Laboratory Experiment pubs.acs.org/jchemeduc
A Molecular Iodine Spectral Data Set for Rovibronic Analysis J. Charles Williamson,*,† Thomas S. Kuntzleman,‡ and Rachael A. Kafader‡ †
Chemistry Department, Willamette University, Salem, Oregon 97301, United States Department of Chemistry, Spring Arbor University, Spring Arbor, Michigan 49283, United States
‡
S Supporting Information *
ABSTRACT: A data set of 7,381 molecular iodine vapor rovibronic transitions between the X and B electronic states has been prepared for an advanced undergraduate spectroscopic analysis project. Students apply standard theoretical techniques to these data and determine the values of three X-state constants (B̃ ″e , ν̃″e , D̃ ″e ) and four B-state constants (B̃ ′e, ν̃′e, D̃ ′e, T̃ ′e). Morse potentials for the two states may be calculated from these seven constants.
KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Laboratory Instruction, Computer-Based Learning, Hands-On Learning Manipulatives, Iodine, UV-VIS Spectroscopy, Quantum Chemistry
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determined could be compared with literature values. Disadvantages of this assignment included the time each student needed to pick out specific transitions from a spectral atlas comprised of many overlapping vibronic bands. There were also fundamental limitations on the accuracy to which some spectroscopic constants could be determined. For example, since the atlas is composed of only absorption data from thermally populated vibrational states, the value of the X-state electronic well depth D̃ ″e found using the atlas data was significantly and unavoidably overestimated.9 To improve upon this assignment, we have combined results from published high-resolution absorption12 and fluorescence13 experiments to create a data set of molecular iodine rovibronic transitions that is well suited to an undergraduate analysis project. Students evaluate these data to determine seven spectroscopic constants with good accuracy: the X-state and B-state fundamental vibrational energies ν̃″e and ν̃′e; the rotational constants B̃ e″ and B̃ e′; the electronic well depths D̃ e″ and D̃ e′; and the electronic energy offset T̃ e′ to the bottom of the B-state potential well.14 Together, these seven constants allow students to calculate Morse potential surfaces for both states. Other quantities like anharmonic correction terms and vibration−rotation coupling constants are also found during the analysis, but with less accuracy.
he acquisition and analysis of gaseous molecular iodine 3 + Π0u B ↔ 1∑+g X absorption and emission spectra are staples of the undergraduate physical chemistry laboratory curriculum. This is for good reason, because a comprehensive study of I2 vapor spectroscopy can be a capstone experience in physical chemistry. A complete investigation of the spectra of I2 vapor requires students to be proficient in quantum topics such as absorption and emission spectroscopy; electronic, vibrational, and rotational transitions; the rigid rotor; the harmonic oscillator; and Morse potential energy surfaces. Of course anharmonicity, centrifugal distortion, and vibrational−rotational coupling also play a role in a complete analysis. Since the introduction of molecular iodine spectroscopy into the pedagogical literature,1 a variety of work has been done to improve upon various aspects of its presentation. For example, several experiments and lessons have been implemented to refine and expand analysis,2−5 increase student conceptual understanding,6 streamline data treatment,7,8 clear up confusion surrounding interpretation of results,9 and probe the system in new and less expensive ways.9,10 However, these innovations do not help chemistry departments that lack the experimental resources necessary for recording I2 vapor spectra at a resolution sufficient for vibronic analysis. As an alternative project, students at Spring Arbor University have been guided in a dry lab molecular iodine experiment. Each student searched an online iodine spectral absorption atlas11 to find multiple transitions belonging to one of several vibronic bands, and then class data were pooled for spectroscopic analysis of both the X and B states. In this way, students had the opportunity to apply quantum mechanical theory to real spectral data, and the spectroscopic constants they © 2013 American Chemical Society and Division of Chemical Education, Inc.
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GENERATION OF THE SPECTRAL DATA SET Rovibronic molecular iodine transitions for the data set were calculated from spectroscopic constants determined by Gerstenkorn and Luc12 and Martin et al.13 Both groups of Published: January 24, 2013 383
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Laboratory Experiment
authors report agreement between experimental and calculated transitions at 0.005 cm−1 or better, so the transition energies prepared for this work are listed to the nearest 0.001 cm−1. A total of 121 vibronic bands were generated, spanning a twodimensional array of vibrational quantum numbers v″ and v′ in steps of five on each coordinate (v″ = 0−60; v′ = 0−50). However, 22 vibronic bands in this array were omitted to reflect poor Franck−Condon overlap as predicted by Martin et al.13 Every included vibronic band was composed of 61 rovibronic transitions: 30 P-branch lines with X state rotational quantum number J″ = 1−30, and 31 R-branch lines with J″ = 0−30. The complete data set of 7,381 transitions is available as a spreadsheet in the Supporting Information.
LINEST function in Excel.8 Further details on both methods are described in the Supporting Information. The electronic well depths D̃ e″ and D̃ e′ are estimated to occur at the local maxima of the vibrational term values, G̃ ″(vmax ″ ) and G̃ ′(v′max). For a cubic representation of G̃ , the value of vmax is given by:
OVERVIEW OF STUDENT ANALYSIS Full details on the theoretical treatment of the molecular iodine data set are presented in the Notes for Instructors in the Supporting Information. Briefly, the total energy Ẽ of molecular iodine may be treated as a sum of electronic, vibrational, and rotational contributions: E ̃ = Tẽ + G̃ (v) + F (̃ v , J ) (1)
Table 1. Molecular Iodine X-State and B-State Potential Energy Parameters Determined by Fitting the Full Data Set of 121 Vibronic Bands and a Partial Data Set of 32 Bands
vmax =
1−
3νẽ (νẽ ye ) ⎞ 1 ⎟− 2 (νẽ xe)2 ⎟⎠
(7)
Most likely vmax will not be an integer. Table 1 presents the values of the seven constants B̃ ″e , ν̃″e , D̃ ″e , B̃ e′, ν̃e′, D̃ e′, and T̃ e′ determined using the full spectral data set of 121
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where G̃ and F̃ are the vibrational and rotational term values, respectively. The photon energy of a specific rovibronic transition between the B and X states is equal to: ̃ = E′̃ − E″̃ = T̃ ′e + G̃′(v′) − G̃″(v″) + F ′̃ (v′, J ′) Eph − F ″̃ (v″ , J ″)
⎛ νẽ xe ⎜ 1− 3νẽ ye ⎜⎝
(2)
a
Note that T̃ ″e = 0 since X is the electronic ground state. Students begin their treatment of the data by analyzing the rotational band structure of various vibronic transitions selected from the data set. Neglecting centrifugal distortion effects, the transition energies within a specific vibronic band are given by:
Quantity
Literaturea
B̃ ″e /cm−1 ν̃e″/cm−1 D̃ e″/cm−1
0.037368 214.53 1.25473 × 104
B̃ ′e/cm−1 ν̃e′/cm−1 D̃ e′/cm−1 T̃ ′e/cm−1
0.029001 125.67 4381.2 15769.1
Full Data Set (121 Bands) X State 0.037352 212.94 1.275 × 104 B State 0.028960 127.3 4208 15760.
Partial Data Set (32 Bands) 0.037349 212.9 1.275 × 104 0.028972 127.7 4221 15759
Refs 12, 13, 16−18.
vibronic bands, and also using a 32-band subset representative of what students might have time to analyze in one or two lab sessions. With these seven constants, students have the information they need to plot Morse potential representations of the B and X states.9 Figure 1 shows the Morse potentials
̃ = ν0̃ (v′, v″) + B′̃ (v′)J ′(J ′ + 1) − B″̃ (v″)J ″(J ″ + 1) Eph (3)
where ν̃0 is the vibronic band origin, the energy difference when J′ = J″ = 0: ν̃0(v′, v″) = T̃ ′e + G̃ ′(v′) − G̃″(v″)
(4)
Students find the rotational constants and ν̃0 for each vibronic band using a Fortrat analysis (see ref 15 and Notes for Instructors in the Supporting Information). Data are then pooled. The rotational constant B̃ e′ is determined by fitting the B̃ ′(v′) values found from the pooled data using a polynomial series expansion in v′ + 1/2: ⎛ ⎛ 1⎞ 1 ⎞2 B′̃ (v′) = Be′̃ − αẽ′⎜v′ + ⎟ + γe′̃ ⎜v′ + ⎟ ⎝ ⎝ 2⎠ 2⎠
(5)
Figure 1. X-state and B-state potential energy surfaces for molecular iodine: (solid lines) curves from literature sources13,18 and (dashed lines) Morse potentials calculated using constants determined from the full data set of 121 vibronic bands.
In eq 5, α̃′e and γ̃′e are vibration−rotation coupling constants. The variation of B̃ ′(v′) with v′ is substantial, and v′ should be limited to a maximum of 30 in this fit for reasons discussed in the Notes for Instructors. An identical type of analysis is done to determine B̃ e″. The three constants ν̃″e , ν̃′e, and T̃ ′e are found by fitting the pooled ν̃0 values (the left side of eq 4) as a function of v′ + 1/2 and v″ + 1/2 using cubic representations of the vibrational term values: ⎛ ⎛ ⎛ 1⎞ 1 ⎞2 1 ⎞3 G̃ (v) = νẽ ⎜v + ⎟ − νẽ xe⎜v + ⎟ + νẽ ye ⎜v + ⎟ ⎝ ⎝ ⎝ 2⎠ 2⎠ 2⎠
calculated from the constants in Table 1 found using the full data set. Quantitatively accurate potential energy surfaces derived from literature data are also shown in Figure 1 for comparison.
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SUMMARY This exercise provides students with the opportunity to conduct a comprehensive rovibronic analysis of the 3Π+0u B ↔ 1∑+g X transitions of molecular iodine vapor using a single data set.
(6)
This calculation may be executed either with software capable of surface fitting or by carrying out a least-squares minimization using the 384
dx.doi.org/10.1021/ed300455n | J. Chem. Educ. 2013, 90, 383−385
Journal of Chemical Education
Laboratory Experiment
Although designed for a standalone project, the data set may also be used in conjunction with actual spectroscopy. For example, if students have access to instrumentation for recording the I2 vapor absorption spectrum but not a fluorescence spectrum, then students may use these data to assess the experimental values of B-state constants they do find, and to calculate X-state constants they do not measure. Analysis of the data set might also be assigned as homework or a take-home examination.
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ASSOCIATED CONTENT
S Supporting Information *
Molecular iodine spectral data set; notes for instructors; representative analysis; student guidelines for data analysis. This material is available via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank the many students of physical chemistry at Spring Arbor University who tested various trial versions of this exercise. REFERENCES
(1) Davies, M. J. Chem. Educ. 1951, 28, 474−477. (2) Stafford, F. E. J. Chem. Educ. 1962, 39, 626−629. (3) D’alterio, R.; Mattson, R.; Harris, R. J. Chem. Educ. 1974, 51, 282− 284. (4) McNaught, I. J. J. Chem. Educ. 1980, 57, 101−105. (5) Snadden, R. B. J. Chem. Educ. 1987, 64, 919−921. (6) Long, G.; Sauder, D.; Shalhoub, G. M.; Stout, R.; Towns, M. H.; Zielinski, T. J. J. Chem. Educ. 1999, 76, 841−847. (7) Pursell, C. J.; Doezema, L. J. Chem. Educ. 1999, 76, 839−841. (8) Cooper, P. D. J. Chem. Educ. 2010, 87, 687−690. Note that hot band transition assignments in the visible absorption spectrum of iodine are more complex than as described in this reference. Additional details can be found in the Online Supplemental Information accompanying Ref 9. (9) Williamson, J. C. J. Chem. Educ. 2007, 84, 1355−1359. (10) Williamson, J. C. J. Chem. Educ. 2011, 88, 816−818. (11) Gerstenkorn, S.; Luc, P. Atlas du Spectre d’Absorption de l’Iode 14 800 - 20 000 cm−1. http://www.lac.u-psud.fr/Atlas-du-spectre-dabsorption-de-l (accessed Jan 2013). (12) Gerstenkorn, S.; Luc, P. J. Phys. (Paris) 1985, 46, 867−881. (13) Martin, F.; Bacis, R.; Churassy, S.; Vergès, J. J. Mol. Spectrosc. 1986, 116, 71−100. (14) Throughout this text, single primes will be placed on symbols corresponding to the B state, whereas double primes will be placed on symbols corresponding to the X state. (15) Herzberg, G. Molecular Spectra and Molecular Structure. Vol. 1: Spectra of Diatomic Molecules; 2nd ed.; Van Nostrand: Princeton, 1950; pp 437−443. (16) Gerstenkorn, S.; Luc, P. Laser Chem. 1983, 1, 83−112. (17) Tromp, J. W.; Le Roy, R. J.; Gerstenkorn, S.; Luc, P. J. Mol. Spectrosc. 1983, 100, 82−94. (18) Barrow, R. F.; Yee, K. K. J. Chem. Soc., Faraday Trans. 2 1973, 69, 684−700.
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