In the Laboratory
High-Resolution Vibration–Rotation Spectroscopy of CO2: Understanding the Boltzmann Distribution
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Karen J. Castle Department of Chemistry, Bucknell University, Lewisburg, PA 17837;
[email protected] In this experiment, a small segment of the infrared absorption spectrum of carbon dioxide is used to explore concepts such as the Boltzmann distribution, rotational and vibrational energy levels, normal modes of vibration, spectral lineshape, and isotope effects. Students record a highresolution absorption spectrum of CO 2 and use it to determine whether the rotational–vibrational populations follow a Boltzmann distribution. Students then heat the sample cell and observe how the increased temperature changes the relative state populations. Statistical mechanics is a particularly difficult area of chemistry for undergraduates to grasp. Many are never exposed to the concepts in the laboratory where learning new concepts is often easier. This experiment, which can be tailored to fit into either a three- or four-hour time slot, aims to incorporate some of these difficult topics into the undergraduate laboratory curriculum. The experiment detailed in this document differs from previously published experiments in several significant ways. Sulkes and Osburn report an experiment designed to verify the Boltzmann distribution through infrared spectroscopy and spectral simulation of HCl and I2 (1). Shoemaker et al. detail a statistical thermodynamic investigation of the I2 sublimation using UV–vis spectroscopy (2). More recently, Francl developed a Mathematica notebook to help students explore the Boltzmann distribution computationally (3). However, all of these experiments use diatomic molecules as examples. The present experiment focuses on a polyatomic molecule in order to incorporate more complex discussions of normal modes of vibration and mode degeneracy. Tennis et al. report an experiment where statistical mechanics is used in combination with FTIR and Raman spectra of methane to determine heat capacity and the speed of sound (4). The Tennis et al. experiment does not specifically test the Boltzmann distribution, but rather focuses on determining thermodynamic properties from experimentally obtained vibrational frequencies. A unique feature of the present experiment is that it not only allows students to see what the Boltzmann distribution means in terms of relative state populations, but it also allows them to directly observe changes in these populations when the temperature of the reaction cell is changed. Lasers have become increasingly important tools in chemistry research, and the need to expose undergraduates to this instrumentation has been repeatedly stressed (5, 6). This experiment will serve to expose students to sophisticated, high-resolution spectroscopic instrumentation. Many universities will already have an appropriate laser source for this experiment in their chemistry or physics department. The data presented here were obtained with a tunable lead-salt diode laser, although the procedure could be easily modified for use with another type of tunable infrared laser such as a quantum cascade laser, a frequency-doubled CO2 laser, or various other tunable lasers in combination with frequency www.JCE.DivCHED.org
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doubling or mixing crystals. Lead-salt diode lasers produce beautiful results because of their high resolution (better than 0.005 cm᎑1). The experiment may be modified for use with a high-resolution FTIR spectrometer, though the results will be somewhat less impressive. Because research-grade FTIR spectrometers are typically capable of 0.1 cm᎑1 resolution, only some rotational–vibrational states will be resolvable. This experiment was specifically designed to expose students to commercial spectral simulation and spectral fitting software, which is either freely available or already present at most institutions. This will give practical experience to students pursuing careers in chemistry, since these software packages are typically used by researchers in the fields of analytical, physical, and atmospheric chemistry. However, instructors who do not wish to use commercial software packages may easily adapt the experiment to avoid this exposure. Introduction The ν3 (asymmetric stretch) absorption bands of CO2 in the 2300 cm᎑1 spectral region are very intense and easily detected. The relative population of a rotational–vibrational state, Nrel, is proportional to absorbance through the following relationship Nrel ∝
A int gvib g rot ia
(1)
where Aint is the experimentally measured integrated absorbance of a given transition, gvib is the vibrational degeneracy of the initial state, grot is the rotational degeneracy of the initial state, and ia is the fractional isotopic abundance. The vibrational degeneracy is one for all vibrational states except for the bend-excited states, which have a vibrational degeneracy of two. Rotational degeneracy is calculated by grot = 2 J + 1, where J is the rotational quantum number. If a system follows the Boltzmann distribution, Nrel depends on sample temperature according to the following expression Nrel = c e
−
E lower kBT
(2)
where c is a constant, Elower is the energy of the initial rotational–vibrational state (in cm᎑1), kB is the Boltzmann constant (0.695 cm᎑1兾K), and T is the sample temperature (in K). If the populations of several states are measured, the sample temperature can easily be obtained by plotting ln(Nrel) versus Elower, yielding a straight line with slope ᎑1兾kBT. Experimental Detailed information regarding the experimental setup used to collect the data presented in this document can be found in the Supplemental Material.W Briefly, CO2 ν3 ab-
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In the Laboratory
Figure 1. CO2 absorption spectrum obtained by a student using a lead-salt diode laser and InSb detector. Vibrational and rotational states involved in some of the transitions are labeled. The absence of an asterisk indicates a transition of the 16O 12C16O isotope (0.984294 relative abundance), a single asterisk the 16O13C16O isotope (0.011057 relative abundance), a double asterisk the 18O12C16O isotope (0.003947 relative abundance), and a triple asterisk the 17O12C16O isotope (0.000734 relative abundance) (7).
Figure 3. Student plots of ln(Nrel) vs Elower for two sample cell temperatures. The vertical scales are offset and labeled on separate axes for clarity. If the system follows a Boltzmann distribution, the slope of the best fit line equals ᎑1兾kBT, allowing determination of the sample temperature, Texp. Uncertainties in Texp at the 95% confidence limit are given based on the uncertainties in slopes of the weighted linear regressions. Thermocouple readings taken at the time of data collection, Tact, are given for comparison.
Relative rotational–vibrational state absorbances may be determined using either absorption peak heights or integrated peak areas. Integrated peak areas are a slightly better reflection of the relative populations since peak widths vary from state to state. The data in this document were fit to Voigt lineshapes using PeakFit v4.12 (9). This software was then used to determine integrated peak areas and associated uncertainties. Hazards Figure 2. A magnified segment of the CO2 absorption spectrum presented in Figure 1. The circles represent student data points while the solid line is the best fit to Voigt lineshapes obtained with PeakFit v.4.12 (9).
sorption spectra over a 2 cm᎑1 spectral region were scanned with a diode laser at sample temperatures of 301 and 332 K. The resulting absorption peaks were carefully assigned to the correct rotational–vibrational transitions for proper data analysis. Comparison of experimental data with the predictions of the high-resolution transmission molecular absorption (HITRAN) database is a convenient way for students to make their own peak assignments (7). The HITRAN database is a collection of spectroscopic parameters on the infrared properties of atmospherically-relevant molecules and is widely used by the atmospheric science community. HITRAN is used in combination with the Java-encoded HITRAN Atmospheric Workstation (JHAWKS) software to simulate infrared emission spectra. HITRAN and JHAWKS are available free of charge and may be requested through the HITRAN Web site (8). 460
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Diode lasers typically produce only a few milliwatts of radiation, and the mid-infrared photons are nearly all absorbed by plastic laboratory safety glasses. However, directly pointing any type of laser at the eyes should be avoided, especially for a prolonged period of time. The experimental setup should be maintained well below eye level to minimize accidental exposure. Caution should be used when working with high pressure gas cylinders. Results and Discussion A room temperature absorption spectrum of CO2 obtained with a tunable, lead-salt diode laser (Laser Components) and liquid N2 cooled InSb infrared detector (InfraRed Associates, Inc.) is shown in Figure 1. A sample pressure of about 200 millitorr was used in a 1-m gas cell. This spectrum was obtained with 64x averaging by an oscilloscope and contains 10,000 data points. Note that in this small spectral range four isotopes of CO 2 , ( 16 O 12 C 16 O, 16 O 13 C 16 O, 18 12 16 O C O, and 17O12C16O) and six initial vibrational states [(0000), (0110), (1000), (0220), (1110), and (0330)] were easily detected. Figure 2 shows a magnified region of the spectrum along with the fit to Voigt lineshapes.
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In the Laboratory
Table 1. Integrated Peak Areas, Uncertainties, and Lower Rotational–Vibrational State Energies for the Spectral Transition Shown in Figure 1. Peak Area /10᎑4 cm2
Transition Energy/cm᎑1
Assignment
2300.815
*(0110) J=44 → (0111) J=45
2300.874
(0220) J=27 → (0221) J=26
2300.999
(0330) J=13 → (0331) J=12
2301.031
**(00 0) J=37 → (00 1) J=36
2301.064
*(00 0) J=24 → (00 1) J=25
2301.096
***(00 0) J=44 → (00 1) J=43
2.05 (± 0.04)
2.51 (± 0.04)
0749.161
2301.146
(1110) J=17 → (1111) J=16
4.22 (± 0.04)
9.12 (± 0.04)
2052.024
2301.428
(10 0) J=30 → (10 1) J=29
2301.681
(01 0) J=39 → (01 1) J=38
2301.814
(02 0) J=26 → (02 1) J=25
2301.855
(03 0) J=12 → (03 1) J=11
2301.974
**(0000) J=36 → (0001) J=35
2302.107
***(0000) J=43 → (0001) J=42
2.09 (± 0.04)
2.45 (± 0.04)
0715.886
2302.253
(11 0) J=14 → (11 1) J=13
2.57 (± 0.04)
5.52 (± 0.06)
2159.031
2302.309
*(00 0) J=26 → (00 1) J=27
2302.394
(1000) J=28 → (1001) J=27
300 K
0
2.87 (± 0.04)
88.2 (± 0.5)
2.98 (± 0.04)
4.82 (± 0.04)
31.4 (± 0.5)
1
0
449.(± 4)
50.1 (± 0.5)
3
1
58.4 (± 0.5)
350.(± 3)
2
3
155.(± 3)
31.7 (± 0.4)
1
2
41.3 (± 0.4)
183.(± 3)
0
1
6.57 (± 0.04)
36.7 (± 0.4)
0
0
92.7 (± 0.5)
3.11 (± 0.04)
0
0
4.51 (± 0.04)
52.8 (± 0.4)
0
0
330 K
35.1 (± 0.6)
184.(± 2)
0
162.(± 2)
22.2 (± 0.4)
41.2 (± 0.5)
Elower/cm᎑1 1422.609 1631.153 2074.654 0517.433 0234.095
1648.421 1276.448 1610.014 2064.454 0490.211
0273.881 1704.942
NOTE: These integrated peak areas were used to generate the data presented in Figure 3.
Relative initial state populations for each rotational– vibrational transition were calculated from the absorption spectrum in Figure 1. Integrated peak areas were used as a measure of relative absorbance and may be found in Table 1. The values of Nrel were calculated from these peak areas using eq 1. Student plots of ln(Nrel) versus E lower for two sample temperatures are shown in Figure 3. The values of Elower were taken directly from the HITRAN database and error bars were derived from the uncertainties in individual integrated peak areas. SigmaPlot v. 9.1 was used to perform a weighted linear regression to determine the slope of ln(Nrel) versus E lower while accounting for the varying sizes of the error bars in ln(Nrel) (10). From the spectral data, the room temperature sample was measured (Texp) at 303 ± 3 K. The ambient laboratory temperature, as measured with calibrated thermocouples (Tact), was 301.0 ± 0.2 K. The same type of analysis was performed on a sample at an elevated temperature. From spectral data, the temperature was measured to be 334 ± 5 K, compared with the thermocouple readings of 331.6 ± 0.2 K. Conclusion This laboratory experiment is designed to give students experience with topics in statistical mechanics, the HITRAN spectral simulation database, commercial spectral fitting software, and high-resolution infrared instrumentation. Student results of the quality presented in this article are typical when using a diode laser for data collection. A larger number of transitions can be measured if a decrease in uncertainty of the derived temperature is desired. www.JCE.DivCHED.org
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Supplemental Material
Instructions for the students and notes for the instructor are available in this issue of JCE Online. Literature Cited 1. Sulkes, Mark; Osburn, Jean E. J. Chem. Educ. 1987, 64, 720. 2. Shoemaker, David P.; Garland, Carl W.; Nibler, Joseph W. Experiments in Physical Chemistry, 6th ed.; McGraw-Hill: Boston, 1996; pp 514–528. 3. Francl, Michelle M. J. Chem. Educ. 2005, 82, 175. 4. Tennis, R.; Bailey, R.; Henderson, G. J. Chem. Educ. 2000, 77, 1634. 5. Steehler, J. K. J. Chem. Educ. 1990, 67, A37. 6. American Chemical Society. Undergraduate Professional Education in Chemistry: Topical Supplement to Guidelines. http:/ /www.chemistry.org/portal/resources/ACS/ACSContent/education/ cpt/ts_physical.pdf (accessed Dec 2006). 7. Rothman, L. S.; Barbe, A.; Benner, D. C.; Brown, L. R.; Camy-Peyret, C.; Carleer, M. R.; Chance, K.; Clerbaux, C.; Dana, V.; Devi, V. M.; Fayt, A.; Flaud, J.-M.; Gamache, R. R.; Goldman, A.; Jacquemart, D.; Jucks, K. W.; Lafferty, W. J.; Mandin, J.-Y.; Massie, S. T.; Nemtchinov, V.; Newnham, D. A.; Perrin, A.; Rinsland, C. P.; Schroeder, J.; Smith, K. M.; Smith, M. A. H.; Tang, K.; Toth, R. A.; Vander Auwera, J.; Varanasi, P.; Yoshino, K. J. Quant. Spectrosc. Radiat. Transfer 2003, 82, 5–44. 8. HITRAN Database Home Page. http://cfa-www.harvard.edu/ HITRAN (accessed Dec 2006). 9. SeaSolve Software, Inc. PeakFit, version 4.12,2003. 10. Systat Software, Inc. SigmaPlot, version 9.01, 2004.
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