Lab1 I2 absorption 182-202001 17 PDF

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EPS/CHEM 182, Lab 1 ELECTRONIC SPECTRUM OF I2 GAS: BAND SPECTRA AND MOLECULAR PARAMETERS (revised 1/17/2020) INTRODUCTION The energy of a molecule is distributed among several degrees of freedom. These are: E total = E translation + ( E electronic + E vibration + E rotation ).

(1)

The translation term is referred to as external energy (related to pressure and temperature), and the three terms in parenthesis are collectively referred to as the internal energy. The internal energy states can be probed by electromagnetic radiation, which causes transitions between them when absorbed or emitted. Diatomic molecules are simple enough that detailed descriptions of the electronic, vibrational, and rotational states can be readily constructed by spectroscopic measurements taken over the appropriate wavelength range. Transitions between the ground and excited electronic states of diatomic molecules involve large amounts of energy (~50 kcal/mole or 17,500 cm-1), which corresponds to absorption or emission of radiation lying in the visible or ultraviolet regions of the spectrum. Pure vibrational and rotational transitions involve much lower energies (in the infrared and microwave regions), but transitions between a given pair of electronic states can involve a wide range of vibrational and rotational states, so information about these states is also present in spectra from electronic transitions. Because of this, the electronic, vibrational, and rotational energies of a diatomic molecule can all be determined from a visible absorption spectrum if the instrument has sufficient spectral resolution. At the resolution used in this experiment, the rotational components of the transitions will not be observed, but there will be sufficient information to develop a detailed description of the electronic and vibrational states of I2. THEORY Absorption spectra are able to yield detailed information about energy states by using models that have been developed to describe electronic, vibrational and rotational degrees of freedom. For atomic absorption spectra, the Bohr model assumes quantized electronic energies and allows calculation of the electronic energy levels of hydrogenic atoms that quantitatively match the observed line spectra. In the same way, models that include quantized electronic, vibrational, and rotational energies are able to describe energy levels that produce the band spectra observed for molecular absorption spectra. In addition to requiring that energy be quantized, molecular models include a hierarchy of energy levels. The molecule has a number of possible electronic states and each electronic state has a set of vibrational sublevels. The energy between vibrational levels is from 10−1 to 10−2 of the energy between electronic levels. Each vibrational level then has a set of rotational sublevels with separations of about 10−2 of the energy between vibrational levels. Figure 1 schematically illustrates these energy levels, where A and B are two electronic states, v is the quantum number of the vibrational sublevels, and J is the quantum number labeling the rotational sublevels. A double prime 1

('') indicates the ground state while a single prime (') indicates the excited state. As a consequence of this energy level scheme, absorption or emission of energy that results in a transition between the A and B electronic states can involve a wide range of vibrational and rotational energy levels. In this experiment we will be dealing with only two electronic states, so we do not require a model to describe how their energies are related. There are, however, a large number of vibrational states. The simplest mathematical model for a describing the energy of a vibrating molecule is the harmonic oscillator, and the potential energy U is given by the equation of a parabola, with k representing the force constant, so that U ( x) = 1 kx 2 (2) 2 Plugging U(x) into the Schrödinger equation gives:

∂ 2ψ 8π 2 µ 1 + ( Ev − kx 2 )ψ = 0 , 2 2 ∂x 2 h

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and the solution of this equation is:

1 E v = ω e (v + ) , v = 0, 1, 2, 3, … (4) 2 The vibrational energy levels, Ev, depend on the vibational quantum number v and are equally spaced. There exists a zero-point energy of 1/2 ωe, where ωe is the harmonic constant expressed in wavenumbers (cm−1). The harmonic oscillator provides a good conceptual model for vibrational energy, but it is a poor model for actual molecular vibrations. Actual potential energy curves correspond to anharmonic vibration, in which energy level spacings decrease at larger internuclear distances and molecular dissociation is possible for large amplitude vibrations. The Morse function is able to approximate the anharmonic potential observed in real molecules with only two parameters, and thus serves as a better model for fitting spectroscopic data. The Morse potential is given by: U ( R) = De (1 − e −β (

R −R ) 2 e

)

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where R is the internuclear separation, De is the depth of the vibrational well and β is a measure of the curvature of the well, related to the force constant ke

ke = 2De β 2

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A Harmonic potential (eq. 2) and Morse potential (eq. 5) with the same force constants are plotted in Figure 2. The Schrödinger equation for the Morse potential can also be solved exactly, giving: Ev = ω e[(v + 1) − x e (v + 1 )2 + ye (v + 1 )3 − . . . ] 2 2 2 Dropping all terms higher than squared leads to: E v = ωe (v + 1 ) − xe ωe (v + 1 ) 2 2 2 The harmonic oscillator term, ωe, and the anharmonic term xe ωe are related to De and β by:

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(8)

1/ 2

 h ω e = β  De  π  2 µc 

2

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and

ω e xe =

hβ 2 8 π 2c µ

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where µ is the reduced mass of the molecule. These expressions can be combined to give: D e=

ω e2 4ω e xe

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The many rotational states of a diatomic molecule can be modeled with the rigid rotor approximation. With this model, each vibrational level has associated with it rotational levels given by E r = BJ ( J + 1) , J = 0, 1, 2, 3, …. (12) where B is a rotational constant. In this experiment the spectrometer will be unable to resolve the individual Er, so they will not be included in the analysis. Taking into account only the electronic and vibrational energy, the energy of electromagnetic radiation ( ω, in cm−1) that causes a transition from the ground electronic state to an excited electronic state is given by:

ω = ω el + E v' − E v'' = ωel + [ωe' (v'+ 1) − ωe' x e' (v'+ 1) 2 ] − [ω 'e' (v' '+ 1 ) − ω e'' xe'' (v' '+ 1 ) 2 ] 2

2

2

(13)

2

Again, the excited electronic state is designated by a prime and the ground state is designated by a double prime. In the transition, there is no restriction on the change in v, i.e., the transition can be from any v'' to v'. ωel is the energy difference between the potential well minima. Figure 3 is a schematic representation of the electronic ground and excited state. If several transtions all originate at the same v’’ and go to different v’, then the separation between successive vibrational energy levels in the excited state (v’ to v’+1) is given by:

ω ' v'→ v'+ 1 = ω e ' − 2(v' + 1)ω e' xe'

(14)

A graph of ω'v'→v'+1 vs. v' is a straight line (known as a Birge Sponer plot). The intercept at v' = 1 is ωe' and the slope is −2 x'e ωe'. The intercept at ω' = 0 is vc' and represents the quantum number where the band spectrum passes from discrete to continuous, i.e., the molecule dissociates. Figure 4 shows a representative Birge-Sponer plot. The thermodynamic dissociation energy is just the summation of all the vibrational energy levels from v = 0 to v = vc: D0 = ∑ E v v

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or the integral under the Birge-Sponer plot: vc D 0 = ∫ ω dV ≈ 1/2 ωe vc 0

3

(16)

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A different heat of dissociation, known as the spectroscopic heat of dissociation, De, can also be defined. It is the height from the minimum of the potential well to the asymptote and is given by: D e = D0 + 1/2 ω e

(17)

Figure 3 is a schematic representation of the ground state and excited state for the I2 molecule and shows the associated parameters. The electronic band origin, ωo→o, represents the transition between the lowest vibrational levels of the electronic states. E* is the energy difference between the lowest ground state vibrational state and the dissociation limit of the excited state. E(I*) is the difference in dissociation limits for the ground and excited states. In this experiment, you will determine the numerical values for these and related properties by analyzing the I2 absorption spectrum and use them to construct a detailed diagram representing the vibrational states for the two lowest electronic states of I2.

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ABBREVIATED PROCEDURE for Chem/EPS 182 In this experiment, you will use the Shimadzu UV-2600 spectrophotometer. Instructions for its use are in Appendix I of this document. Obtain the I2 sample in a sealed cell from your GSI. The constant temperature bath circulates heated water around the cell holder. Make sure there is sufficient water and allow ample time for the sample cell to reach the appropriate temperature before collecting spectra. Instrument settings for all scans are listed below First, to get an overview of the I2 absorption bands, collect a low-resolution spectrum at 90oC. Do not label peaks for the low resolution spectrum. Next, after checking the temperature again, collect a high resolution spectrum for detailed analysis at 90oC. You should be able to resolve about 100 bands. Include peak labels on the spectrum.

INSTRUMENT SETINGS: LOW RESOLUTION SCAN at 90oC Range: 700-400nm Sampling Interval: 0.2 nm Slit width: 0.2 nm HIGH RESOLUTION SCAN at 90oC Range: 700-500nm Sampling Interval: 0.05 nm Slit width: 0.2 nm

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CALCULATIONS 1.

First determine the ground state vibrational level in which each transition originated. The set of bands for transitions arising from the same ground state level to successive levels of the excited state (e.g. v" = 0 → v' = 0, 1, 2, 3, ....) is called a progression. Visual inspection of the spectrum is a good place to begin identifying progressions. The most prominent transitions belong to the v" = 0 progression. At somewhat longer wavelengths, bands from the v" = 1 progression begin to appear and the v" = 0 progression fades out. The v" = 2 progression becomes visible at still longer wavelengths. Assign each band in the high resolution spectrum to a progression. Be sure to read References 2 and 5 (listed below and available on the 182 bspace website under Resources/Lab 1 Materials) for help in making some initial assignments from which to work from.

2.

Create a Deslandres table to assign v' for each band in a progression and verify v" assignments. Detailed descriptions of Deslandres tables are given in References 2 and 5 and an example is shown in Appendix II. Briefly, the energy of each transition in the spectrum is placed in the table at its (v", v') coordinate. Alternating columns and rows are used to list the difference between adjacent v" or v'. When the data are presented in this way, (a) the difference in the frequencies between adjacent columns are approximately constant and (b) the difference in frequencies between adjacent rows vary uniformly (recall how the anharmonic vibrational level spacings are expected to vary). Note that wavelengths are converted into cm−1. For I2, the table in Appendix III is helpful for setting the register of v' for each v" progression. The observed transitions for I2 will not necessarily have v' at or near 0.

3.

From ω''0→1 and ω''1→2 obtain an average value for ω''e assuming x''e = 0.

4.

Create a Birge-Sponer plot by plotting values of ω'v'→v'+1 vs. v' for v'' = 0, 1, 2. Label each of these three progressions (Does each progression fall on the same straight line?) From this BirgeSponer plot, find v’c, ω'e, and ω'ex'e.

5.

Calculate D0' from the integral under the Birge-Sponer plot (note: 1 cm−1 = 2.8589 cal/mol).

6.

In addition to the method in (4), there is another way to determine the parameters of interest for I2. In this different approach, assume x''e = 0 and use eq. 13 to calculate ωel by multiple least squares analysis using the data from your Deslandres table. Choose variables such that ω 'e, ω'ex'e , and ω''e are the coefficients. Use a program (Excel works fine) to do the multiple linear regression analysis by letting ω be the y-variable, with three coefficients (for this capability in Excel you may have to install the Analysis Tool Pack from the original installation disk). See documents on bspace under Resources/Lab 1 Materials for guidance on how to use Excel for a multiple linear regression.

7.

Using your values of ωel, ω 'e, ω'ex'e , and ω''e from either method in (4) or (6) above, calculate β', β'', D0', D0'', De', De'', ke', ke'', ω 0→0 , E*, and the zero point energies for the ground and excited states (don’t forget to include units). Figure 3 may be useful. E* can be determined using v0→0 and Do'. Do'' should be determined from the difference between E* and E(I*). E(I*) is a constant of 7598 cm−1. CALCULATE ERRORS for all values reported here (see Harris text handout under RESOURCES/MULTI-LAB RESOURCES and other resources under Lab 1 for help with 7

error analysis). Compare your calculated values (and your error estimates for them) to literature values reported in Reference 5. Do they match within expected/estimated experimental errors? If they do not, can you think of experimental reasons this might be (problems with the experimental procedure, etc)? Finally, be able to explain the meaning/physical significance of each of the parameters you calculated. 8.

From the literature, Re'' = 2.667 Å and Re' = 3.016 Å. Plot the Morse potential for the ground and excited states of I2 using your data. Set the minimum of the ground state to zero and offset the excited state by ωel.

9.

Calculate the energies for v' = 0, 1, 2, 3, 4, 5 and v'' = 0, 1, 2, 3, 4, 5. Include ωel where appropriate and add them to your Morse plot.

10. Thermal energy can excite vibrational transitions. Calculate the energy for v'' = 0 → v'' = 1 and for v'' = 1 → v'' = 2. Use the Boltzmann factor to predict the probability of these transitions at 25°C and 90°C. How do you predict a spectrum taken at room temperature would be different from that which you collected at 90oC? 11. Look up the rotational constant for I2. What resolution would be required to clearly see the rotational contribution in this absorption spectrum? (Hint: compare the wavelengths of electronic transitions that differ by one rotational level) QUESTIONS 1. 2. 3. 4. 5.

What is the Franck-Condon principle and how does it relate to this experiment? Derive eq. 14. Why does the I2 spectrum obtained consist of bands and not sharp lines? Does the translational energy of I2 have any effect on the absorption spectrum? Explain. How do you think the spectroscopy learned in this laboratory experiment is relevant to the atmosphere?

REFERENCES 1. 2. 3. 4. 5. 6. 7.

Barrow, G. “Molecular Spectroscopy,” McGraw-Hill (1962). CHEM: QC451 .B33 1962 D’alterio, R. et. al. J. Chem. Ed. 51 (4):282-284 (1974) Herzberg, G., “Spectra of Diatomic Molecules,” 2nd ed., Van Nostrand, (1950). CHEM: QC451 .H455 1950 v.1 --in the “classics” section (noncirculating) Lessinger, L., J. Chem. Ed. 71 (5):388-391 (1994) McNaught, I., J. Chem. Ed. 57 (2):101-105 (1980) McQuarrie and Simon. "Physical Chemistry," University Science Books (1997). Snadden, R.B., J. Chem. Ed. 64 (11):919-921 (1987).

Revised Spring 2014 (2/8/2014)

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APPENDIX I: OPERATION INSTRUCTIONS FOR THE SHIMADZU UV-2600 UV-VIS SPECTROPHOTOMETER

Power Up: 1. Turn on the spectrophotometer. Facing the front of the instrument, the power switch is on the right side towards the front. 2. Turn on water bath and make sure water is circulating. Check that the temperature is set to something reasonable for your first measurement. 3. Open the UVProbe 2.42 program from the desktop. 4. Make sure you are in “Spectrum” mode. The program header should read “UVProbe – [Spectrum].” If you are in Kinetics or Photometric mode, click on the prism icon in the top toolbar to enter spectrum mode. 5. Connect to the instrument by selecting the “Connect” icon near the bottom right of the window. When setup tests are passed, click “OK.” Setting up a Measurement: 1. Click Edit > Method to setup a Measurement Method. 2. In the “Measurement” tab, set the following parameters to change the resolution of your scans: a. Wavelength Range (nm): 400-700 b. Scan Speed: Slow-Fast c. Sampling Interval (nm): 0.05-0.2 3. In the “Instrument Parameters” tab, set the following parameters: a. Measuring Mode: Absorbance b. Slit Width: 0.2 4. Click “OK.” 5. Each time you change the method, you must correct your baseline. With no sample in the instrument, click the “Baseline” icon in the instrument control panel at the bottom of the screen. When the wavelength selection window appears, make sure the scan range matches that which you set in the Measurement Method and click “OK.” You can view the correction scan’s progress in the white window near the bottom of the screen. It displays the current wavelength and absorbance values and will return to the start wavelength when the correction is complete. 6. Use a Kimwipe to clean the transparent sides of the cuvette and open the lid of the instrument. Take off the cap of the front slot of the spectrophotometer and insert your sample. Make sure the cuvette is orientated such that the light beam passes through the transparent sides. Replace the cap for the cuvette holder and close the lid of the instrument. Wait 1 to 2 minutes for the I2 to heat up. 7. Click the “Start” icon in the instrument control panel. You can monitor the progress of your scan in the “Overlay” window. When the scan is complete the “New Data Set” window appears. Enter a new filename, and click OK.

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Saving and Printing your Data: 1.

2. 3.

4.

5.

6. 7.

To adjust the scales of the graph such that the spectrum fills the plot area, right click within the plot area and click “Customize.” You can adjust the X- and Y-axis scales under the “Limits” tab. To save spectrum file, click File > Save As. Save as type: Spectrum File (*.spc) For the high resolution scans, you need to label your peaks. Click Operations > Peak Pick to bring up the peak picking window. A table of the peak values is displayed. To label them on the graph, right click within the table and click “Properties.” In the “Peaks” tab, select only X-value in the Labels box. Right click within the plot area and click “Copy > Picture.” You can paste this into Paint or WordPad and save it. Be sure to choose your limits so that you can see all your peak labels. You should also save the raw data for your scans. To do this, click Operations > Data Print to bring up the raw data table. Right click on the table and select Properties. Under the General tab, change the table parameters to match the experimental parameters (start, end, interval). Click File > Save As. Save as type: Data Print Table (*.txt). If you would like to save the peak pick table, repeat step 5 for this data (Operations > Peak Pick to bring up the peaks window). Save as type: Peak Pick Table (*.txt). You should save everything to a flash drive or e-mail it to yourself or your lab partner for analysis after your lab session. NOTE THAT THIS LAB REQUIRES VERY EXTENSIVE AND VERY TIME-CONSUMING DATA ANALYSIS, SO TRY TO START YOUR DATA ANALYSIS IMMEDIATELY AFTER LAB!

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