Absorption and Quantization
Copyright (c) Lant, Montgomery, and Mazziotti 2023 This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Learning Goals:
References
By the end of this lesson, students will be able to:
understand electrons and energy levels in atoms and molecules
predict the energy of absorbed or emitted photons during transitions between energy levels
In these lessons on "The Chemistry of Art" we explore the relationship between matter and color. In order for a substance to be colored, it must absorb certain colors or wavelengths of visible light. The reflected or transmitted light that we observe would therefore be the complementary color. So, "why are some substances colored and others not?"
To answer this question, we have to look at the atomic and molecular structure of matter. We begin with the basic concepts using a simple hydrogen atom and then apply those concepts to more complicated molecules, such as dyes and pigments.
Let's begin with the atom. Recall, atoms of an element are composed of three subatomic particles: protons (positive charge), neutrons (neutral charge), and electrons (negative charge). In the Rutherford-Bohr model of the atom, electrons are confined to move in circular orbits at fixed distances around the nucleus, which contains the protons and neutrons. In the most modern view of the atom, the quantum mechanical model, the electron is considered to have wave-like properties, and the idea of an orbit is replaced with an "orbital" describing a region of space with high probability of finding an electron.
Consider the simplest atom: hydrogen. The Rutherford-Bohr model of the atom restricts the orbit of the electron (Figure 1a), and this leads to a restriction in the allowed values of energy that the electron in an atom can have (Figure 1b)! It turns out that for the hydrogen atom, the Rutherford-Bohr model and the quantum mechanical model (more complex) yield the exact same energy levels! Therefore, we tend to think of electrons as occupying "energy levels" in an atom. In Figure 1b, the energy levels are indexed by the letter n.
Figure 1: a) The Bohr-Rutherford model of the hydrogen atom. b) The first 5 energy levels of the hydrogen atom with the electron (black circle) in the ground energy level.
The atom depicted in Figure 1a is said to be in the ground state. However, if the electron can gain just enough energy, it can be excited to a higher energy level. The energy to excite can come from heat, electrical energy, collisions with other particles, and from light! As we saw in the Previous Knowledge section, light is composed of photons, or packets of energy with E = hn. If a photon with an energy that exactly matches the energy difference (ΔE) between two energy levels collides with the atom, then the photon can be absorbed, and the electron can be excited to a higher energy level! Figure 2a and 2b depict this process for the excitation of an electron in the hydrogen atom from the n = 1 energy level to the n = 2 energy level.
Figure 2: a) Ground state hydrogen atom absorbing a photon with the correct energy, ΔE = hν. b) Excited hydrogen atom.
The requirement that the photon of light have the exact energy as the difference in energy between two states can be expressed as
ΔE = hn,
or in terms of wavelength:
ΔE=hcλ,
where ΔE is the difference in energy between two states. This can be rearranged to determine the wavelength of light that would be absorbed:
(Equation 1) λhcΔE
If the wavelength of light absorbed is in the visible spectrum, then the atom will be colored!
While we do not discuss the solution to the hydrogen atom here, the energy levels are given by
E(n) = −2.18×10−18n2 J,
where n is a positive integer (1, 2, 3, ...). Use the Maple commands below to calculate the first 5 energy levels of the hydrogen atom and answer the follow-up questions to determine if hydrogen is colored!
for n from 1 by 1 to 5 do En≔−2.18ⅇ−18n2⋅Units:-Unit'J' end do
E1≔−2.18×10−18J
E2≔−5.450000000×10−19J
E3≔−2.422222222×10−19J
E4≔−1.362500000×10−19J
E5≔−8.720000000×10−20J
Let's calculate the energy (ΔE) and wavelength (Equation 1 above) of the photon required to excited the electron in the hydrogen atom from the n = 1 state to the n = 2 state, as depicted in Figure 2.
First we use the ScientificConstants package to import precise values for the speed of light (c) and Planck's constant (h).
c ≔ evalfScientificConstants:-Constant'c', units;
c≔2.99792458×108ms
h≔ evalfScientificConstants:-Constant'h', units;
h≔6.626070040×10−34m2kgs
Now we can calculate the energy difference between the two states
n1≔1:
n2≔2:
ΔE≔En2−En1
ΔE≔1.635000000×10−18J
Finally, we can use Equation 1 to determine the wavelength absorbed during the transition
wavelength≔h⋅cΔE
wavelength≔1.214951574×10−7Jm2kgsms
We can simplify and convert the units to nanometers:
wavelength ≔ combinewavelength, units
wavelength≔1.214951574×10−7m
wavelength≔convertwavelength, units, 'nm'
wavelength≔121.4951574nm
(a) Is the wavelength in the visible spectrum? Would hydrogen in its ground state be colored?
(b) What if an electron in an excited state absorbed a photon? Calculate the wavelength of photon required to excite the electron from the n = 2 to the n = 3 energy level. Would this transition correspond to a photon in the visible spectrum? Can you find other transitions that might correspond to a photon in the visible spectrum?
1. Christie, R. The Physical and Chemical Basis of Colour. In Colour Chemistry. 2nd Ed. Royal Chemical Society: Cambridge. 2001. pp. 12-21. 2. Stockman, A., MacLeod, D. I., & Johnson, N. E. (1993). Spectral sensitivities of the human cones. Journal of the Optical Society of America, A, Optics, Image & Science, 10(12), 2491–2521.
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