Trends in Bond Dissociation Energies for the Homolytic Cleavage of

Communication Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX

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Trends in Bond Dissociation Energies for the Homolytic Cleavage of Successive Molecular Bonds Julie Donnelly† and Florencio E. Hernań dez*,†,‡ †

Department of Chemistry and ‡CREOL/The College of Optics and Photonics, University of Central Florida, P.O. Box 162366, Orlando, Florida 32816-2366, United States

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S Supporting Information *

ABSTRACT: The students in our physical chemistry course pointed out an interesting trend in the bond dissociation energies (BDEs) of successive homolytic cleavages of hydrogens from methane and water. Namely, while there is an increase in BDE from CH4 to CH3, there is a decrease in BDE from H2O to OH. In order to explain this trend, we employed a theoretical approach that could be used in an undergraduate setting. We compare our theoretical results with experimental and highly accurate theoretical values readily available in the literature. Having validated our theoretical approach, we use equilibrium molecular structure as evidence of the change in dominating attraction and repulsion effects of the central atom nucleus and nonbonding electrons and explain the anomaly in BDEs in undergraduate chemistry terms. KEYWORDS: First-Year Undergraduate/General, Upper-Division Undergraduate, Physical Chemistry, Misconceptions/Discrepant Events, Thermodynamics

C

AB → A + B

onsidering that the making and breaking of chemical bonds is fundamental to the discipline, an important concept covered in all general and physical chemistry texts is that of bond energy. However, this concept is associated with several critical misconceptions1−5 likely due to the confusing and misused terminology used in the instruction of the topic.4 For example, the phrase “energy stored in chemical bonds” can be confusing to students who may infer that this energy will be released as a result of bond breaking.2 On the other hand, use of the term bond “strength” is purely qualitative but is sometimes used as a synonym for bond energy.4 The challenge increases when bond dissociation energy (BDE) is introduced. BDEs are particularly important for physical chemistry students to understand when they are introduced to radical chain reaction mechanisms in kinetics. Benson gives a clear example of how seemingly insignificant differences in bond energies lead to huge differences in rates of reactions,6 illustrating the importance for students to understand this topic. The definition of BDE has been sufficiently addressed in the literature. While some discrepancies exist (e.g., whether enthalpy is the correct measure of bond energy),7 there is a consensus that BDE differs from bond energy. Knox defined bond energy as the “contribution of the bond between a particular pair of atoms...in a molecule to the total binding energy present in the molecule”.4 On the other hand, BDE corresponds to the energy change for the homolytic cleavage of a specified bond in the molecule: © XXXX American Chemical Society and Division of Chemical Education, Inc.

(1)

It is important to note that for the purposes of this explanation we use Ruscic’s recent definition of BDET as “the change in standard enthalpy at temperature T that occurs upon cleavage of a particular chemical bond, assuming ideal gas behavior of the dissociating chemical species and its dissociation products”.8 All of the BDEs discussed herein, theoretical and experimental, are reported at 298 K and are denoted as BDE. Trends in BDE

An interesting trend was pointed out to us by our physical chemistry students concerning the BDEs of methane and water. Our students were not surprised that the BDE for each C−H bond was not equivalent to the average bond energy. They reasoned that when the first hydrogen dissociates, the remaining electrons are pulled closer to the nucleus as the carbon attempts to maintain electron density. By using this explanation, they are referring to the increase in attraction of the carbon nucleus, which results from the increase in s character of the central atom with each successive hydrogen cleavage. Thus, they correctly predicted that the BDE would increase for the removal of the second hydrogen. However, Received: December 14, 2017 Revised: June 25, 2018

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DOI: 10.1021/acs.jchemed.7b00962 J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

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Table 1. Experimental Bond Dissociation Energies BDE (kJ/mol) species

BDE (kJ/mol)5,9

species

H−R bond5

C−R bond10

species

BDE (kJ/mol)5

H−OH H−NH2 H−CH3 H−F H−Cl H−Br H−I

499 453 439 570 432 366 298

CH3−OH CH3−NH2 CH3−CH3

435 427 423

385 356 377

PhO−H PhNH−H PhCH2−H

361 374 372

Finally, an increase in BDE is observed with increased s character of carbons in C−H bonds. This trend has been identified in the ethane, ethene, ethyne series9 and amidines11 and has been discussed in terms of overlap between MOs of heavy atoms, but has not been considered in smaller molecules (e.g., methane and water). Many trends of BDE have been discussed in the literature thus far, but the successive dissociation of bonds from a single molecule has not. Recently, Ruscic reported the BDEs of sequential bond dissociations of methane, ethane, and methanol obtained using Active Thermochemical Tables (ATcT) and values obtained from experimental and highly accurate theoretical methods.8 However, the trends therein are not interpreted, leaving our students’ question unanswered.

they did not observe the same trend for water and could not explain why. Trends in BDE began to be discussed in the literature in the early 2000s as experimental BDEs started to become available.5,9 There is a large body of literature on these trends, but for the purposes of this paper, we will review only those involving species similar to the ones we are interested in (e.g., methane and water). The relevant BDEs are reported in Table 1. Several important factors have been identified as important for explaining trends in the BDE. These include, but are not limited to, the electronegativity of the heavy atom, hyperconjugation and direct conjugation, the geometry of the resulting species, and the hybridization of the heavy atom. The decreasing trend in BDE in the water, ammonia, methane series as well as the H−F, H−Cl, H−Br, H−I series has been attributed to the decrease in Pauling electronegativity of the heavy atom.5,10 This decrease is associated with a lengthening of the heavy atom hydrogen bond.5 When larger substituents are substituted for H (e.g., CH3−OH, CH3−NH2, and CH3−CH3), decreases in the BDE of the H−R bond are observed for molecules in the series as well as decreases in the difference among the BDEs of all species.5,9 Ingold and Wright attribute this trend to hyperconjugation that occurs between the p orbital containing the resulting radical electron and the σ orbitals in the methyl group. The delocalization of the radical electron decreases the BDE.5 Further increases in delocalization by a phenyl substituent not only decrease the BDE of the entire series but also reverse the order of the trend. The O−H bond on a phenyl substituent has a lower BDE than does the N−H or C−H bond.5 In 2009, van Zeist and Bickelhaupt reiterated the previously observed decreasing BDE and increasing bond length (BL) for the water, ammonia, and methane series but pointed out the small difference between the BDEs of H−NH2 and H−CH3 (about 14 kJ/mol) in comparison to that between H−OH and H−NH2 (46 kJ/mol).10 Noting the same anomaly in the CH3−OH, CH3−NH2, and CH3−CH3 series, they turned their attention to the C−R bond to help explain the trend. They highlighted the initial decrease from CH3−OH to CH3−NH2, but there is a subsequent increase from CH3−NH2 to CH3− CH3. They considered two factors important to the total bond energy that could explain this trend: (i) the strain associated with the resulting species’ optimized geometry and (ii) the interaction between the resulting fragments. They identified a significantly higher strain in the H−CH3 and CH3−CH3 radical fragments than in the other species in each series. In addition, the trend in interaction energy showed the same anomaly as that in BDE. They attributed these effects to the fact that the methyl radical is the only one in the series that will be deformed from its planar geometry to pyramidal geometry as a result of the radical electron repulsion.10

Instructor Responses to Student Questions

Student questions provide benefit to both the learner and the teacher.12 They can foster discussion between students and increase interest in concepts.12 In addition, they can inspire future coursework in the form of inquiry investigations and problems.12 Thus, there is a substantial effort to both encourage and improve the quality of student questions in the classroom.12−16 A dualistic view of the types of questions students ask is to consider text-based and knowledge-based questions.12,17 Text-based questions are usually simple, based on the content delivered, and often clarifying. On the other hand, knowledge-based questions utilize higher order thinking skills (e.g., analyzing, evaluating, and creating) and lead to learning that is more meaningful.12,13,17 As in our case, these questions may address some discrepant event that the student experiences when learned information conflicts with newly presented information. Questions that contest what has been learned can stimulate argumentative discussion between students.13 However, there is less in the literature about how instructors respond to student questions, especially in the case that they cannot immediately answer them. Teacher responses to questions that are not immediately answerable result in both emotional and practical outcomes. In the case that the question is not directly aligned with learning objectives, the instructor may feel threatened and perceive the questions as disruptive and annoying.12 Further, the instructor may try to come up with an answer on the spot, ask students to suggest answers, avoid and ignore the question, or make the question an assignment.12 Our philosophy of teaching and learning involves considering the teacher as a life-long learner. Thus, in an effort to both encourage student inquiry, and to answer the question for ourselves, we changed our students’ question into a theoretical study that can be implemented in an undergraduate course. B



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Figure 1. BDEs and BLs of the methane and water series of homolytic cleavages.

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THEORETICAL METHODS AND LITERATURE VALUES In order to discuss the trends in BDE of methane, water, and other species, students can run relatively low-level calculations that reasonably accurately predict equilibrium structures and BDEs or collect literature values obtained from experiment or highly accurate theoretical methods. Our theoretical predictions along with literature values for BDE and BL are presented in Figure 1. All of the calculations were performed in Gaussian 0918 and will be described next. The BDE literature values reported here were obtained from compilations of experimental values9 and networks of experimentally and accurately computed thermodynamic data.8 BLs were obtained from the NIST Computational Chemistry Comparison and Benchmark Database.19 In order to explain the trends discussed here for the series of methane radicals, we started by performing geometry optimizations and frequency calculations using Density Functional Theory (DFT)20 with the Becke’s three-parameter exchange, Lee, Yang, and Parr correlation (B3LYP) hybrid functional21−23 and the 6-311++G(d,p)24,25 basis set. We also ran molecular orbital calculations using the same level of theory in order to visualize natural bonding orbitals. Highest occupied molecular orbital (HOMO) visualizations are provided in the Supporting Information. This method yielded BLs and BDEs that reasonably accurately predict experimental observations of radicals in the methane series. Before interpreting trends, we note the agreement between theory and experiment. For BDE, the theory and experiment match well using this theoretical method. For BL, the trend is clearly reproduced even if the values are not in exact agreement. The only species in the series for which we noted a difference in the equilibrium structure from experimental values was the methyl radical. Using our method, the equilibrium structure is planar with bond angles of 120°, which is in contrast to the slightly distorted structure discussed in the literature.10 This is not

surprising since DFT is known to overestimate planarity (e.g., ref 26.). We did find that using a higher level of theory (discussed for water next), yielded a slightly distorted structure with bond angles of about 117°. However, the BLs and BDE were not significantly affected. We maintain that for the purposes of discussing this trend in an undergraduate setting, the lower level of theory can be used, but if the instructor would like to discuss the differences between the methods, both theories can be used to compare and contrast the results. The method described above yielded the trends reported experimentally, but differing theoretical values for water, the hydroxide radical, and the hydroxide anion. In particular, the BDE of water is underestimated by about 17 kJ/mol, and BLs are slightly overestimated (by less than 0.01 Å). While these small differences do not prevent students from interpreting trends in BL and BDE, we sought to find a more accurate theoretical method. A 2001 study used a coupled-cluster calculation with single, double, and perturbative triple excitations (CCSD(T))27 with the cc-pCVQZ core−valence set28 to accurately predict several molecular equilibrium structures, including water.29 On the basis of their results, we decided to use the same theory with the cc-pVQZ basis set27 since the core−valence set is not available in the current version of our software. Calculations using this method yielded results that match very well with available experimental data (maximum 7 kJ difference in BDE and less than 0.01 Å differences in BL).

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AN EXPLANATION

Methane Series

Experimental and theoretical BDEs and BLs of methane (1CH4), the methyl radical (2CH3), the methylene radical (3CH2), and the methylidyne radical (2CH) are presented in Figure 1. Next, we discuss the trends in both properties according to our theoretical predictions. First, we consider the resulting species of the homolytic cleavage of the first hydrogen from CH4, which results in CH3, a species with a single C



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Figure 2. Natural bonding orbitals (NBOs) of the methane series.

The first, the increasing effective electronegativity of carbon with increasing s character, has already been discussed. The second is the electrostatic repulsion of the resulting lone pairs and unpaired electrons on the bonding electrons. The BDE of CH2 is less than that of CH3 at 423.8 kJ/mol, a value that agrees well with previously reported values.8,9 The decrease can be explained by the increase in electrostatic repulsion by the nonbonding electrons. This is demonstrated by the change in the structure from planar (CH3) to bent (CH2) and the increased electron density from the two unpaired electrons shown in Figure 2. However, the BL of CH2 (1.080 Å) is only 0.001 Å shorter than that of CH3, a difference that, calculated using DFT, should not be interpreted as meaningful. Interestingly, while the BL does not change from CH3 to CH2, the BDE still decreases from 461.9 to 423.8 kJ/mol. We suggest that both the lack of change in BL and the decrease in BDE result from a change in the dominating effect from attraction in CH3 to electrostatic repulsion in CH2, which destabilizes the bond. With the repulsion effect dominating, the energy required to remove the remaining hydrogens is reduced. Finally, the species resulting from the cleavage of a hydrogen from CH2 results in CH. The NBOs, as shown in Figure 2, are more localized over the carbon center, with the red lobe

unpaired electron (2CH3). The main influence on the change in BDE in this case is the increased attraction on the bonding and nonbonding electrons. While there is not a complete change in hybridization from sp3 to sp2, there is an increase in the s character of the carbon atom, thus increasing the electronegativity of the carbon atom. This change is evidenced by the visualization of the natural bonding orbitals (NBOs) in Figure 2. It can be seen that the electron density is more localized over the carbon center in CH3 and less spread out than in CH4. The red lobe envelops more of the carbon atom, and the blue lobes are smaller in CH3 than in CH4. This effect causes a reduction in the length of the remaining C−H bonds (from 1.091 to 1.081 Å) and an increase in the BDE from 430.9 to 463.5 kJ/mol when CH3 dissociates to 3CH2. Considering only the increasing effective electronegativity of the carbon with increasing s character and without analyzing theoretical or experimental data, students might predict that the BDEs will continue to increase with each successive broken bond. However, the cleavage of a hydrogen from the CH2 requires less energy than from the methyl radical, as demonstrated by both theoretical predictions and experimental measurements.8,9 To explain this, it is useful to consider that in the series of homolytic cleavages of CH4 two opposing effects influence the equilibrium structures and BDEs of each species. D



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BL diagram for a diatomic molecule (Figure 3). At r0, the C− H bond is stable and the energy required to break the bond is slightly less than the potential energy (e.g., V0 − ZPE).6 As the BL increases, the energy required to break the bond decreases as is shown schematically. Thus, we conclude, for example, that removing the first hydrogen from methane decreases the average BL and increases the BDE of the resulting species as electron density is maintained closer to the carbon nucleus due to the increase in s character. Water

As for the opposing trend in BDE for water (H2O), we consider the homolytic cleavage of the first hydrogen. Considering only the two effects discussed previously (the opposing effects of attraction and repulsion), students might predict that the species resulting from the removal of the first hydrogen from water (OH) has a shorter BL and a larger BDE. However, obviously there are important differences between carbon and oxygen as well as between methane and water. First, oxygen is more electronegative than carbon, so the bonding electrons in water will already be closer to the oxygen nucleus than they are in methane. This property causes the resulting unpaired electron to be pulled even closer to the oxygen nucleus than the electron in the methane example. This slight change in electron density is apparent in the NBOs (Figure 4). The electrostatic repulsion of the unpaired electron on the bonding electrons along with the two lone pairs on the OH outweighs the attraction of the nucleus resulting from the increase in s character of the O atom. Thus, the BL is increased from 0.958 to 0.970 Å rather than decreased, and the BDE is decreased from 491.3 to 422.7 kJ/mol. Furthermore, as evidenced by the NBOs of both species, a better comparison can be made between H2O and OH with CH2 and CH since these species are electronically more similar than H2O and OH with CH4 and CH3. Thus, the BDE follows the same trend as the last two species in the methane series.

Figure 3. Potential energy diagram depicting the change in BDE of the C−H bond with an increase in internuclear distance. At rest in the ground state, the bond has zero-point energy (ZPE). The BDE (ΔH) is the energy required to separate the atoms an infinite distance. With an increase in energy, oscillations between the nuclei are such that the displacement from equilibrium in the ground state increases, increasing the length of the bond.

entirely enveloping the carbon atom. The nonbonding orbitals show the significant electron density of the unpaired electron and lone pair, causing significant repulsion of the bonding electrons. This results in a lengthening of the bond to 1.126 Å and a decrease of the BDE to 338.5 kJ/mol. Relationship between BL and BDE

It is important for this discussion that students be able to use BL as evidence to support the change in BDE. For a conceptual discussion of the relationship between BL and BDE, students may refer to the familiar potential energy versus

Figure 4. Natural bonding orbitals (NBOs) of the water series. E



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(6) Benson, S. W. III-Bond energies. J. Chem. Educ. 1965, 42 (9), 502. (7) Treptow, R. S. Bond Energies and Enthalpies: An Often Neglected Difference. J. Chem. Educ. 1995, 72 (6), 497. (8) Ruscic, B. Active thermochemical tables: Sequential bond dissociation enthalpies of methane, ethane, and methanol and the related thermochemistry. J. Phys. Chem. A 2015, 119 (28), 7810− 7837. (9) Blanksby, S. J.; Ellison, G. B. Bond dissociation energies of organic molecules. Acc. Chem. Res. 2003, 36 (4), 255−263. (10) van Zeist, W.-J.; Bickelhaupt, F. M. Trends and anomalies in H−AH n and CH 3−AH n bond strengths (AH n= CH 3, NH 2, OH, F). Phys. Chem. Chem. Phys. 2009, 11 (44), 10317−10322. (11) Bordwell, F. G.; Ji, G. Z. Effects of structural changes on acidities and homolytic bond dissociation energies of the hydrogennitrogen bonds in amidines, carboxamides, and thiocarboxamides. J. Am. Chem. Soc. 1991, 113 (22), 8398−8401. (12) Chin, C.; Osborne, J. Students’ questions: a potential resource for teaching and learning science. Stud. Sci. Educ. 2008, 44 (1), 1−39. (13) Aguiar, O. G.; Mortimer, E. F.; Scott, P. Learning from and responding to students’ questions: The authoritative and dialogic tension. J. Res. Sci. Teach. 2010, 47 (2), 174−193. (14) Hofstein, A.; Navon, O.; Kipnis, M.; Mamlok-Naaman, R. Developing students’ ability to ask more and better questions resulting from inquiry-type chemistry laboratories. J. Res. Sci. Teach. 2005, 42 (7), 791−806. (15) Middlecamp, C. H.; Nickel, A.-M. L. Doing science and asking questions II: An exercise that generates questions. J. Chem. Educ. 2005, 82 (8), 1181. (16) Middlecamp, C. H.; Nickel, A.-M. L. Doing Science and Asking Questions: An Interactive Exercise. J. Chem. Educ. 2000, 77 (1), 50. (17) Chin, C.; Brown, D. E. Student-generated questions: A meaningful aspect of learning in science. Int. J. Sci. Educ. 2002, 24 (5), 521−549. (18) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; et al. Gaussian 09, Revision A.1. Gaussian, Inc.: Wallingford, CT, 2009. (19) National Institute of Standards and Technology. Computational Chemistry Comparison and Benchmark Database; NIST Standard Reference Database Number 101; https://na01.safelinks. protection.outlook.com/?url=https%3A%2F%2Fcccbdb.nist. gov%2F&data=02%7C01%7CFlorencio.Hernandez%40ucf. edu%7C003c7f20a1c14943fe4f08d5d7b7beca%7Cbb932f15ef3842 ba91fcf3c59d5dd1f1%7C0%7C0%7C636652106540417792&sdata= eSkuXipl5DlCQbM75Ar1ySxreLbNYZa2puMV8pjOpME%3D&re served=0 (accessed June 2018). (20) Runge, E.; Gross, E. K. U. Density-Functional Theory for Time-Dependent Systems. Phys. Rev. Lett. 1984, 52, 997−1000. (21) Becke, A. D. Density-Functional Exchange-Energy Approximation with Correct Asymptotic Behavior. Phys. Rev. A: At., Mol., Opt. Phys. 1988, 38, 3098−3100. (22) Lee, C.; Yang, W.; Parr, R. G. Development of the ColleSalvetti Correlation-Energy Formula into a Functional of the Electron Density. Phys. Rev. B: Condens. Matter Mater. Phys. 1988, 37, 785− 789. (23) Becke, A. D. Density-Functional Thermochemistry. III. The Role of Exact Exchange. J. Chem. Phys. 1993, 98, 5648−5652. (24) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. SelfConsistent Molecular Orbital Methods. XX. A Basis Set for Correlated Wave Functions. J. Chem. Phys. 1980, 72 (1), 650−654. (25) Clark, T.; Chandrasekhar, J.; Spitznagel, G. W.; Schleyer, P. V. R. Efficient Diffuse Function-Augmented Basis Sets for Anion Calculations. III. The 3-21+G Basis Set for First-Row Elements, Li−F. J. Comput. Chem. 1983, 4 (3), 294−301. (26) Alparone, A.; Millefiori, A.; Millefiori, S. Non-planarity and solvent effects on structural and polarizability properties of cytosine tautomers. Chem. Phys. 2005, 312, 261−274.

Theoretical and experimental results confirm that the BL and BDE of the hydroxyl anion (OH−) are between those of H2O and OH. This observation can be reconciled using the same line of reasoning. The BL of this species is longer than that of water for the same reason described for the radical species. However, contrary to the OH radical, there are three lone pairs and no unpaired electrons in OH−. Consequently, there is more repulsion between the nonbonding electrons than there is in the radical species, which is illustrated in the visualization of the NBOs (Figure 4). This repulsion shifts the nonbonding electrons away from the nucleus, shortening the BL (in comparison to that of the hydroxyl radical) to 0.964 Å and decreasing the BDE to 456.3 kJ/mol.

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CONCLUSIONS Trends in BDE for the homolytic cleavage of successive molecular bonds can be explained using concepts introduced in general chemistry (e.g., the effects of attraction and repulsion of nuclei and electrons and periodic trends) and expanded upon using concepts such as the relationship between BL and BDE, electronegativity, hybridization, and the effect of the geometries of resulting radicals on BDE. Undergraduate students can use computational chemistry to observe these trends or find experimental and highly accurate theoretical data in the literature in order to interpret them.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.7b00962. Highest occupied molecular orbitals (HOMOs) of the methane and water series (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: fl[email protected]. ORCID

Florencio E. Hernández: 0000-0001-7753-6995 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge the students in the fall 2017 section of CHM3410 at UCF for starting this conversation. Additionally, we acknowledge our reviewers for their thoughtful critiques, which greatly improved the quality of this paper.

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REFERENCES

(1) Bain, K.; Moon, A.; Mack, M. R.; Towns, M. H. A review of research on the teaching and learning of thermodynamics at the university level. Chem. Educ. Res. Pract. 2014, 15 (3), 320−335. (2) Galley, W. C. Exothermic bond breaking: A persistent misconception. J. Chem. Educ. 2004, 81 (4), 523. (3) Greenbowe, T.; Meltzer, D. Student learning of thermochemical concepts in the context of solution calorimetry. Int. J. Sci. Educ. 2003, 25 (7), 779−800. (4) Knox, B. E.; Palmer, H. B. The uses and abuses of bond energies. J. Chem. Educ. 1961, 38 (6), 292. (5) Wright, J. S.; Ingold, K. U. Understanding trends in CH, NH, and OH bond dissociation enthalpies. J. Chem. Educ. 2000, 77 (8), 1062. F



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(27) Dunning, T. H., Jr Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90 (2), 1007−1023. (28) Woon, D. E.; Dunning, T. H., Jr Gaussian basis sets for use in correlated molecular calculations. V. Core-valence basis sets for boron through neon. J. Chem. Phys. 1995, 103 (11), 4572−4585. (29) Bak, K. L.; Gauss, J.; Jørgensen, P.; Olsen, J.; Helgaker, T.; Stanton, J. F. The accurate determination of molecular equilibrium structures. J. Chem. Phys. 2001, 114 (15), 6548−6556.

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Trends in Bond Dissociation Energies for the Homolytic Cleavage of

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