An Integrated, Statistical Molecular Approach to the Physical

In the Classroom

An Integrated, Statistical Molecular Approach to the Physical Chemistry Curriculum Stephen F. Cartier Department of Chemistry, Warren Wilson College, Asheville, NC 28815; [email protected]

Chemistry is ultimately the study of change. Changes of state, the mixing of substances, chemical reactions, and spectroscopic transitions are manifestations of change on an atomic and molecular level. Indeed, many students are drawn to the study of chemistry because of a change they have witnessed. Ultimately, the tendency of a system to change, whether it is through a transition state or to a final state, is governed by the relative probability of existence of these states and quantitatively accounted for with the tools of statistical mechanics. To comprehend and investigate this dynamic nature it is essential to understand the second law of thermodynamics. Unfortunately, common classical expressions of the second law do not convey insight into, or even any connection to, atomic and molecular behavior. As a result, in physical chemistry courses in which the second law is presented from the macroscopic perspective, its significance and relevance to atomic and molecular systems is often lost on students of chemistry. Regardless of whether a “thermodynamics first” or “quantum first” approach is adopted, a disconnect exists from the outset in students’ comprehension and ability to relate (apparently unrelated) concepts that comprise the remainder of the curriculum. Why, then, do we continue to typically teach physical chemistry from such a compartmentalized perspective? As Atkins (1) states, “The importance of the molecular partition function is that it contains all the information needed to calculate the thermodynamic properties of a system of independent particles.” Implicit in this statement is the intrinsic importance of probability, in general, and the partition function, in particular, in elucidating the nature of chemical change. Thus, in an effort to address the pedagogical challenges associated with the traditional approaches and to emphasize the probabilistic nature of chemical change, a novel, statistical molecular approach to the physical chemistry curriculum has been developed. Rather than simply being a reordering of curricular topics, the proposed curriculum represents a more cohesive approach in which statistical mechanics is introduced early and then integrated throughout. As a result, the curriculum naturally enables and encourages the consistent, quantitative analysis of entropy changes accompanying a wide variety of chemical and physical transformations. In this curriculum, the Boltzmann distribution, partition functions, and statistical definition of entropy are introduced in the first days of the course. Students quickly gain an appreciation and preliminary understanding of statistical behavior on an atomic and molecular level and readily grasp the power of applying statistical mechanics in predicting, understanding, and interpreting the macroscopic manifestation of most probable atomic and molecular behavior. As opposed to the more traditional practice of teaching physical chemistry from the macroscopic perspective, this curriculum (without loss of quantitative rigor) represents an integrated alternative that provides students with the conceptual foundation and mathematical skills required to treat systems of ever-increasing complexity. Rather than being a collection of seemingly disparate topics, the curriculum is

presented and learned as a cohesive whole, much greater than the sum of its parts. Hanson and Green (2) have recently written an introductory-level molecular thermodynamics text in an effort to address this specific issue early in students’ chemistry educational experience. In this text, the authors introduce the role probability plays in predicting outcomes and emphasize the application of statistical concepts to interpreting chemical, physical, and electrochemical changes. Additionally, in this Journal, Kozliak (3) and Novak (4) have presented alternative quantitative molecular approaches to entropy and the second law. As all of these authors convey, the statistical definition of entropy is conceptually more accessible, logical, and relevant to chemistry students than the classical definition. Kozliak’s approach, in particular, relies on a semiquantitative introduction to the Boltzmann distribution and the molecular basis of entropy before the more thorough treatment of statistical and quantum mechanics later in the course. Novak has presented a semiquantitative introduction to thermodynamics from the microscopic point of view. This integrated, logical approach is based on introducing elementary, particle concepts before discussing the thermodynamic behavior of bulk systems and more complex statistical mechanical concepts. To reiterate Novak’s sentiment, the macroscopic approach has no pedagogical justification and it is not just anachronistic, it hinders the development of “rational concept networks”. From a more qualitative perspective, in a series of articles Lambert (5–7) has elucidated the molecular interpretation of entropy changes as a function of the dispersal of energy accompanying a variety of processes. Lambert effectively makes the argument for the dispersal of energy, as opposed to a system’s dispersal in space (disorder), as the basis for interpreting the nature of entropy changes and these examples can be quantitatively validated through the application of the appropriate statistical methods. The intent of this article is not to provide conceptual insight into all of the subtleties associated with various physical chemistry topics or to advocate a rigid, prescriptive approach to the physical chemistry curriculum. Rather, this work provides the context for a unique, alternative approach to the curriculum. Using a combination of resources that individual instructors can adapt according to their own personal preferences and priorities,1 the dispersal of energy and the role it plays in enabling increases in entropy can be emphasized. Virtually all of the concepts treated here are treated in standard physical chemistry texts and many of them have been discussed and clarified in a thorough qualitative and quantitative manner in this Journal (3–12). However, no currently available text presents these concepts in the integrated manner proposed here. Several different texts (13–15) have been used in teaching the course from this new perspective with equal success. Regardless of the text used, this curriculum requires both extensive integration of material between chapters and supplementation from other sources (3–12, 16–25).

© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 86 No. 12 December 2009 • Journal of Chemical Education

1397

In the Classroom

Nor is the purpose of this article to suggest that the proposed curriculum is the appropriate approach for all students of physical chemistry. This particular curriculum has been delivered to classes composed of 10–15 chemistry and biochemistry majors at two different liberal arts institutions a total of five times. For chemistry majors required to take two semesters of physical chemistry, the curriculum is appropriately rigorous and thorough in scope and provides abundant opportunity to emphasize the integrated nature of the field. For biochemistry majors who are often required to take only one semester of physical chemistry and find themselves at institutions where a physical biochemistry course is not offered, the curriculum requires some adjustments, although its integrated, statistical molecular emphasis would certainly be of value to these students as well.2 For engineers, the proposed statistical molecular emphasis has the least value and the traditional approach would perhaps still best serve their needs. First Semester Statistical Nature of Entropy Even before entering a physical chemistry course, students know that atoms and molecules exhibit a quantized nature and that the basis for all spontaneous chemical change is an increase in entropy or “change towards a condition of maximum probability” for the system of interest and its surroundings. Fundamentally, for an isolated, quantized system, the condition of maximum probability corresponds to the configuration that can be achieved in the greatest number of ways, or alternatively, has the greatest number of microstates, W. Employing the introductory examples from various general and physical chemistry texts, students quickly grasp the concept of the dominant configuration in the context of the number of microstates associated with each configuration. Statistically, the Boltzmann equation, S = kB ln W, provides the quantitative link between the number of ways a particular configuration can be achieved and the associated entropy of the system. In other words, the greater the number of microstates associated with a particular distribution of particles among energy levels, the greater the probability of its observation and the greater its entropy. By extension, it is easy and worthwhile to demonstrate at this point that the change in entropy that accompanies the spontaneous change of an isolated system is related to the change in the number of available microstates, and under a given set of conditions the most probable state of the system is that which corresponds to the greatest number of microstates. Although it would be impractical to calculate the change in entropy as a function of the change in number of microstates for a given spontaneous process, the Boltzmann equation can be used as the basis for briefly introducing the second law and beginning to rationalize spontaneity from a molecular, probabilistic perspective. The Boltzmann Distribution and the Partition Function The Boltzmann distribution represents the most probable distribution corresponding to the configuration with the greatest number of microstates (and therefore the greatest absolute entropy) and can be derived in a manner consistent with the preceding qualitative introduction. This derivation, whether it is based upon Lagrange’s method of undetermined multipliers (1, 20, 24) or an alternative approach (8, 10, 17, 18), demonstrates 1398

the relationship between the most probable distribution and the maximum value of W, reinforcing the intent of the curriculum to develop the students’ ability to integrate concepts and quantitatively justify these relationships. Alternatively, depending on the level of mathematical preparation of the students, the derivation can be postponed until later in the curriculum and the Boltzmann distribution equation simply presented in accord with the presentation of Davies (16). However, as this is the first opportunity in the curriculum to introduce quantitative rigor and mathematical justification for observed behavior, there is a certain pedagogical value to including the derivation at this point. The average number of accessible states depends quantitatively on both the spacing of energy levels and the energy available to the system and is given by the “molecular partition function”, or the normalization constant (denominator) of the Boltzmann distribution when expressed as a fractional population. As the total energy of the system is defined by the sum of the translational, vibrational, rotational, electronic, and nuclear contributions, the complete molecular partition function, as a product of partition functions for these individual modes of energy dispersal, can be derived. Each of the individual partition functions depends uniquely on the corresponding energies and can be treated independently (thus yielding an approximate molecular partition function), while also being integrated into the discussion of the translational, vibrational, rotational, and electronic nature of atoms and molecules. Quantum Mechanics, Spectroscopy, and Statistical Mechanics Once all of these statistical mechanical concepts have been introduced, the goal becomes to statistically quantify atomic and molecular quantum mechanical behavior to deduce macroscopic properties of matter. To achieve this goal, rather than treating quantum mechanics as a stand-alone topic and then relating statistical mechanical concepts in retrospect as is the approach taken in virtually all texts, the two topics are developed concurrently as each of the different degrees of freedom is individually addressed. Thus, the quantum-mechanical nature of microscopic systems and the statistical basis for the populations of the quantum-mechanically defined states and their related spectroscopic manifestation are all integrated. Most probable translational, vibrational, rotational, and electronic distributions are defined as a function of quantum and statistical-mechanical principles and the appearance of their corresponding spectra are quantitatively rationalized. Following a conventional overview of the historical development of quantum mechanics, the particle in a box, and the related problems of a particle on a surface, on a sphere, or in a cube, the translational contribution to the molecular partition function is derived and thoroughly studied. Relying on the introductory principles, students are encouraged to rationalize translational behavior, the magnitude of the translational partition function and its dependence upon mass, temperature, and volume, and the magnitude of atomic absolute entropies. The Sackur–Tetrode equation can then be introduced (although the derivation is reserved until later in the curriculum) and serves to validate and quantify this behavior. Rationalizing trends in absolute entropies of atomic systems at this point is a valuable introductory exercise in integrating knowledge. Furthermore, Lowe’s article (11) is an excellent introductory resource empha-

Journal of Chemical Education • Vol. 86 No. 12 December 2009 • www.JCE.DivCHED.org • © Division of Chemical Education

In the Classroom

A

Frequency / cm∙1 2900

Frequency / cm∙1

B 2800 0

3100

3000

2900

2800

2700

2600

2500 0

Relative Intensity

Relative Intensity 1

1

Figure 1. Simulated ro-vibrational spectra of H35Cl demonstrating the spectroscopic manifestation of quantum and statistical mechanical behavior of a diatomic system: (A) 25 K and (B) 500 K.

sizing these qualitative connections and expanding the discussion to alternate systems that the students are able to intuitively grasp and appreciate. The quantum mechanics of molecular rotation and vibration are then treated from the conventional perspective and the corresponding spectroscopic principles (transition energies, intensities, peak spacing, selection rules) are introduced and discussed. To further develop students’ understanding of the relevance of the population of states with respect to the appearance of spectra, a rigorous treatment of the rotational and vibrational contributions to the molecular partition function and the corresponding fractional populations of states is undertaken. The discussion of ro-vibrational spectra then naturally builds on the quantum mechanical interpretation of vibration and rotation and the statistical analysis of spectra developed in those contexts. Similarly, the electronic spectroscopies of absorption, fluorescence, and phosphorescence are then discussed and, as part of this discussion, the contribution of (typically) only the ground state to the electronic partition function is emphasized. By discussing each of the contributions to the molecular partition function3 in detail, students can then be challenged to visualize and predict how changes in temperature and molecular properties influence the populations of vibrational and rotational states and, consequently, the appearance of spectra. To facilitate the understanding of this relationship between the quantum and statistical mechanical concepts and how they manifest themselves in molecular spectra, a graphical model has been developed that enables students to visualize the changes in spectra under various conditions. Additionally, an invaluable resource for visualizing the effect of temperature and molecular properties on the appearance of ro-vibrational spectra, in particular, can be found on the ChemCollective Web site (26). Using these interactive models, students can study the appearance of a variety of spectra as a function of temperature and different molecular parameters, which permits them to visually relate and quantitatively rationalize quantum mechanical, statistical mechanical, and spectroscopic properties. Examples that we have developed of model ro-vibrational spectra generated using the Microsoft Excel application appear in Figure 1. Canonical Partition Function and Ensembles Applying the same pedagogical principle to the presentation of partition functions as applied to the entire curriculum, and emphasized by Novak (4) in his microscopic-to-macroscopic approach, the presentation of the canonical partition function fol-

lows that of the molecular partition function.4 In particular, the canonical ensemble and the corresponding partition functions for systems of distinguishable and indistinguishable noninteracting particles are introduced and applied. More generally, the conceptual distinction among the microcanonical, canonical, and grand canonical ensembles is discussed and their significance in calculating thermodynamic properties is alluded to. However, such applications are reserved until the field of statistical thermodynamics is introduced later in the curriculum. Second Semester Kinetic Theory and the Statistical Interpretation of Internal Energy and Heat Capacity The second semester begins with a thorough discussion of the kinetic theory of gases and naturally leads to a discussion of the classical equipartition of energy. The classical notion of the equipartition of energy is discussed and assessed on the basis of predicted heat capacities. Students understand quickly that the classical notion of the dispersal of energy among the various degrees of freedom fails for any systems more complex than those that are monatomic. Thus, a more accurate description of the dispersal of energy is sought. To explain these discrepancies between the classical predictions and the behavior of systems more complex than those that are monatomic, it is necessary to invoke a statistical treatment of the dispersal of energy. The concept of the average energy of a system is introduced and is simply presented as a weighted average of populated states. Students are familiar with calculating average atomic masses and thus readily understand that the average internal energy, 〈U 〉, of an atom or molecule is given by

U

=

∑ pi εi = i

∑ i

gi e −βεi εi q

where pi represents the fractional population (probability) of the level i (given by the Boltzmann distribution in which q is the partition function) and gi and εi the degeneracy and energy of that level, respectively. When summed over all energies, the average internal energy is obtained. The importance of the partition function is again evident here. From this introduction to the average internal energy, combined with the previously presented definition of the canonical partition function, expressions for the average molar internal energy and, by extension, the constant volume heat capacity are derived. These expressions then serve as


1399

In the Classroom

the basis for an immediate discussion of the partition of energy and, eventually, thermodynamics. Monatomic and polyatomic systems are analyzed thoroughly from this statistical perspective. By applying the appropriate equations to the previously derived expressions for the individual contributions to the partition functions, the students derive expressions for the internal energy and heat capacity for a variety of systems of increasing complexity. Through applying these equations for heat capacity, the students acquire an appreciation and understanding of the extent of the true nature of energy dispersal among the translational, rotational, and vibrational degrees of freedom of a system. (Depending upon how the zero-point electronic energy is defined, this contribution can also be taken into account.) Classical thermodynamic principles

Relative Energy / (10∙20 J)

A 1.2

energy level: 20 ⋮ 1

1.0 0.8 0.6 0.4

〈U〉

0.2 0.0 0

2

4

6

8

Fractional Population (%)

10

12


B 1.2


1.0 0.8 0.6 0.4

〈U〉

0.2 0.0 0

1

2

3

4


5

6


C 1.2


1.0 0.8 0.6 0.4

〈U〉

0.2 0.0 0.0

0.5

1.0

1.5

2.0


2.5

3.0

Figure 2. Fractional populations plotted as a function of energy (and labeled by quantum state) for three different volumes for the model system undergoing isothermal (300 K) expansion into vacuum: (A) L, (B) 2L, and (C) 4L, where L is the initial length of the containing box. The average internal energy, 〈U〉, for each distribution is indicated.

1400

can then be treated in this same integrated manner invoking the partition function to complement the traditional macroscopic interpretation of thermodynamic properties. Classical Thermodynamics The laws of thermodynamics are treated, albeit briefly, as presented in most standard physical chemistry texts and applied to a variety of processes and fields, including temperature changes, isothermal–adiabatic, reversible–nonreversible (and combinations thereof ) expansions of gases, phase changes, and thermochemistry. As Novak emphasizes in his article (4), however, understanding thermodynamics requires familiarity with macroscopic and microscopic concepts simultaneously. Relying on topics presented earlier, this curriculum thus permits a more thorough, microscopically intuitive and quantitative interpretation of common thermodynamic phenomena. In particular, students are expected to be able to independently interpret the statistical nature of thermodynamic changes and apply the Boltzmann distribution to graphically represent the corresponding population changes. To further facilitate this approach to the teaching of classical thermodynamic processes, a graphical model has been developed to statistically model such phenomena.5 Through this model, the concepts of statistical mechanics, quantum mechanics, and classical thermodynamics are applied in the analysis of the (a) constant volume heating, (b) constant pressure heating, (c) isothermal expansion into vacuum, (d) adiabatic irreversible expansion, and (e) adiabatic reversible expansion of an ideal gas. Students study, discuss, and graphically present changes in the density and population of states of a system upon expansion to acquire a quantitative, statistical understanding of the driving force behind such processes, namely, an increase in the dispersal of energy on an atomic or molecular scale. The changes in internal energy and entropy accompanying such processes are then quantitatively assessed from a discrete, statistical perspective and correlated with changes in temperature and volume. Whether used as a tool in lecture or as a hands-on computational exercise, the model is a powerful quantitative and visual tool that enables students to more readily grasp and integrate seemingly disparate microscopic and macroscopic concepts. An example of one such process, treated from the classical and statistical perspectives, is the isothermal expansion of a perfect gas. Upon the isothermal reversible or irreversible expansion of a perfect gas into vacuum, the temperature does not change and no heat flows. Although there is no net increase in the internal energy of the system, the process does result in an increase in the density of states due to the increase in volume. To clarify the statistical nature of isothermal expansions, a statistical analysis of a perfect gas initially confined and then expanded to 8 times and then 64 times its original volume is presented. To simplify the analysis and eliminate the need to account for degeneracy (as well as the distinction between levels and states), the expansion is considered in one-dimension so that the length of the confining “box” increases to two times and then four times its original size. As a result of the compression of states, a greater number of higher energy levels must contribute to the discrete sum that yields the average internal energy and consequently, the fractional populations of lower levels decrease relative to their initial values as seen in Figure 2. To maintain constant internal energy, the population distribution shifts towards higher energy levels. As a result, as the average internal energy remains constant and


In the Classroom

the volume increases, the temperature remains the same but the average energy corresponds to increasingly higher energy level. As a result of state compression and the corresponding increase in the number of populated states (the partition function) upon expansion, the number of microstates corresponding to each Boltzmann distribution also increases. The Sackur–Tetrode equation accounts for this increase in the number of microstates and quantitatively predicts an increase in the translational entropy of the system. Once thermodynamics has been treated from the classical perspective and the statistical interpretation of heating and expansion processes has been mastered, the next phase of the curriculum is to derive thermodynamic properties as a function of the partition function. In particular, beginning with the Boltzmann equation, the fundamental statistical definition of entropy as a function of the canonical partition function, Q, can be derived (1, 20) and reconciled with the Boltzmann equation, U − U0 S = + kB lnQ T where U and U0 represent the internal energies at some temperature T and at 0 K, respectively. A thorough, integrated discussion of temperature changes, energy availability, the population of states, the number of microstates, and the magnitude of the partition function and entropy naturally follows and reinforces conceptual networks. This formal, statistical definition of entropy can then be applied in conjunction with the definition of the Helmholtz energy, A = U − TS, to derive statistical expressions for the pressure, enthalpy, chemical potential, and Gibbs energy. Developing the statistical interpretation of these thermodynamic properties along with their classical application builds on knowledge acquired in the preceding months and provides a sound conceptual and quantitative grasp of their fundamental, microscopic basis. Rather than only emphasizing their natures as macroscopic state functions, the integrated approach further quantitatively defines them in the context of the populations of states as represented by the partition function. By emphasizing the statistical nature of these properties in conjunction with their practical application, students simultaneously develop the insight and skills to interpret and solve realistic thermodynamic problems. Ultimately, they are well-prepared to interpret the statistical behavior manifested in the dynamics of equilibrium process and rates of reaction. Chemical Equilibrium and Kinetics The last segment of the course is devoted to chemical equilibrium and kinetics. Both topics are treated thoroughly from a classical point of view. However, equilibrium constants and rate constants are also derived using statistical mechanics and can be compared to the classical expressions. At this point in the curriculum students are expected to readily rationalize the magnitude of the equilibrium constant or rate constant of a reaction from a statistical and classical perspective. In particular, the classical and statistical changes in entropy from reactants to products (for the equilibrium constant) and reactants to intermediate (for the rate constant) can be assessed and reconciled. As an understanding of the populations of states and partition functions must be invoked to rationalize the observed behavior from a statistical perspective, this provides yet another opportu-

nity for the integration of concepts. Thus, in the last days of the course, students are still applying the basic concepts introduced on the first day of the course two full semesters before. Laboratory As this curriculum is initially fairly abstract, in the associated laboratory the first several weeks are devoted to computational projects. For example, methods of statistical analysis; manipulation and application of partition functions and the Boltzmann distribution; and molecular modeling are emphasized early on. The students are adequately prepared to begin independently investigating the properties of these functions through computational exercises because they have been exposed to the Boltzmann distribution and partition functions during the first week of lecture. Subsequently, as an example of how these concepts are incorporated in the understanding of chemical systems, the molecular modeling (using Spartan) of one-dimensional potential energy surfaces are plotted as a function of dihedral angle. Example systems include, in order of increasing complexity, ethane, n-butane, and cyclohexane. Once the relative stabilities of conformers of these individual molecules have been determined, the Boltzmann distribution is used to plot relative populations. Students are then expected to rationalize observed computational energetic behavior with the statistical predictions. Following the introduction to these computational techniques, it is then expected that students will incorporate aspects of these studies towards elucidating the results obtained in the more traditional experiments conducted throughout the remainder of the laboratory portion of the course.6 Certainly, there are numerous avenues in which statistical methods and molecular modeling can be incorporated in the laboratory curriculum to demonstrate the integrated nature of the field. As is typical, the laboratory experience is a work in progress and the most significant outstanding goal of this endeavor in physical chemistry educational reform is the continued development of a more thoroughly integrated laboratory experience consistent with the lecture curriculum. Results of these efforts will be disseminated in another article. Assessment As with any major curricular modification, the necessity of incorporating material from a variety of sources causes the course preparation to be initially cumbersome. However, the payoff is a more conceptually accessible, cohesive, and integrated presentation of the material without any loss of rigor. To quantitatively validate these claims, the ACS Combined Semester Physical Chemistry exam has recently been introduced as the final exam for the course. To date, only one group of students has taken the exam, but the class average was a respectable 74%. The ACS Physical Chemistry Combined Semester Exam comprised five sections: quantum mechanics, thermodynamics, kinetics, statistical mechanics, and dynamics. Unfortunately, national exam statistics are not available for the combination (all five sections) administered here. Thus, it is currently not possible to draw comparisons to national norms. However, in future years, this exam will continue to serve as the final assessment measure and student performance data will be collected and analyzed. Ideally, other faculty will adopt a similar approach to the curriculum and data from different student populations at different institutions can be compiled and compared to students exposed to a more traditional curriculum.


1401

In the Classroom

Anecdotally, students’ attitudes towards physical chemistry have changed as a consequence of implementing this curriculum and are reflected in their annual evaluations of the course. Before developing this approach, most students’ sentiments were reflected in the student comment, “Physical chemistry is the black hole into which all interest in chemistry disappears.” Now, student sentiment (generally) mirrors comments such as “Physical chemistry wasn’t nearly as bad as I thought it would be.” Considering the reputation of the course, this is a significant achievement. Granted, these are anecdotal examples from different groups of students having used different texts at different universities in different years. However, response to and success in the course for students of average ability in particular have both improved greatly since this curriculum was adopted and quantitative measures of assessment of such claims are being developed. Conclusion Classical treatment of the laws of thermodynamics provides little or no insight into the behavior of chemical systems on an atomic and molecular level. To provide future chemists with the ability to intuit chemical behavior, it is more practical and effective to present physical chemistry from a statistical molecular perspective. The proposed curriculum does not sacrifice essential components or the rigor associated with the traditional physical chemistry course. In this age of ever increasing complexity (and miniaturization) and cross-disciplinary studies, one responsibility we have as chemistry educators is to develop our students’ understanding of atomic and molecular behavior while nurturing their ability to integrate knowledge across disciplines. The curriculum proposed herein represents a curricular evolutionary step towards these goals. Eddington (27) poetically stated, “Entropy is time’s arrow.” Less eloquently, but consistent with this sentiment, statistical entropy ought to serve as physical chemistry’s arrow. Notes 1. An outline of topics covered in this curriculum accompanies this article in the online material. 2. For example, rather than the quantum mechanics of atomic and molecular systems being presented during the first semester, chemical thermodynamics and kinetics could be substituted to meet these students’ needs while still exposing them to the statistical molecular perspective. 3. The nuclear contribution to the partition function is not currently addressed in the lecture portion of the curriculum. However, there is no pedagogical reason why it could not be treated as a formal aspect of the lecture. Depending on the time available, it could be effectively presented in a similar integrated manner at this point in the curriculum. (Currently, NMR is treated through the laboratory curriculum.) 4. This microscopic-to-macroscopic approach to introducing the canonical partition function is contrary to that adopted in several texts (12, 22), but similar to that in others (1, 14, 23). Both types of text have been used equally effectively in teaching this curriculum, although the former do require greater adaptation with respect to the presentation of this particular concept. 5. The details of this statistical graphical model of expansion processes will be presented in a forthcoming article to be submitted to this Journal. 6. For example, currently experiments investigating the particle in a box model, ro-vibrational spectrum of HCl, heat capacity ratio of 1402

gases, kinetic versus thermodynamic control of reactions, applications of NMR, thermochemistry, entropy of vaporization, and quantum dots all include significant computational (statistical or molecular modeling) aspects.

Literature Cited 1. Atkins, P.; de Paula, J. Atkins’ Physical Chemistry, 8th ed.; W. H. Freeman: New York, 2006. 2. Hanson, R. M.; Green, S. Introduction to Molecular Thermodynamics; University Science Books: Sausalito, CA, 2008. 3. Kozliak. E. I. J. Chem. Educ. 2004, 81, 1595–1598. 4. Novak, I. J. Chem. Educ. 2003, 80, 1428–1431. 5. Lambert, F. L. J. Chem. Educ. 2002, 79, 187–192. 6. Lambert, F. L. J. Chem. Educ. 2002, 79, 1241–1246. 7. Lambert, F. L. J. Chem. Educ. 2007, 84, 1548–1550. 8. McDowell, S. J. Chem. Educ. 1999, 76, 1393–1394. 9. Jungermann, A. H. J. Chem. Educ. 2006, 83, 1686–1694. 10. David, C. W. J. Chem. Educ. 2006, 83, 1695–1697. 11. Lowe, J. P. J. Chem. Educ. 1988, 80, 403–406. 12. Spencer, J. N.; Lowe, J. P. J. Chem. Educ. 2003, 65, 1417–1423. 13. McQuarrie, D. A.; Simon J. D. Physical Chemistry-A Molecular Approach; University Science Books: Sausalito, CA, 1997. 14. Atkins, P. Physical Chemistry, 6th ed.; W. H. Freeman: New York, 1998. 15. Chang, R. Physical Chemistry for the Chemical and Biological Sciences; University Science Books: Sausalito, CA, 2000. 16. Davies, W. G. Introduction to Chemical Thermodynamics: A NonCalculus Approach; W. B. Saunders: Philadelphia, 1972. 17. Andrews, F. C. Equilibrium Statistical Mechanics, 2nd ed.; Wiley: New York, 1975. 18. Bent, H. A. The Second Law; Oxford University Press: New York, 1965. 19. Nash, L. K. Elements of Statistical Thermodynamics, 2nd ed.; Addison-Wesley: Reading, MA, 1974. 20. McLauchlan, K. A. Molecular Physical Chemistry: A Concise Introduction; Royal Society of Chemistry: Cambridge, 2004. 21. Linder, B. Thermodynamics and Introductory Statistical Mechanics; Wiley: Hoboken, NJ, 2004. 22. Berry, R. S.; Rice S. A.; Ross, J. Physical Chemistry; Oxford University Press: Oxford, 2000. 23. Levine, I. Physical Chemistry, 5th ed.; McGraw Hill: New York, 2002. 24. Laidler, K. J.; Meiser, J. H.; Sanctuary, B. C. Physical Chemistry, 4th ed.; Houghton Mifflin: Boston, 2003. 25. Moog, R. S.; Spencer, J. N.; Farrell, J. J. Physical Chemistry: A Guided Inquiry: Atoms, Molecules, and Spectroscopy; Houghton Mifflin: Boston, 2004. Moog, R. S.; Spencer, J. N.; Farrell, J. J. Physical Chemistry: A Guided Inquiry: Thermodynamics; Houghton Mifflin: Boston, 2004. 26. The ChemCollective Spectroscopic Simulator. http://www.chemcollective.org/applets/spectro.php (accessed Oct 2009). 27. Eddington, A. The Nature of the Physical World; University of Michigan Press: Ann Arbor, MI, 1958.

Supporting JCE Online Material

http://www.jce.divched.org/Journal/Issues/2009/Dec/abs1397.html Abstract and keywords Full text (PDF) with links to cited URLs and JCE articles Supplement Curriculum outline


An Integrated, Statistical Molecular Approach to the Physical

Recommend Documents