CH342 Handin Homework 2
1. What are the quantum numbers for the energy levels that are involved in the lowest energy
electronic transition for the molecule: C=C-C=C-C=C-C=C . Base your answer on the particlein -the-box model.
2. (a). Calculate the wavelength of the light absorbed in Problem 1. Assume the bonds are
equivalent and the bond length is 1.39 Å. (b). Calculate the energy in cm-1.
3. Sketch the wavefunctions for the potentials shown on the next page.
Etch-a-Sketch
E
E
use this function as a start
V
x
E
E
use this function as a start
V
x
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Spectral Deconvolution and Energy level Diagrams
The electronic energy level diagram for a typical molecule is shown in Figure 1. Molecules
have many possible excited states. Absorption transitions in the UV/Visible portion of the
spectrum correspond to transitions from the ground electronic state to the various excited
electronic states. The closely spaced horizontal lines represent the different vibrational states of
the given electronic state. These diagrams are called Jablonski diagrams.
third excited state
E
second excited state
absorbance
first excited state
ground state
Figure 1. Typical electronic energy level diagram.
The assignment is to construct such a diagram, carefully and to scale, for bromothymol blue. The
UV/Visible absorption spectrum for bromothymol blue in given in Figure 2.
0.9
0.8
0.7
A
0.6
0.5
0.4
0.3
0.2
0.1
0
210
310
410
510
610
λ (nm)
Figure 2. UV/Visible absorption spectrum for bromothymol blue in water.
Example Problem: Here is an example that will help you draw the energy level diagram from
your spectrum. A typical example spectrum is given in Figure 3.
0.45
0.4
Absorbance
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
200
250
300
350
400
450
500
w ave le ngth (nm )
Figure 3. Example spectrum
The first step is to convert the wavelengths to energy units or units like cm-1 that are directly
proportional to energy, Figure 4. Then each transition is resolved by approximating each
transition as a simple Gaussian peak. This process is often done by least squares fitting
programs, which in this context is called spectral deconvolution. For the purposes of this
exercise, the deconvolution process can just be done by eye with a pencil. Often the actual
number of transitions is not completely clear, but you do the best you can with the information
available. Each transition is to a different electronic state. For each electronic state the electrons
are in different sets of molecular orbitals.
0.45
0.4
0.35
Absorbance
0.3
0.25
0.2
0.15
0.1
0.05
0
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
E (cm -1 )
Figure 4. Spectrum with the wavelength axis converted to wavenumbers (cm-1).
The process of drawing the energy level diagram can be illustrated simply by rotating the
absorbance spectrum on its side and using the spectral transitions to delineate the energy levels
into bands. It is common for the transitions to overlap. Table 1 provides the energies that are
needed for this process from Figure 4. The wavelengths or wavenumbers at the start and end of
each band are read by eye directly from the deconvoluted spectra. The resulting energy level
diagram is shown in Figure 5.
Table 1. The start and end of each band are read from the deconvoluted spectrum. The values are
approximate and are often read in nm from the original spectrum and converted to wavenumbers.
Transition
First excited state
Second excited state
Third excited state
Fourth excited state
Start of absorption band End of absorption band
cm-1
cm-1
λ (nm)
λ (nm)
440
22700
340
29400
350
28600
280
35700
295
33900
250
40000
270
37000
235
42600
45000
4
4
40000
3
3
35000
2
2
Energy (cm-1)
2
E (cm-1)
30000
1
25000
2
1
20000
15000
10000
5000
ground state
0
0
0.1
0.2
0.3
0.4
0.5
A bsorb ance
Figure 5. The process for drawing the energy level diagram can be illustrated by picturing the
spectrum tilted on its side. The different excited state bands are offset for clearity (they are all
singlet states if the ground state is a singlet).
In this example, the original spectrum was converted to a plot of absorbance versus
wavenumber. In actual use, the start and end wavelengths are often read directly from the
spectrum plotted versus wavelength. The intermediate step of converting the spectrum to a
wavenumber axis is useful for demonstrating the relationships involved, but the conversion is not
necessary in practice.
Each electronic transition is really a set of transitions to different vibrational states of the same
electronic state. The set of vibrational transitions to a given electronic state form a band of states
given by the width of the electronic transition. The vibrational bands are often drawn as a series
of lines, Figures 1 and 5. These lines correspond to the different vibrational transitions. For our
current purposes, the spacing between the lines is arbitrary since the wavenumber resolution in
solution UV/visible spectra is usually not sufficient to discern the vibrational lines.
For homework purposes, the process of deconvoluting a spectrum can be done by hand with a
pencil. No complicated calculations are necessary. However, if you don’t have some prior
experience, the process of determining the number of transitions and their widths can be difficult.
Two Excel spreadsheets are available on the PChem Homework Web page to help you explore
the deconvolution process. The deconvolution can be done on spectra as a function of
wavelength or as a function of wavenumber. These spreadsheets do Gaussian deconvolution for a
spectrum plotted as a function of wavelength. Try the Excel spreadsheet example:
http://www.colby.edu/chemistry/PChem/homework/spectraldeconvolutionexample.xlsm
to test your skills. On the PC, the following message will appear in Excel below the top icon bar.
Click on “Options”:
In the subsequent Security dialog box, click on “Enable this content” and then click “OK”:
On the Mac, a single dialog box will appear in which you click on “Enable Macros.” In the
spreadsheet, use the up and down arrows to change the center, width, and area settings for each
absorption band to get a good fit. You can judge the fit by looking at the difference spectrum in
the bottom plot. You will only need five components to fit this example spectrum, even though
six are available. The best parameter values are listed below, so that you can check your work.1
This example spectrum is actually calculated from overlapping Gaussians, so the fit can work out
to be perfect, which is not possible with experimental spectra. The spectrum for bromothymol
blue in Figure 2 is also available loaded into the same spreadsheet on the Homework Web page:
http://www.colby.edu/chemistry/PChem/homework/spectraldeconvolution.xlsm
You then need to convert the start and end wavelengths to wavenumbers before constructing
your energy level diagram. Use the following table to organize your measurements.
Transition
Start of absorption band
cm-1
λ
End of absorption band
cm-1
λ
First excited state
Second excited state
Third excited state
Fourth excited state
Fifth excited state
1. Parameters to fit the example spectrum in Figure 3:
center
width
area
cmp 1
200
15
20
cmp 2
240
20
15
cmp 3
350
30
5
cmp 4
450
30
7
cmp 5
580
40
6
nm
nm