrational frequencies. In any event, the ΔrEelec values increase steadily in the manner: HKrF < HArF < HNeF. A simple rationalization of this trend relates to size; because the noble gas must expand its octet to bond, a larger size reduces repulsion between electron pairs about the noble gas. In addition, neon, being a second-row element, is presumably tyrannized by the octet rule with only s and p orbitals in its valence shell, but the HNeF compound is quasi-stable, with distances about the Ne that are still much closer to bonded limit than the non-bonded limit. No compound containing the element neon has ever been observed, however. Another factor that may underlie this trend is the extent of shielding of the valance electrons (i.e., ionization energies decrease in proceeding down a group); charges indicate that the noble gas gives up electron density in every interaction among the compounds discussed herein. Thus, more bonding should ensue when these electrons are held less tightly. A trend in the charge distribution is also apparent, and it reinforces this rationale; the charge on the noble gas increases (Ne < Ar < Kr) with size.
Figure 3. Structure parameters, atomic charges (NPA), and ΔrEelec values, via M06/aug-cc-pVTZ, for HNeF, HArF, and HKrF. Rcov and Rvdw are distances predicted by summing covalent and van der Waals radii, respectively. 44
2. Stability Increases with the Polarity of the X–Y Bond. Two illustrations of this trend are apparent from the results shown in Figure 4, together with the HArF results (Figure 3). In the hydrogen-halide series (H-Ar-X, X=F, Cl, Br), the ΔrEelec values steadily increase with the size of the halogen. Apparently, this reflects the electronegativity of the halogen; a greater tendency for X atom to withdraw electron density from the noble gas seems to stabilize the compounds. An analogous trend is apparent for the 2nd row series (H2N-Ar-H vs. HOAr-H vs. H-Ar-F).
Figure 4. Structure parameters, atomic charges (NPA), and ΔrEelec values, via M06/aug-cc-pVTZ, for HArCl, HArBr, HArOH and HArNH2. Rcov and Rvdw are distances predicted by summing covalent and van der Waals radii, respectively. 45
3. Stability Decreases with X−Y Bond Strength and/or Over Stability of XY. Within the constraint of the bond-insertion mechanism conveyed by equation 10, it would follow that ΔrEelec will increase when the noble gas is inserted into a stronger bond, or otherwise reacts with a more stable (lower energy) compound overall. A good illustration is the fact that ΔrEelec is much higher for HArOH than for HArCl; though the O-H bond is more polar, it is also stronger, and the resulting compound has a higher energy relative to its respective fragments. The highest energy compound ever designed by students is displayed in Figure 5; the result of inserting Ar into the triple bond in N2. A Lewis structure is also included; we found it challenging to draw this during the group meeting in which this compound first arose. The energy is high, and it epitomizes the trend noted immediately above, it a relatively small noble gas (the analogous neon compound is truly stable), and the XY unit has a strong, non-polar bond. The Ar-N distances are actually short relative to the covalent prediction, but the best Lewis structure for NArN has double bonds (it is isoelectronic with SO2). The assignment for Part 2 of the module consists of several parts. First, students must construct a spreadsheet that shows the calculation of the ΔrE, as well as a high-quality graphic with a caption, just as in Part 1 for HArF. In addition, there is a partial lab report. In general, we try to scaffold the development of manuscript writing skills by breaking the process down into parts, so we assign specific sections of partial reports early in the course. In this project, we focus on the introduction, which builds on the literature session they did at the outset of Part 2. I typically suggest a rough outline as to guide them through this process. In addition, they also write a paragraph of discussion in which they compare and contrast their new noble gas compound with at least one analogous compound, and discuss apparent stability trends and their underlying rationale. The broader intent is to guide them in the process of making valid, insightful comparisons, and writing an effective discussion with a genuine chemical content. A computational methods paragraph would be a worthy addition, and will most likely be included in the next revision.
Figure 5. Lewis structure, geometry parameters, atomic charges (NPA), and ΔrE via M06/aug-cc-pVTZ for NArN. The Ar-N distances are actually shorter than the sum of covalent radii (Rcov = 1.72 Å), as is displayed in Figure 4 for H-Ar-NH2. 46
The bigger question perhaps, is whether or not engaging in an open-ended activity such as this actually manifests some enhancement of students’ critical and creative thinking skills. I have never collected any formal assessment data that addresses this issue, and it seems that designing the outcomes and rubrics to try to measure such progress would be more challenging than probing mere “content”. Nonetheless, I’d argue that offering such an opportunity to utilize and develop these skills must have a greater impact than confining the course to “canned” experiments in which all the questions are asked for the students, and the goal is strictly to measure some “value” while discussing only errors and/or experimental shortcomings. In addition, I can say informally that students do engage this process; some are enthusiastic about this freedom from the outset, some are apprehensive. Often, students in the latter group become quickly aware of their shortcomings and embrace the need to develop these skills. The other observations, in regard to the subsequent designbased activity (see below), are: i) Apprehensive students tends to approach the follow-up task with greater confidence, and ii) Overall, nearly all projects are rooted in meaningful, valid questions (after varying degrees of feedback from the instructor). To summarize Part 2 of this module, students will: i) Design a “new” noble gas compound of their choosing, subject to a few design constraints. ii) Search the literature to obtain previous reports on their compound and those closely related to it, as well as possibly refining their hypothesis or target compound accordingly. iii) Characterize the compound via computations, and present their preliminary results in a mock group meeting. iv) Prepare a partial lab report, including: A publication-style graphical figure with a descriptive caption, an introduction that conveys the hypothesis or question that motivated the design of their compound as well as the broader intellectual context, and a discussion paragraph that compares and contrasts their compound with at least one analogous system.
Looking Forward: Modeling and Extending Calorimetry Results Later in the term, physical analysis lab students embark on a major calorimetry project which builds directly on the outcomes targeted in the noble gas module. There are two possible tracks for this project, they either measure ΔrH of a reaction by solution calorimetry or ΔrE (ΔrU in most texts) of some reaction via bomb calorimetry. In both cases, they use computations to extend these results to other reactions or compounds and establish a valid comparison or a consideration of some structure-reactivity relationship. In the case of the bomb calorimetry experiments, most students investigate the difference in combustion energy (or enthalpy) of a pair of structural isomers (e.g., n-butanol and diethylether):
There are two minor complications that must be addressed. One is that O2 has a triplet ground state, which must be specified. In turn, this requires an “unrestricted” calculation (6), and this specification should be applied to all reactants and products. Two, by default, the models reflect gasphase reactants and products. Thus to compare between the measured reaction, which involves H2O (l) and a liquid or solid organic compound, one must appropriately construct a thermochemical cycle to correct the modeled ΔrE or ΔrH value with the ΔvapH value of water and some reference value 47
(or estimate) of the sublimation enthalpy of the organic. Nonetheless, students typically can predict the measured reaction energy to within 10 or 20%, and thus having “validated” their model, they can explore the energy difference between with compound and any structural isomers. Because the products of reaction (11) are identical for any set of structural isomers, the difference in combustion enthalpy (or heat of formation) is nearly equal to the difference in Eelec between the various isomers. Ultimately, students can address this difference directly if they so choose, but either way, they can rationalize the effect of various structural features (e.g., types of bonds, types of unsaturation) on the overall energetic stability of the isomeric compounds. At this point, they are truly exploring chemical ideas with computational chemistry. For solution calorimetry, I usually direct students to a study of strong acid (H3O+)-weak base reactions, for which they will compare the reaction enthalpies for pair or weak bases (e.g., pyridine and 3-chloropyridine). The two complications here are: i) the reactions take place in solution, so they must use the E(aq) values from Spartan for Eelec (which are actually free energies but the entropic contribution is relatively insignificant) (6), and ii) the observed reaction enthalpy (ΔrHmeas) reflects two processes, the heat of solution (or dilution) of the base (ΔsolH), plus the enthalpy change for the acid-base reaction (ΔrHAB), which is the modeled quantity. Thus, to compare their measured and modelled reaction enthalpies and thus validate their computational predictions, they must run a second set of experiments to determine solution/dilution enthalpy of the base, and combine the data as follows.
Once they have established confidence in their theoretical (ΔrHAB) values through favorable comparison to their experimental results (again, agreement to within 10-20% is deemed acceptable), they can compare to another weak base (or two) using the computations. Moreover, the comparison is direct and reflects only the acid-base reaction, free of the potentially obscuring effects of the solution enthalpy. The common theme here is to first “validate” a computational model against an experimental observation (often with a liberal definition of success), and then, with a reliable method in hand, students can explore related reactions and thus make comparisons between compounds without doubling or tripling their experimental work. Moreover, they can also investigate compounds that are otherwise inaccessible due to instability or cost. As such, linking this calorimetry project to the noble gas module not only allows them to utilize their practical experience with computational methods, but also utilize and further develop their critical and creative thinking skills. They are provided with the opportunity to formulate a hypothesis or question and test it, and develop a discussion for their reports that has a truly chemical focus. As noted above, students seem to engage this project with a greater degree of confidence after having sought their own direction in the noble gas compound project. 48
Summary The design and implementation of a computational module for a physical chemistry lab course has been described. It occurs in two parts, the first of which involves an analysis of HArF, and the main intent is assessing the energy relative to Ar and HF. Subsequently, students design their own noble gas compound, and intimately, compare and contrast this system with an appropriate analog. Beyond the practical experience using chemical computations, the module provides the opportunity to build literature proficiency, as well as critical and creative thinking skills. In addition, students employ these skills, both the practical application of computational chemistry techniques, as well as creative and critical thinking, in a subsequent calorimetry project.
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