The Wave Model for Light and Electrons - ChemPRIME

# The Wave Model for Light and Electrons

We noted that light, color, and photochemistry owe their existence tothe electronic structure of atoms. It may not be surprising, therefore, that the same wave/particle model that is used for light can be applied to electronic structure, and that we can get some insight into the energy involved in photosynthesis by studying the light that is absorbed. Let's see how this wave model developed, and review the wave model.

## Historical Development

At much the same time as Lewis was developing his theories of electronic structure, the physicist Niels Bohr was developing a similar, but more detailed, picture of the atom.

The Bohr picture of the Na atom. Two electrons orbit very to the nucleus. Eight others move around somewhat farther away, and there is a single outermost electron in an elongated, elliptical orbit.

Since Bohr was interested in light (energy) emitted by atoms under certain circumstances rather than the valence of elements, he particularly wanted to be able to calculate the energies of the electrons. To do this, he needed to know the exact path followed by each electron as it moved around the nucleus. He assumed paths similar to those of the planets around the sun. The figure seen here, taken from a physics text of the period, illustrates Bohr’s theories applied to the sodium atom. Note how the Bohr model, like that of Lewis, assumes a shell structure. There are two electrons in the innermost shell, eight electrons in the next shell, and a single electron in the outermost shell.

Like Lewis’ model, Bohr’s model was only partially successful. It explained some experimental results but was quite unable to account for others. In particular it failed on the quantitative mathematical level. The Bohr theory worked very well for a hydrogen atom with its single electron, but calculations on atoms with more than one electron always gave the wrong answer. On a chemical level, too, certain features were inadequate. There is no evidence to suggest that atoms of sodium are ever as elongated or as flat as the one in the figure. On the contrary, the way that sodium atoms pack together in a solid suggests that they extend out uniformly in all directions; i.e., they are spherical in shape. Another weakness in the theory was that it had to assume a shell structure rather than explain it. After all, there is nothing in the nature of planets moving around the sun which compels them to orbit in groups of two or eight. Bohr assumed that electrons behave much like planets; so why should they form shells in this way?

One way to explain the fact that electron energies are quantized, or to explain why electrons can be said to exist in particular shells, is to suggest that they behave like standing waves. Ever since Pythagorus' "music of the spheres" we've noted that waves on a string (like a guitar string) produce only certain numerically defined pitches or tones. We can use a wave model to explain why that's so, for a string that is fixed at both ends.

## Traveling Waves

If you flick a string, a traveling wave moves down it; if you do this continually, say once a second, you generate a travelling wave train with a frequency of 1 s-1, or one wavelength per second, where the wavelength is the distance between successive peaks (or any other repeating feature) of the wave:

Wavelength.

There is a relationship between the frequency, usually denoted "ν" ("nu"), the wavelength, usually denoted "λ" (lambda) and the speed that the wave moves down the string (or through space, if it's a light wave). If we denote the speed "c" (a symbol used for the speed of light), the relationship is:

$\lambda =\tfrac{c}{\nu}$      (1)

Example 1: Calculate the wavelength of a microwave in a microwave oven that travels at the speed of light, c = 3.0 x 108 m s-1> and has a frequency of 2.45 GHz (2.45 x 109 s-1. of 12.24 cm.

Solution: Rearranging (1) we have:

$\nu =\tfrac{c}{\lambda}$ = $\tfrac{\text{3.0}\times \text{10}^{\text{8}}\text{m s}^{\text{-1}}}{\text{2.45}\times\text{10}^{9}\text{s}^{\text{-1}}} =$ 0.1224 m or 2.24 cm

Microwaves are waves like light waves or radio waves, but their wavelength is much longer than light, and shorter than radio. Waves of this wavelength interact with water molecules make the molecules spin faster and thereby heat up food in a microwave oven.

## Standing Waves

If the string we're flicking is held on one end and tied at the other end, the waves are reflected backwards, and the backward moving interact with the forward waves to create a constructive interference pattern which appears not to move. It's called a standing wave:

 Standing wave 1. Standing wave 2 (of higher frequency) How standing waves (black) are created by interference a forward (blue) and reflected (red) waves[1]

Note that the "nodes" (where there is no motion) don't move, but the "antinodes" vibrate up and down, so the exact position of the string is not fixed. Several more standing waves are shown in the next section; all these have particular frequencies that account for the specific notes produced by a guitar string of particular length, or when the string is "fretted", for example. Videos of another standing wave are shown here.

But some frequencies are not allowed (we don't want the guitar to play all tones at once!). It happens in situations like this:

Impossible standing wave.

So the standing wave pattern goes from Standing wave 1 to Standing wave 2, and can't exist anywhere in between. That's exactly the behavior we find for electrons in shells!. Electrons don't exist anywhere between the shells.

## Light Energy

We usually think of electron shells in terms of their energy. That's because light energy is emitted when an electron falls from a higher shell to a lower one, and measuring light energy is the most important way of determining the energy difference between shells. When electrons change levels, they emit quanta of light called "photons"). The energy of a photon is directly related to its frequency, or inversely related to the wavelength:

$\text{E} = \text {h} \times \nu =\frac{\text{h}\times\text{c}}{\lambda}$      (2)

The constant of proportionality h is known as Planck’s constant and has the value 6.626 × 10–34 J s. Light of higher frequency has higher energy, an a shorter wavelength.

Light can only be absorbed by atoms if each photon has exactly the right amount of energy to promote an electron from a lower shell to a higher one. If more energy is required than a photon possesses, it can't be supplied by bombarding the atom with more photons. So we frequently find that light of one wavelength will cause a photochemical change no matter how dim it is, while light of a neighboring wavelength will not cause a photochemical change no matter how intense it is. That's because photons must be absorbed to cause a photochemical change, and they must have exactly the energy needed to be promoted to the next shell to be absorbed. If they're not absorbed, it doesn't matter how intense the light is (how many photons there are per second).

## Energy for Photosynthesis

Not all light is effective in allowing plants to carry out photosynthesis. Plants generally absorb light in roughly the same region that is visible to the Human eye (350 - 700 nm). Shorter wavelengths are absorbed by the ozone layer, or if they make it through, have enough energy to cause cell damage. Longer wavelengths do not provide enough energy for photosynthesis.

Example 2: The absorbance spectrum of some biomolecules involved in photosynthesis is shown here.
Photosynthetic Action Spectrum [2]
Note that "chlorophyll a" absorbs strongly in the red at ~680 nm and in the blue at ~440 nm. Photosynthesis cannot occur in plants irradiated with light in the yellow region (570-590 nm), because it is not absorbe, no matter how bright it is. What is the energy of photons with a wavelength of 440 nm?

Solution:

$\text{E} = \text {h} \times \nu =\frac{\text{h}\times\text{c}}{\lambda}$
E = [(6.626 × 10–34 J s)(3 x 108)] / 440 × 10-9m = 4.52 x 10-19 J.

The energy supplied by each photon of 680 nm light is 2.92 x 10-19 J, and that of 580 nm light is 3.43 x 10-19 J. Even though it lies between the energies of the two photons that cause photosynthesis, it is completely ineffective, no matter how bright (how many photons per second).

In the early 1900s Frederick Frost Blackman and Gabrielle Matthaei<rf>http://en.wikipedia.org/wiki/Photosynthesis</ref> found that at constant temperature, the rate of carbon assimilation varies with light intensity, initially increasing as the intensity increases. How can this be consisten with the energy of light being related to just its frequency or wavelength?

The paradox is solved by noting that intensity is the number of photons per second. If one photon has enough energy to initiate photosynthesis in one chlorophyll molecule, then many photons will cause more photosynthesis. But if the photons were 580 nm light, they would have no effect, no matter what their intensity.

Some botanists claim that blue light at 440 nm is most effective in promoting leaf growth through photosynthesis, while red light around 680 nm is most effective in causing flowering[3].

## Two Dimensional Standing Waves

Of course the shells for electrons are three dimensional, not one dimensional like guitar strings. We can begin to visualize standing waves in more than one dimension by thinking about wave patterns on a drum skin in two dimensions. Some of the wave patterns are shown below. If you look carefully, you'll see circular nodes that don't move:

## The Shape of Orbitals

Electrons exist around the nucleus in "orbitals", which are three-dimensional standing waves. Electron standing waves are quite beautiful, and we'll see more of them in the next few sections. One example is the flower like "f orbital" below. Here the red parts of the "wavefunction" represent mathematically positive (upward) parts of the standing wave, while blue parts are mathematically negative (downward) parts: