1.E: Spectroscopy (Exercises)

Exercise 1.1 List three reasons why spectroscopy is used as the stepping stone to Quantum Mechanics.

Exercise 1.2 Identify why 1 hertz is said to equal \(6.62608 × 10^{-34}\) J and 1 wavenumber is said to equal \(1.98645 × 10^{-23}\) J.

Exercise 1.3 Show how to obtain the number of electron volts in 1 hartree from the information in the tables.

Exercise 1.4 Show how to obtain the number of wavenumbers in 1 electron volt from the information in the tables.

Exercise 1.5 Identify the physical interpretation of the wavenumber value.

Exercise 1.6 The absorption of infrared radiation excites vibrational motion in molecules. Wave number units, \(\bar {\nu}\), commonly are used to describe this radiation, and a typical value for vibrational motion is \(\bar {\nu}\) = 1000 cm-1. Calculate the frequency, wavelength (in meters and microns), and photon energy (in joules and electron volts) for this radiation.

Exercise 1.7 Use Table 1 and identify an instrument that is used by chemists for each of 8 regions of the electromagnetic spectrum.

Study Questions

1. Identify the key idea/concept that makes spectroscopy an important tool in chemistry. (Write a few sentences.)
2. Describe a distribution function. Use a sketch to make your description clear.
3. What is Beer’s Law and why is it best used at low concentrtions?
4. Sketch them major structural features of a double beam UV-vis spectrometer. Label each part and specify why it is important.
5. Identify five instruments use in you laboratories and the spectral region these instruments employ.
6. What is electromagnetic radiation? Give examples.

Problems

1. An absorption spectrum shows how much light is absorbed at each wavelength. Beer’s Law describes the absorption,

\[I (\lambda) = I_0 (\lambda) 10^{-\epsilon (\lambda) cd} \label {1.7}\]

where \(I_0 (\lambda)\) is the intensity of light incident on the sample, \(I (\lambda)\) is the intensity transmitted by the sample, \(\epsilon (\lambda)\) is the decadic absorption coefficient, which depends on the wavelength of light, c is the concentration of the chemical species that is absorbing light, and d is the path length. Decadic refers to base-10 in the power function. Below we use ε'for the absorption coefficient with base-e. The absorption coefficient is a measure of the strength of the interaction of a molecule with the electromagnetic field. When the units of ε are \(Lmol^{-1}cm^{-1}\), it is called the molar extinction coefficient or molar absorptivity.

Function notation is used here to make explicit that the intensities, I and \(I_0\), and the absorption coefficient, \(\epsilon\), depend on the wavelength, \(\lambda\). Beer’s law can be derived by considering the absorption of an infinitely thin slice of sample and then integrating over the total thickness. Since any continuous function is linear over a sufficiently small range, the change in light intensity, dI, by an infinitely thin sample must depend linearly on the absorption coefficient, \(\epsilon '\), the concentration of the absorbing species, c, the intensity of the light, I, and the sample thickness, dx. This statement can be expressed mathematically.

\[dI = -\epsilon ' c I dx \label {1.8}\]

1. What does the minus sign indicate physically?
2. What would this equation be telling us if the minus sign were replaced by a plus sign?
3. Show the mathematical steps that produce Beer’s Law from this equation.
4. Derive the relationship between the decadic absorption coefficient (\(\epsilon\)in base-10 Beer’s Law) and the Napierian absorption coefficient (\(\epsilon '\)in the base-e Beer’s Law).
5. Show why the Napierian and decadic logarithms are related by a factor of 2.303 and not some other value.

2. What dye concentration must you dope into a plastic film that your sun glass company is manufacturing for use in ultrathin (0.1 mm) sun glasses that absorb 75% of the visible light? The decadic absorption coefficient of the dye is \(10^5\) L/mol cm across the visible region of the spectrum.

3. Use Mathcad, or some other program, to convert several of the values in the table into all the units given in the top row. A completed sample is given in the second row. Completing this problem will give you practice with using units and also some skill with unit conversions.

J/photon	kcal/mol	eV	hartree	cm-1	nm	Angstrom	Hz
\(3.44 x 10^{-20}\) J	4.94 kcal/mol	0.214 eV	\(7.88 x 10^{-3}\) Hartree	\(1730 cm^{-1}\)	5.78 x 103 nm	5.78 x 104 Angstroms	\(5.19 x 10^{13}\) Hz
\(2.6 x 10^{-19} \)J (green light)
	100 kcal/mol (typical bond energy)
		13.6 eV (ionization energy for H-atom)
			1.0 hartree (atomic unit for energy)
				3600 \(cm^{-1}\) (O-H bond stretch)
					280 nm (absorbed by ozone)