Absolutely Small - Michael D. Fayer [49]
The question is, what wavelengths of light will be absorbed by a molecule? This is a very complex question for any given molecule. Large quantum theoretical calculations are performed to determine the absorption spectrum of a molecule. However, we can learn about important aspects of molecular absorption of light from the particle in a box problem. As an exceedingly simple model of a molecule, we will consider a single electron in a molecular-sized box. Later we will put in numbers. The electron will be in its lowest energy state, called the ground state, when no light shines on the electron in the box (the molecule). For the particle in the box, the lowest energy has the quantum number, n = 1. For n = 1, the energy is:
When light shines on a molecule, a photon can be absorbed. If the photon is absorbed, the energy of the photon is lost from the total energy of all of the light. Energy must be conserved, which happens by an electron going into a higher energy state, that is, it goes from the ground state, the lowest energy level, to a higher energy level. However, this higher energy level cannot have any energy value because the energy levels of the particle in a box (and molecules) are quantized. The lowest energy state above the ground state has quantum number n = 2. This is called an excited state. The electron has been excited by absorption of a photon from the ground state to the first excited state. The energy of the first excited state, the n = 2 state, is:
Energy must be conserved. This is true in classical mechanics, and it is true in quantum mechanics. We start with the electron in the ground state. When the photon is absorbed, the electron is in an excited state. Therefore, to conserve energy, the photon energy must equal the difference between the excited state energy and the ground state energy. Only a photon with this energy can be absorbed by the system. The photon energy determines the wavelength of the light. Therefore, only certain colors of light can be absorbed.
Figure 8.7 illustrates absorption of a photon. The arrows show two allowed paths for photon absorption. These are called transitions. The transitions from n = 1 to n = 2, and n = 1 to n = 3 are shown in the figure. For a photon to be absorbed, the photon energy must equal the difference in energy of two of the quantum levels. If the photon energy does not match the difference in energy between two levels, it cannot be absorbed.
The difference in the energy, ΔE, between the n = 2 first excited state energy level and the n = 1 ground state energy level is:
FIGURE 8.7. Particle in a box energy levels. The quantum number is n. E is the energy plotted in units of h2/8mL2. The arrows indicate absorption of photons that can take an electron from the lowest energy level, n = 1, to higher energy levels, n = 2, n = 3, etc. For a photon to be absorbed, its energy must match the difference in energy between two energy levels.
This is the energy that the photon must have to cause the electron to make a transition from the ground state to the first excited state. We can use Planck’s relation, E = hν, for the photon energy to see that the energy ΔE corresponds to a certain frequency of light. Also, because λν = c, the wavelength times the frequency equals the speed of light, we can determine the wavelength (color) of the light that will be absorbed.
The Color of Fruit
Let’s put in numbers. h = 6.6 × 10-34 J-s. The electron mass, me = 9.1 × 10-31 kg. For the length of the box, let’s take L to be that of a medium-sized molecule, that is, L = 0.8 × 10-9 m (0.8 nanometers, 0.8 nm). Then,
Converting this energy to a frequency by dividing by h gives ν = 4.25 × 1014 Hz, which corresponds to the wavelength of the light that will be absorbed, λ = 7.06 × 10-7 m = 706 nm. Light with wavelength 706 nm is very deep red.