In the Laboratory
Integrating Statistical Mechanics with Experimental Data from the Rotational–Vibrational Spectrum of HCl into the Physical Chemistry Laboratory Bret R. Findley Department of Chemistry and Physics, Saint Michael’s College, Colchester, VT 05439 Steven E. Mylon* Department of Chemistry, Lafayette College, Easton, PA 18042; *
[email protected] The acquisition and interpretation of vibrational–rotational spectra have become vital laboratory activities in introductory physical chemistry courses (1–4). High-resolution gas-phase IR spectra can now be acquired using relatively inexpensive FTIR spectrophotometers, making these types of laboratory activities even more accessible to students from nearly any institution. The beauty of these laboratory exercises lies in the quantity of information one can obtain from the IR spectra. In addition to demonstrating the application of quantum mechanics, students are able to obtain high quality structural information about heteronuclear diatomic molecules (1–4) and more complex polyatomic gases such as acetylene (1) and sulfur dioxide (2). Interpreting the vibrational–rotational spectra of HCl and DCl is a typical laboratory exercise that incorporates these concepts. Students usually determine the spectroscopic constants of these molecules. Additionally, they can employ Boltzmann statistics and partition functions to explain spectral transition intensities (5, 6). However, in many physical chemistry courses, these calculations may be the closest most instructors get to any discussion of statistical mechanics. Here we describe an approach for extending the study of the vibrational–rotational spectrum of HCl in which students use statistical mechanics to calculate bulk thermodynamic properties from their own previously recorded spectroscopic data. This extension of the traditional HCl lab exercise (1, 2) requires only a computer and an appropriate spread sheet application (we use MS-Excel). By constructing a spreadsheet using MS-Excel, students make calculations from their original data. With this exercise students are forced to think deeply about the meaning of bulk thermodynamic properties in order to tie the microscopic with macroscopic properties. After a thorough literature and Web search, we were unable to find any exercises that extend the analysis of the HCl vibrational–rotational spectrum in this manner. Theory The theory used in this exercise can be found in most physical chemistry texts. All of the formulas and supporting materials are also provided in extensive detail as part of student instructions in the online material. In light of this, we present only a brief outline of our approach here. To determine bulk thermodynamic properties from microscopic properties one must first consider the partition function for each degree of freedom of the molecule; however, some of these degrees of freedom do not play a role at room temperature. For instance, excited electronic states are unoccupied at room temperature and the electronic partition function for HCl, qelec, is equal to 1. Similarly, at the temperatures considered, the 1670
contribution of the nuclear spin partition function, qnuc, can also be ignored (7, 8). Therefore only the translational, rotational, and vibrational terms need to be considered in the molecular partition function, qtot: qtot qelec qnuc qtrans qrot qvib qtrans qrot qvib (1) While approximations can be used to calculate the rotational and vibrational partition functions, a greater degree of accuracy is obtained by using the full set of spectroscopic constants that the students will have obtained previously in their analysis of IR spectrum data of HCl. These spectroscopic constants include ~νe, the fundamental vibrational constant; ~ ~ νe χe, the first anharmonic correction constant; Be , the rotational ~ , the vibrational–rotational coupling constant; and constant; α e ~ De, the centrifugal distortion constant. Students thus begin this exercise by using their spectroscopic constants to calculate the vibrational–rotational terms, and proceed to calculate the rotational–vibrational and translational partition functions under normal laboratory conditions. When this has been completed, they will be able to determine the internal energy of HCl, Um° − Um°(0); the molar entropy, Sm°; and the molar enthalpy and Gibbs energy, Hm° − Hm°(0) and Gm° − Gm°(0), respectively. Because the standard molar enthalpy and Gibbs energy of formation, Δf H° and Δf G°, respectively, are much more commonly used, these values should be calculated as well. However, to correctly complete these calculations one needs to calculate both Hm° − Hm°(0) and Sm° for 35Cl2 and 1H2 gases. This can be accomplished in a similar fashion to the method used for HCl, but here students should use the literature values for the appropriate spectroscopic constants. One benefit of making these calculations is the opportunity to introduce the role symmetry plays in populating the rotational states of 1H2. Here students can gain a deeper understanding of nuclear spin statistics, which is particularly important for 1H2. To calculate Δf H° and Δf G°, one must account for the difference in the bond energies from the ground states of all three species. This requires the dissociation energy from the ground vibrational state for each substance, D0 (5). For HCl, Δf H° can be calculated by % f H HClp
1 2
NA
Hmp Hmp 0
Hmp Hmp 0
HC Cl
Cl2
Hmp Hmp 0
H2
(2)
1 D0, Cl2 D0, H2 D0, HCl 2
These values can be found in the literature, or in the case of HCl, D0 can be determined from the HCl infrared spectrum.
Journal of Chemical Education • Vol. 85 No. 12 December 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
In the Laboratory
Experiment Students will need individual computer work stations and access to MS-Excel or analogous spreadsheet program to complete this exercise. To provide a suitable framework, students should be required to review their results from their original HCl laboratory, as they will need the spectroscopic constants they determined. Additionally, students will need to use tabulated data for 1H2 and 35Cl2 from Herzberg (9) or another source for some of the calculations required. Once the students have completed their calculations, they can compare their results for the thermodynamic properties with values given in Herzberg (9) or their physical chemistry texts. In principle, students can complete this exercise for all four of the isotopes in the original experiment (1H35Cl, 2H35Cl, 1H37Cl, and 2H37Cl). For the sake of time, we suggest working with only isotope, 1H35Cl, although in larger classes different isotopic molecules can be performed by each group of students providing the class with additional opportunities for comparison and discussion on this method. This laboratory exercise has been tested during the course of a physical chemistry laboratory with three different sets of chemistry majors in their third and fourth years. These students have regularly commented that this exercise has significantly enhanced their understanding and appreciation of the power of statistical thermodynamics. The complete exercise can be accomplished in one three hour laboratory course, although some students may need some extra time outside of the laboratory to complete the full set of calculations required for their laboratory report. Hazards
Results and Discussion A typical set of results from a student’s calculations are reported in Table 1. The result for the molar internal energy, Um° − Um°(0), of HCl, 6160.6 J mol–1, is less than 0.6% lower than (5/2)RT. The typical result for the absolute molar entropy, Sm°(298.15), 186.6 J mol–1 K–1, differs from published values (6) for the standard molar entropy of HCl by 0.15%. As expected, our result for Δf H°, ‒92.10 kJ/mol, shows excellent agreement with the value listed in a typical introductory physical chemistry text (8). The molar Gibbs energy of formation then is calculated and also yields an excellent result, ‒92.72 kJ mol–1, which differs from the literature value (10) by 2.71%. As mentioned previously, the calculations presented in this article are specific to the isotopes, 35Cl and 1H. However, the values for bulk thermodynamic properties presented in physical chemistry texts, as listed in Table 1, often reflect an average value Table 1. Data From a Representative Student’s Report Calculated
Literaturea
Error (%)
6160.6
6197.0b
0.59
S m /(J mol–1 K–1)
186.6
186.9
0.15
ΔfH /(kJ mol–1)
–92.10
–92.31
0.23
–1
–92.72
–95.30
2.71
Quantity [U m – U m (0)]/(J mol–1) o
o
o
o
ΔfG /(kJ mol ) aValues
Conclusions In summary this exercise helps students learn because
• It provides students with an additional opportunity to review the fundamental concepts of rotational–vibrational spectroscopy.
• It bridges spectroscopy and thermodynamics, which are two generally distinct concepts in introductory physical chemistry courses.
• It shows that basic experimental spectroscopic data can be used to determine bulk thermodynamic properties.
• It introduces a more in-depth study of statistical thermodynamics than many undergraduate courses provide.
• It demonstrates the inter-relationship between quantum mechanical states and bulk properties of a gas.
• It deepens students’ understanding of nuclear spin statistics.
Based on discussions and assessments, students find this computer exercise instructive and many have commented that they enjoyed making the connection with their original HCl laboratory exercise. This exercise has been a nice complement to the physical chemistry laboratory curriculum at our institutions. Acknowledgments The authors wish to acknowledge John Winn of Dartmouth College and the anonymous JCE reviewers for their helpful suggestions, which have improved this exercise. Literature Cited
There are no hazards with this experiment.
o
weighted by the natural abundances of the isotopes involved. Students should be expected to understand the sources of these differences and to comment on them.
taken from a typical physical chemistry text (10) and are not specific to 1H35Cl as explained in the text. bHere (5/2)RT is used as a comparison.
1. Garland, C. W.; Nibler, J. W.; Shoemaker, D. P. Experiments in Physical Chemistry, 7th ed.; McGraw-Hill: Boston, 2003; pp 403–411. 2. Halpern, A. M.; McBane, G. C. Experimental Physical Chemistry: A Laboratory Textbook, 3rd ed.; W. H. Freeman and Company: New York, 2006; Chapter 36. 3. Schwenz, R. W.; Polik, W. F. J. Chem. Educ. 1999, 76, 1302–1307. 4. Zielinski, T. J.; Schwenz, R. W. Chem. Educator 2004, 9, 108–121. 5. Tellinghuisen, J. J. Chem. Educ. 2005, 82, 150–156. 6. Francl, M. M. J. Chem. Educ. 2005, 82, 175. 7. McQuarrie, D. Quantum Chemistry, University Science Books: Mill Valley, CA, 1983. 8. Winn, J. Physical Chemistry, Harper Collins College Publishers: New York, 1995. 9. Huber, K. P.; Herzberg, G. Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules; van Nostrand Reinhold Company: New York, 1976. 10. Atkins, P. W.; de Paula, J. Physical Chemistry, 7th ed.; W. H. Freeman: New York, 2002.
Supporting JCE Online Material
http://www.jce.divched.org/Journal/Issues/2008/Dec/abs1670.html Abstract and keywords Full text (PDF) with inks to cited JCE articles Supplement A full set of instructions for students as well as notes and a copy of typical MS-EXCEL results for the instructor
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 12 December 2008 • Journal of Chemical Education
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