8) Rotation-Vibration Spectrum of HCl and DCl

A) ABSTRACT1

In this experiment infrared spectroscopy is used to obtain a rotation-vibration spectrum of the diatomic molecule HCl as well as of its isotopes. Through an analysis of this spectrum, information on the molecular structure and the vibrational potential function are obtained.

B) THEORY

The simplest model for dealing with rotation-vibration spectra of diatomic molecules is the harmonic oscillator-rigid rotor model. To a fair level of approximation, the potential energy function of nuclear motion for a diatomic molecule may be approximated by a quadratic function in the vicinity of its minimum. The harmonic force constant for vibration relative to the molecule's equilibrium internuclear distance is the second derivative of this potential energy function evaluated at its minimum

1 Updated version of the lab handout. We included numerous figures, notes about how to use the data acquisition program, and added more hints to the “problems part” at the end of the draft. You should now be able to write a decent lab report without a “master copy” of prior classes. U.Bgh-

k = [d2V(r)/dr2]r=re.

The classical vibrational frequency of a harmonic oscillator with this force constant is given by

ν = [1/(2π)] (k/µ)1/2 with µ = m1m2/(m1 + m2).

with µ for the reduced mass. The allowed quantum mechanical energy levels of the diatomic harmonic oscillator are given by

E(n) = hν(n + 1/2) with n = 0,1,2,3,...

The simplest model for a rotating diatomic molecule is the rigid rotor model in which one considers the two point masses to be joined by a rigid weightless bond. The allowed rotational energy levels for such a rigid rotor are

E(J) = [h2/(8π2I)] J (J+1) with J = 0,1,2,3,....

where I is the moment of inertia of the molecule. It is related to the reduced mass and the internuclear distance by the equation: I = µr2. For a real molecule which is undergoing both vibration and rotation simultaneously, the situation is somewhat more complicated. A more complete expression for the energy levels of a molecular vib-rotor is given below with the energy levels expressed as term values T in cm-1 units rather than as energies in ergs:

T(n,J) = E(n,J)/hc = ν~e (n+1/2) - χeν~

e (n+1/2)2 + BeJ(J+1)-DeJ

2(J+1)2 - αe(n+1/2)J(J+1).

The ~ symbols over the νe symbols signify

that units are in wavenumbers (cm-1). We will not derive this equation, as it involves

complicated perturbation theory calculations. It is important, however, to understand the physical significance of each term:

The first and third terms correspond to the harmonic oscillator-rigid rotor approximation. The quantity Be (sometimes called B) is the rotational constant, and is related to the moment of inertia I of the molecule by:

Be = h / (8π2Ic).

Note that this formula gives Be in wave number units; to get true energy units, multiply by hc.

The second term is the anharmonicity correction; this accounts for the fact that a true potential for nuclear motion is not a perfect harmonic oscillator. The anharmonicity constant is, using cumbersome but standard notation, called χeν

~e. This constant is usually positive; the

result is that vibrational energy levels are not equally spaced but rather converge with increasing vibrational quantum number n, as shown in Figure 1.

The fourth term with the constant De is called the centrifugal distortion correction. This takes into account the fact that the bond stretches as the rotation rate is increasing. The centrifugal distortion constant is usually quite small, and this term may be ignored in your data analysis. The fifth and last term is the vibration-rotation interaction term. This accounts for changes in the moment of inertia of the molecule as it vibrates.

The selection rules for the harmonic oscillator and the rigid rotor are ∆n=±1 and ∆J=±1, respectively. For an anharmonic oscillator the ∆J=±1 selection rule is still valid, but transitions corresponding to ∆n=±2, ±3,... are also weakly allowed; these are called overtone bands. Since we are interested in analyzing only the most intense absorption band (the fundamental), we will only be concerned with transitions from J" levels of the vibrational ground state (n"=0) to J' levels in the first excited vibrational state (n'=1). From the selection rules we know that the transitions must be

Fig. 1 Vibrational energy levels.

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Fig. 3: Notation for R and P branches. Note that the lines for

larger ν~ values are assigned to positive m values. The plots obtained in the experiment might be the other way around. Use the notation of the figure since the equations discussed here apply that notation.

Fig.4: The gas cell

from J" to J'=J"±1. The frequencies of these transitions are given by the difference in the terms, T(n',J')-T(n",J"). When ∆J=+1, (J'=J"+1), and ∆J= -1, (J'=J"-1), we find, respectively, that

ν~R = ν~

0 + (2Be - 3αe) + (2Be - 4αe)J" -

αeJ"2 J"=0,1,2,...

ν~P = ν~

0 - (2Be - 2αe)J" - αeJ"2

J"=1,2,3,... where ν~0, the frequency of the forbidden transition from n"=0, J"=0 to n'=1, J'=0, is: ν~0 = ν

~e - 2 χeν

~e. We have ignored

anharmonic and centrifugal distortion corrections in the derivation of these equations, but have retained the vibration-rotation interaction term.

The two series of lines are called R and P branches, respectively. These transitions are indicated in Figure 2. It is convenient to introduce a new quantity m, where m = J"+1 for R branch transitions and m = -J" for P branch transitions as shown in Figure 3. It is then possible to combine the two equations for P and R branches into one equation:

ν~e = ν~

0 + (2Be - 2αe)m - αem2

where m takes all integer values except zero. The separation between adjacent lines is given by:

∆ν~(m) = ν~(m+1) - ν~(m) = (2Be - 3αe) - 2αem.

Note that a plot of ∆ν~(m) versus m can be used to determine both Be and αe.

B) Experimental

C1 Filling the infrared gas cell. Depending on the setting of the measuring program you may need to record first a background spectrum, i.e., you may need an empty gas cell first. • The infrared gas cell has NaCl

windows and must be handled with care (Fig. 4). Never touch the windows with your fingers, and never allow the windows to come into contact with water. A certain amount of fogging due

to room humidity is inevitable, but will not severely reduce infrared

Fig. 2: R and P branches.

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Fig. 5: Filling the gas cell.

transmission. The vacuum line you will be using is also very fragile, and is easily broken. Always turn the valves slowly, using two hands. Do not force them or they will break. If you need help, ask the T.A.

• The manifold pressure may be read by the Baratron capacitance manometer; it is calibrated for full scale (10 volts) equals 1000 torr. You need roughly 100 torr of HCl in your sample cell; this

corresponds to 1.0 volt on the Baratron (the exact pressure is not crucial). “Zero” pressure (perfect vacuum) corresponds in theory to 0.0 volts on the meter, although small voltage offsets are common.

• Turn on the mechanical pump if it is not already running. Evacuate the manifold shown in Figure 5 by opening valve 1. When the pressure has reached near zero (after ~1 minute), evacuate the sample cell by opening valves 2 and 5.

At this time you should also open valve 4, but do not open valve 6 yet.

• Next place a styrofoam cup filled with liquid nitrogen (this liquid can burn your skin: handle with care!) over the test-tube end of the round bottom storage flask to freeze out the small amount of HCl which is contained in this flask. Note that the gas is a mixture of different isotopes. After the HCl has been frozen out (about 1 minute), you

may open valve 6. • Next, isolate the manifold from the

pump by closing valve 1. Remove the liquid nitrogen from the HCl flask, and carefully watch the pressure meter as the HCl evaporates and the pressure increases in the manifold. When the desired sample pressure is reached (~100 torr), close valve 5, and then place the liquid nitrogen back on the test-tube end of the HCl flask. The manifold pressure should return to zero

again as the remaining HCl is again frozen out (this should take about a minute). Close valve 6. Open valve 1 to evacuate any trace gasses left in the manifold. Close valve 2 and remove the gas cell from the manifold by rotating the ball joint to break the vacuum seal. Close valve 4. This completes the cell filling operation. Note that this procedure

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Fig. 6: Parameter menu.

Fig. 7: Example about how the labeling should look like. (That was a contaminated sample, i.e., do not take the peak positions too seriously.) Add the m values to these plots.

wastes very little HCl by freezing the contents of the vacuum line back into the storage bulb, and also prevents significant amounts of corrosive HCl gas from reaching the vacuum pump.

C2 Handling of the spectrometer You can now collect an infrared spectrum using the Fourier-Transform infrared spectrometer (Nexus 470 Ft-Ir from thermo Nicolet). Place the sample cell in the sample compartment of the spectrometer. Be sure that the cell is aligned properly to allow the infrared light to pass through. The spectrometer is operated by the attached PC-AT computer. Your T.A. will assist you with the details of operation. However, consider the following: 1) Purging the IR with

nitrogen, the TA will help. Keep the nitrogen gas running all the time, close at the end of your measurements.

2) Start the OMNIC E.S.P. 52.a program. An IR library program with a similar name is also available on that computer, i.e., use the right program.

3) Type in the password which is burghaus1 and the user name which is CHEM471

4) Set the measuring parameter by activating the experimental set up menu (collect experimental setup).

5) Use the collect button for starting a measurement. Depending on your set up parameters, you have to collect a

background spectra (empty IR cell) before or after taking the spectra or by choosing a stored background

spectra. 6) After averaging four scans, a spectra

will be displayed on the screen. It’s better not to interfere with the running program. When the whole scans are finished (ca. 10 min) a message box should pop up on the screen. Save the final spectra which is an average of a large number of single scans. Switch from the collect window to a data window. Print the overview spectra which shows all the lines (include that in your lab report).

7) Select different regions of the spectra e.g. by drag –draw and clicking in the box. Scale the plot. Go to analyze data and find peaks

to determine the exact peak positions. You have to define a threshold for the peak find algorithm for distinguishing the background and the peaks. Click on replace to activate the labeling tools.

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Label the peak positions in a decent way. Print and store the result. Repeat that procedure for all regions where a set of IR peaks has been detected.

D) CALCULATIONS / PROBLEMS

Especially for this experiment, we expect a clear structure of your lab report. Please, explain to us how you have analyzed your spectra. Use clear and well structured graphs and tables. Label the graphs, explain the meaning of your variables and parameters. Otherwise, it is hard to understand your writing. If we do not understand it … your grade might be affected … Include an overview spectra in the lab report as well as “blow ups” of the spectra. Label and number the attachments.

We have included more hints about how to analyze the spectra than in older versions of the lab handout, since this experiment might be the most difficult one of the class. However, do not just copy old lab reports. Do it by yourself!

Please, answer the following questions and include the calculations as detailed below in the lab report. Please use the following numbering also in your lab report (discussion section). 1) You should observe a splitting of the

lines (Fig. 7). Assign the major lines and the minor ones to the different isotopes of HCl contained in the gas cell. Is the line splitting related to the deuterated gas or to the different Cl isotopes? Note, consider the difference in the reduced mass of different isotopes.

2) You should observe different groups of lines. Assign them to the different isotopes as well. Some lines might be related to impurities of the gas. Try to assign them also. Label the overview spectra accordingly.

3) Index the lines in the spectrum according to the m values, using Figure 3 as a guide. Construct a table of wave numbers versus m for as many lines as

you can. Use different tables for the different isotopes.

4) Use the equation ∆ν~(m) = ν~(m+1) - ν~(m) = (2Be - 3αe) -

2αem which is discussed above, and plot ∆ν~(m) versus m. Use a linear least squares fit. From the slope and intercept, determine the rotational constant, Be, and the rotation-vibration coupling constant, αe. Include a discussion of that procedure and the equations in your lab report. See the note of the figure caption of Fig. 3. A common mistake is to assign the m values with the wrong sign.

5) Next, use the equation ν~e = ν

~0 + (2Be - 2αe)m - αem

2 (see above) to determine the value of ν~0, the frequency of the n"=0 to n"=1 vibrational transition. That equation can be rewritten as

ν~e = A(m) + ν~

0 .

Thus, ν~0 can be determined from the

intercept of a plot of ν~e vs. A(m). Again, include a discussion of that procedure in your lab report. A(m) is a function of m.

6) From the rotational constant as determined above and

Be = h / (8π2Ic)

determine the value of the equilibrium internuclear distance, r, for HCl and DCl. Note I = µr2, with µ denoted to the reduced mass.

7) From the value obtained for ν~0, calculate the vibrational force constant, k, for HCl and DCl in millidynes per angstrom. Note

ν~0 = 1/(2πc)sqrt(k/µ). 8) Be sure to include estimated

uncertainties in each of your derived parameters; since the parameters are

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related to your plots, it is not a large effort to do so. Compare your results with literature values. Include the references.

9) Discuss what additional information would also be needed to determine the value of the anharmonicity constant χeν

~e. How might this information be

obtained experimentally? 10) Briefly, how is a Fourier-Transform IR

spectrometer working? What is the advantage as compared with a gratings spectrometer?

• Your lab report should include the following spectra (peaks labeled with the m number and wave number) as well as the corresponding tables: overview spectra, HCl35 (with labels), HCl37 (on a second paper sheet and labeled, otherwise you cannot read anything), DCl35, DCl37. Tables with the slope and intercept of the linear fits you have used to determine the spectroscopic parameters.

E) References 1. “Experiments in Physical Chemistry”

from D.P. Shoemaker, C.W. Garland, J.W. Nibler, McGraw-Hill.

2. Copies of the manual of the IR spectrometer and some more information is included in the info folder.

3. A power point presentation about FTIR is available from the lab class web site at http://www.uwe.burghaus.de or through the chemistry department web site.

Web sites http://www1.nicolet.com/labsys/Applications/theory.html http://www.columbia.edu/ccnmtl/draft/dbeeb/chem-udl/Spectrometer_doc.html

Safety notes Liquid nitrogen can lead to severe injures.

Grading and how to write the lab

report.

This may be the laboratory experiment with the most difficult data discussion section. The following structure is recommended for the HCl/DCl laboratory report. 1) Title of the experiment, date, your name 2) Abstract (approx. 1/3 of a page) – 5% • What is what in your spectra? Peak

assignments. • All parameters obtained including

measuring errors 3) Introduction/Theory (one page) – 5% Look through your quantum mechanics notes (or the web site for this class) and summarize the theory of this experiment in one page. • Harmonic oscillator • Rigid rotor • Rigid - rotor harmonic oscillator

approximation • Improvements on these approximations 4) Experimental procedures (one page) – 5% • How is a FT-IR system working? The

idea, the concepts. • What is the advantage of FT-IR as

compared with dispersion type spectrometers.

We have discussed this in detail in class, pull out your notes. 5) Data presentation (briefly) – 5%

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http://www1.nicolet.com/labsys/Applications/theory.htmlhttp://www1.nicolet.com/labsys/Applications/theory.html

Just one sentence and your data plots. Number the spectra. Fig. 1, Fig. 2, Fig. x. 6) Discussion (a number of pages) 6.1 Peak assignment (1/3 page) – 5% What is what and why? We have HCl, DCl, and their isotopes. Larger mass smaller wave number, why? Look at the reduced mass. Perhaps add a table and a small Fig. 6.2 Label the peaks (1/4 page) – 5% What is the m quantum number? What is an R and P branch? Label the spectra accordingly. Double check this; otherwise all the following plots will be wrong. See D1-D3. 6.3 Rotational constant and rotation-vibration coupling constant - 20% Read p.25 of this lab handout, it is explained how to do this on p.25. You need several tables and plots here for the different isotopes. Problems? Stop by in my office. (See D4) 6.4 ν0 – 20% Plots & table. See D5 6.5 Equilibrium internuclear distance – 5 %

This is a simple equation, plug in your numbers. See D6. 6.6 Force constant – 5% This is a simple equation, plug in your numbers. See D7. 7. Summary – 10% Make a table with all the spectroscopic parameters and add some sentences. Estimate the measuring errors for one of the isotopes. The errors are obtained from your plots. This is part of the discussion. – 20% Decent structure of your lab report - 20% Think about it by yourself. What would be a good structure of the draft; use sub headings. Some people mix up the data presentation with the discussion part which leads typically to a chaotic outline. Use the standard structure outline above; it's the simplest way to "write a paper". Use a formula editor for typing equations. If you add up the percentages you could obtain in principle 125%.

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Fig.1: Potential energy curve.

12) Band Spectrum and Dissociation of the Iodine Molecule A) ABSTRACT

The analysis of the visible band spectrum of iodine leads to the spectroscopic determination of molecular energy levels and thermodynamic quantities. In this experiment, the visible absorption spectrum of iodine is observed using a recording spectrometer. The wavelengths and energies

(cm-1) of the bands are tabulated and the dissociation energy Do is calculated using the graphical Birge-Sponer extrapo-

lation2 and the known excitation energies of the product atoms. A formula is developed for the location of the band heads, and the spectroscopic parameters of the excited electronic states are calculated.

2 Birge-Sponer example: Most students last year had difficulties with the Birge-Sponer technique. Therefore I was looking for an example which is given at the end of the lab handout.

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Fig.2: Absorption spectrum of I2. Your spectra will look quite different, but the figure helps for assigning the peaks. Here, energy increases ( λ decreases) from right to left.

B) THEORY In Fig. 1 the potential energy, E, is shown as a function of the internuclear distance, r, for a diatomic molecule. The lowest curve (X) represents the combination of two iodine atoms to form a stable molecule in their lowest ground electronic state. The letter X is the accepted nomenclature among spectroscopists to designate the lowest energy (ground) electronic state of the molecule. The next electronic states are labeled A, B, C, etc. in order of increasing energy. Note that the level A is also due to the combination of two iodine atoms in their atomic ground states. This combination is in a slightly different configuration, resulting in a slightly higher molecular electronic energy. The state A is responsible for absorption in the infrared region. In this experiment we will be

measuring spectra at a higher energy and will thus not be interested in the state A. The upper curve in Fig. 1 is

obtained when an excited iodine atom, I*, combines with a ground state iodine atom, I, to form a stable molecular state labeled B. The observed light absorption in this experiment is a result of electronic transitions from X to B. The spectroscopic parameters with which we will be dealing are labeled in Fig. 1. In discussing the potential energies of the electronic states, we will refer to traditional spectroscopic practice and distinguish the parameters for the upper state with a prime (′) and the parameters for the ground state with a double prime (′′). De and Do represent the dissociation energies from the equilibrium position (bottom of the potential well) and from the ground vibrational state (v = 0),

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Setup of the spectrometer.

respectively. The energy difference between the minima of electronic states X and B is represented with νe. The vibrational states within each electronic state are labeled v = 0, 1, 2, etc. Energy differences between vibrational levels within the same electronic state are labeled with ω. Associated with each vibrational level are many rotational levels whose energy spacing is much smaller than the ω′s and are therefore not conveniently shown in the diagram. When indicated, they are usually designated with the letter J. One possible change in the molecule’s internal energy is indicated in Fig. 1 by the arrow labeled ν. The transition indicated is from the zeroth vibrational level of X to the third vibrational level of B. In spectroscopic shorthand this would be written: (B,v′=3 ← X,v′′=0), where the direction of the arrow indicates absorption. If it is understood which two electronic states are involved in the transition, then one may adopt a simpler notation: (3′ ← 0′′). The energy of this transition or any transition is just the energy of the upper

state minus the energy (in cm-1) of the lower state: ν = Eupper – Elower. The energy of the upper state is: Eupper = Te′ + G(v′) where Te represents electronic energy and G(v) represents vibrational energy (rotational energy will be neglected). Similarly, the energy of the lower state is: Elower = Te′′ + G(v''). Thus we may write:

ν = {Te′ + G(v′)}- {Te′′ + G(v′′)}. Rearranging this equation yields:

ν = {Te′ - Te′′} + {G(v′) - G(v′′)} Now from Fig. 1 we see that the difference in the electronic energies (Te′ - Te′′) = νe, so we may write: ν = νe + {G(v′) - G(v′′) } (1)

Quantum mechanics tells us that the energy of a vibrational state is:

G(v) = ωe(v + 1/2) - ωeχe(v + 1/2)2, (2)

where ωe is the characteristic or fundamental vibrational frequency. The ωe

χe term corrects the energy for the fact that the potential energy surface is not truly harmonic. We therefore call ωeχe the anharmonicity constant. Using equation (2) we can rewrite equation (1) as:

ν = νe + [{ωe′(v′ + 1/2) - ωe′χe′(v′ + 1/2)2}

- {ωe′′ (v′′ + 1/2) - ωe′′χe′′ (v′′ + 1/2)2}].

(3) This equation relates the physical observable, ν, to fundamental quantities of the system. You will need (see below) to calculate energy differences according to ω′ = ∆ν = ν(v′+1) - ν(v′). (4)

A representative absorption spectrum for I2 is shown in Fig. 2. Each small bump corresponds to a transition between two vibrational levels (v′ ← v′′) and is called a band. When observed with a spectrometer of very high resolution, these bands are seen to consist of many lines corresponding to transitions between individual rotational states of the two vibrational states involved. You will notice that there are actually three series of absorption lines which overlap in the spectrum. These are labeled according to whether they arise from v” = 0, 1, or 2 of the ground electronic state. Part of your job in assigning the spectrum will be to identify which lines belong to which transitions. You will also notice that beyond a certain minimum wavelength (maximum energy,

cm-1) the spacing between bands appears to decrease to zero. This is referred to as the convergence limit, and the energy

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corresponding to this transition is shown in

Fig. 1 as E*. Beyond the convergence limit, the spectrum is continuous. One of the purposes of this experiment is to locate this convergence limit precisely by means of a Birge-Sponer extrapolation. C) EXPERIMENTAL Place a few crystals of I2 in the 10 cm quartz cell and heat the cell gently until iodine vapor saturates the cell. Do not heat the cell near the windows. Place the sample and a blank cuvette in the sample compartment as shown in the figure below.

Start the program (click on scan button), go to the setup menu and obtain the visible spectrum of I2

from 650 nm to 490 nm on the CARY500 UV/Vis spectrometer using the following settings:

Scale = 1 nm/cm; Scan speed = 4 nm/min;

Band width (SBW) = .1 nm Double beam

For printing and analyzing the data (using these settings), you will have to split the spectrum into 6 pieces to cover the entire wavelength range. Use the peak label tool and rescale the graph for getting good plots.

In this experiment, you will first assign the observed bands to different spectroscopic transitions. In order to extract the maximum possible data from the spectrum, you will need to observe progressions (see Fig. 2) from at least three different vibrational levels of the ground electronic state. This means that the temperature of the I2 gas in the cell should be elevated during the data collection. This may be accomplished by wrapping the middle of the cell with heating tape, or using a heat gun and gently heating the cell to ca. 40 C. D) ANALYSIS OF THE DATA This might be the most difficult lab report you have to write, but do not just copy old lab reports, try following the outline below. I) Excited state potentials a) First, the bands must be identified and

numbered according to the v’ and v” to which they correspond. A good reference point is the (v′=29 ← v′′=0) transition which occurs at 536.87 nm. Use Fig. 2 for finding the peak assignment for v’ v’’ > 0. E.g. peak v’ = 27 v’’ = 1 is at the lower wavelength side of 24’ 0”, etc.

b) Make a table of those transitions corresponding to (v′=? ← v′′=0). Tabulate the wavelength in Angstroms, the energies (ν) in cm-1 and the differences in energy (also in cm-1), see Eq.(4).

c) Starting with some chosen band, vn′, plot the energy difference ∆ν vs. the quantum number v′. This is known as a Birge-Sponer plot. The area under this plot (in cm-1) plus the energy of the band (νn) from which you started the

plot is equal to the energy E* on Fig. 1. This is the amount of energy needed for

the reaction: I2 → I + I*.

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d) Calculate the value of E* and use it to

calculate Do′′. The value of E(I*) has

been found experimentally to be 7598 cm-1.

e) The intercept of the Birge-Sponer plot with the vertical line v’ = -1 gives the value of ωe′. The slope of the plotted line is -2ωe′χe′. Using equations (3) and (4) show that this is true by deriving an expression for ∆ν in terms of ωe′ and ωe′χe′

f) and calculate their values. Be sure to indicate the method used to find the slope of the graph.

g) As a check, repeat your calculations by doing a similar Birge-Sponer plot and analysis with the data on the transitions originating in the v′′ = 1 band.

h) Using the equation you have derived and information from your plot, you can calculate Do′.

i) Having Do′ we are in a position to calculate De′. Using equation (2) and Fig. 1, derive an expression for De′ in terms of Do′, ωe′ and ωe′χe′ and calculate its value.

II) Ground state potentials

j) We now have a fairly detailed picture of the excited state potential. The only information we have so far about the ground state is Do′′. As you have seen from the previous analysis, we use energy differences, (the ω′ s in Fig. 1), to characterize the excited state potential. We would like to do the same thing for the ground state. Unfortunately, we only have two differences (ω′′ s) available. You should expect then that the error in calculating the ground state parameters will be larger than the error in the excited state parameters. Choose a v′ band (i) which has transitions from each of the three

ground state (v′′) bands and find the energies: (v′=i ← v′′=0) , (v′=i ← v′′=1), (v′=i ← v′′=2)

and calculate the energy differences (ω′′) between them.

k) Using equation (2), write down expressions for the energy differences you have just calculated and use these expressions to find ωe′′ and ωe′′χe′′. You should not put too much faith in the value of ωe′′χe′′ you obtain (why?). Note, that is a tricky question. One possible result would be ωe’’ = 2∆ν(01) – ∆ν(12) and ωe’’χe’’=(∆ν(01) ∆ν(12))/2.

l) Repeat this procedure for a few more values of v′ which have transitions from v′′ = 0, 1 and 2.

m) You should now be able to calculate De′′ using the expression you derived previously relating De, Do, ωe and ωeχe. (See question i above and Fig. 1).

n) Finally, calculate the value of νe. III) Discussion Be sure that all of your derivations are well labeled and your calculations are easy to follow. You may use the labels of the exercise given above (a, b, c…) in your lab report. Your discussion should indicate that you understand the physical meaning of the parameters you have calculated. Specifically:

α) Compare your values to literature values. Literature values may be found in a paper by R. D. Verma, J. Chem. Phys. 32, 738 (1960). β) Explain what is meant by "zero-point-energy". γ) Explain why the spectrum becomes continuous beyond a certain energy. What

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would the energy levels be if the anharmonicity constants were zero? δ) Is it possible to observe the (v′=0 ← v′′=0) transition in this experiment? Why or why not?

Literature [1] J.I. Steinfeld, Molecules and Radiation – An Introduction to Modern Molecular Spectroscopy, Harper & Row (1974). Chapter 4.5, p. 105, Department library code is Chem. QC 454 M6 S83

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Birge-Sponer example

Most students last year had difficulties with the Birge-Sponer technique. Therefore I was looking for an example which is given below. Question For HgH the vibrations, νn, have been determined experimentally: v = 0 --> 1 1203.7 cm-1 v = 1 --> 2 965.6 cm-1 v = 2 --> 3 632.4 cm-1 v = 3 --> 4 172.0 cm-1 Calculate the dissociation energy and extrapolate the missing data by a linear interpolation by means of the Birge-Sponer technique. Answer According to the figure we have E E E E E E Ediss = − + − + = +1 0 2 1 1 2... ∆ ∆ +…

or E E E Gdiss n nn

n= − =+=

∞

∑ ∑( )10

.

A linear regression (or linear least squares fit) would yield ν[ ]( ) . . ( )cm n cm n cm− − −= − +1 1 114291 342 8 1 2 . Now Ediss corresponds simply to ½ of the area, A, of the square defined by A = lx*ly with ly obtained from the fit with n = 0 which equals 1429.1cm-1 and ly obtained from the fit with ν[ ]( )cm n− =1 0 which amounts to ly= 1429.1/342.8 = 4.16. Hence, A = ½*1429.1*4.16 cm-1 = 2972.5 cm-1. Or directly

ν nn

cm cm+=

∞− −∑ = + + + =1

0

1 11203 7 965 6 632 4 172 2973 7. . . .

Other techniques use a parabola instead of a linear function.

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I hope that example which illustrates the general idea of the Birge-Sponer technique helps a little.

Grading – Band Spectrum and Dissociation of I2 NOTE: Be very precise with your symbol notation throughout your report! 1. Abstract – 5% a. What was done/purpose b. Results for Do”, Do’, E*, νe c. Results for ωe”, ωe’, ωe”χe”, ωe’χe’ d. Uncertainties & literature comparison 2. Introduction – 10% (consider at least the following topics) a. What was done/purpose b. Briefly, how does a UV-Vis spectrophotometer work? c. What kinds of quantum state transitions are observed in the band spectrum of I2? d. Why does the cuvette (sample cell) need to be heated? e. What is the difference between Do and De? f. Derive equation 3 of the lab manual 3. Experimental Procedure – 5% a. Cite lab manual & list any changes made b. Name the instrument used c. Give a short summary of the procedure 4. Results – 20% a. Text section describing what is shown b. Table of transitions arising from v” = 0 (include extrapolated values) c. Birge-Sponer plot for v” = 0 d. Table of transitions arising from v” = 1 (include extrapolated values) e. Birge-Sponer plot for v” = 1 f. Table of transitions ending at the same v’ (part j) g. Table with calculated parameters, including uncertainties for each and the corresponding

literature value i. ωe’, ωe’χe’, Do’, De’, ωe”, ωe”χe”, Do”, De”, E*, νe

h. Attach labeled spectra to the end of the report 5. Discussion – 60% a. Briefly explain how you numbered the bands in the spectrum b. Show the conversion of nm to Angstroms and cm-1; show an example of an energy

difference (ω’) calculation c. Show how you calculated the area under the Birge-Sponer plot and subsequently E*; what

does E* correspond to? d. Show the calculation of Do”; what is its physical significance? e. Derive the indicated expression relating ∆ν to ωe’ and ωe’χe’; define these quantities f. Show how you calculated ωe’ and ωe’χe’ g. Mention the results of calculating the above parameters using the Birge-Sponer plot for v”

= 1; how do they compare with the v” = 0 case?

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h. Show how to calculate Do’; what is its physical significance? i. Derive the indicated expression for De’; show how to calculate De’ j. State which v’ bands you are using to determine the ground state parameters; show an

example energy difference (ω”) calculation; why should the error in the ground state parameters be larger than for the excited state parameters?

k. Derive the indicated expressions for ω e” and ω e”χe”; show their calculation; why should you not put too much faith in your ω e”χe” value?

l. Mention the results of calculating the ground state parameters with other v’ values; how do they compare to each other? If they are similar, averaging them might be reasonable.

m. Show the calculation of De” n. Show the calculation of νe; what is its physical significance? o. Compare your values to literature values found in Verma’s paper and the NIST Chemistry

Webbook p. Answer question β q. Answer question γ r. Answer question δ s. Are there any notable differences between the ground and excited electronic state

parameters? t. Show example error propagation calculations for all parameters u. Discuss sources of experimental error and error due to approximations 6. Summary (10%) a. What was done b. Major results c. Comparison with literature & overall conclusion

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A) ABSTRACT0FB) TheoryB) ExperimentalC1 Filling the infrared gas cell.D) Calculations / ProblemsE) ReferencesA) ABSTRACTC) EXPERIMENTALD) ANALYSIS OF THE DATA