Waves are a vibrating disturbance in a medium by which energy is transferred. Waves can have a wide variety of mediums. For example, you can make a wave in a string held taught between your hands. Or you can drop a pebble into a pond. Ocean waves and sound waves are still other familiar examples. As it happens, one way to describe light is by thinking about its wavelike characteristics. Mysteriously, light does not require a medium in order to travel through space.
All waves have several features in common. Waves have wavelength, frequency, amplitude, and speed. The wavelength of a wave is the distance between repeated identical points on the wave. In the diagram at right one wavelength is the distance from the point marked A to the point marked E. The same distance is found by measuring between B and F. As a numerical quantity, wavelength has a symbol and units. The symbol used for wavelength in equations is the Greek letter lambda (λ). The basic unit is the meter (m) but for light certain other length units are useful. For example, visible light is most easily measured in nanometers (nm) and infrared light in micrometers (μm). One meter is equal to 1 × 109 nm and 1 × 106 μm.
The frequency of a wave is closely related to its wavelength. Frequency is a general term that can be applied to a variety of things. For example, in music the metronome marking on a score tells the musician how fast or slow to play the piece. It is given in units of beats per minute. For a slow piece of music there may be 60 beats per minute. This quantity tells the frequency with which a particular kind of note is produced. For a wave, frequency is a quantity that tells you how many full wavelengths pass by per second. In our musical example, the frequency is 60 beats per minute or one beat per second. If you were making a wave in a string then the frequency would depend on how long you took to move the string up, then down, then all the way up again. As a mathematical quantity, frequency is symbolized by the letter f. Frequency is measured in hertz (Hz). This unit is named after Heinrich Hertz, a German scientist who was a pioneer in the production and use of radio waves. One hertz is equal to one beat per second and can also be written as 1/s (or s-1). Other units used for frequency include gigahertz (GHz), megahertz (MHz), and kilohertz (kHz). One GHz is equal to 1 × 109 Hz; one MHz is equal to 1 × 106 Hz; one kHz is equal to 1 × 103 Hz.
In a particular medium all waves travel at the same speed. But not all waves carry the same amount of energy. Think about shaking a string. It takes more energy to shake it up and down several times per second than when you do it more slowly. This makes a difference in how the wave looks. When you wave the string at a slow pace the wavelengths are long and the frequency is low. When you wave it at a fast tempo the wavelengths are short and the frequency is high. So the higher the frequency, the shorter the wavelength. There is a useful simulator available online which you can use to demonstrate this. It’s found here: https://goo.gl/dP2jEv. For all waves in a given medium it is true that the wavelength times the frequency equals the speed of the wave: speed = (wavelength)·(frequency) or s = λf. This speed is measured based on how fast the crests and troughs of the wave move through space.
The amplitude of a wave is the height of the wave from the midline to the top of a crest or from the bottom of a trough to the midline (see the diagram). It will not play an important role in this lesson.
Waves have certain familiar properties such as reflection, refraction, and diffraction. If you have ever watched a child move forward and backward in the bathtub, then you have seen waves begin to reflect off the front and back of the tub. Echoes are another example of wave reflection. Refraction is what happens to a wave that moves out of one medium and into another. For example, if a water wave is coming in at an angle to the shore into a swampy area. The waves will both slow down and change direction when they get into the swamp. Light exhibits these characteristics: it reflects off of surfaces and this is how we see objects. Light refracts when it passes from air into water or glass. Diffraction is when waves interfere with one another and light shows this property in the colors seen on the back of a DVD or in a soap bubble.
In 1864 a scientist named James Clerk Maxwell (1831 - 1879) published a very concise theory of electricity and magnetism. This theory included an equation that suggested that electro-magnetic energy would be able to travel through space as a wave. The speed of this wave was the same as the already-known speed of light. At that time, light was definitively shown to be nothing other than electro-magnetic radiation: waves of electro-magnetic energy.
Just like any other wave, light travels at some speed and has a wavelength and a frequency. The speed of light depends on the medium it travels through but to keep things simple we will always take the speed of light to be the same as its speed in a vacuum. This is a constant of nature and all light everywhere in the universe travels at this speed and no other. The symbol c is used to mean the speed of light and it equals 3.00 × 108 m/s. The wavelength and frequency of a light wave can be calculated one from the other using the equation c = λf.
Prior to Maxwell’s work it had been thought (by Isaac Newton (1642 - 1727), among others) that light consisted of little tiny particles that flew around bouncing into things. This despite the fact that light acted like a wave when it refracted through a prism to make a rainbow. No one quite knew how to settle the matter until the 19th century, when other strong evidence for the wave nature of light was found, particularly the double-slit experiment of Thomas Young (1773 - 1829). But then, near the very end of the 19th century, a scientist by the name of Max Planck (1858 - 1947) discovered a property of light no one had until then suspected. In short, he found that light was quantized. In other words, the energy of light was not smoothly tunable to any value. Instead, the energy of light came in quanta, which are tiny and unbreakable quantities of energy. The word quanta is the plural of quantum and this work of Planck’s is the first step on the road to developing the quantum theory of matter and energy. These quanta of light were later named photons and it was found that photons can only be produced in whole-number multiples of a very small amount of energy. This energy is embodied as the constant now known as Planck’s constant, which uses the symbol h. Planck’s constant equals 6.626 × 10–34 J·s. All photons have an energy of either 1h, 2h, 3h, 4h, or some larger multiple of h.
As discussed above, waves carry energy and the amount of energy is proportional to the frequency. Higher frequency means more energy went into making the wave. This is true of light and the energy of a single quantum, that is, a single photon is equal to the frequency of the light times Planck’s constant. This is expressed in what we will call Planck’s Formula: E = hf. This equation says that the energy per photon of light equals a constant times the frequency of the light. Energy per photon is given the symbol E and its standard units are joules (J). Another unit used for energy is the electron-volt (eV), which is much smaller and handy for talking about single atoms, electrons, and photons. One electron-volt equals 1.602 × 10–19 J.
When all is said and done the most important thing to understand about light is that it is a form of energy. Photons are truly particles and are truly the embodiment of light but they have no mass: they can be thought of as bundles of pure energy. These bundles of pure energy are exchanged by particles with electric charge, such as electrons and protons. When an electron gains energy it does so by absorbing a photon. The photon’s energy causes a change in the behavior of the electron but the photon itself no longer exists. When an electron loses energy it does so by emitting a photon. The photon that comes into existence carries off the energy lost by the electron.
What does all of this have to do with atoms? Quite a lot, actually. The electrons in an atom occupy different shells called energy levels. The higher the shell of a particular electron, the higher the amount of electronic potential energy it has. When a boulder rolls down a hill or you drop your iPod the gravitational potential energy is changed into kinetic energy: the boulder rolls, the iPod speeds toward doom. When an electron loses potential energy the energy is not changed into motion: it is changed into light. Specifically, a single electron moving from a higher level to a lower level within an atom emits a single photon. The energy of that photon exactly equals the difference in potential energy between the two levels.
Electrons lose potential energy (or gain it) whenever they move from one shell to another. This is analagous to moving from one step to another on a staircase. If an electron is struck by a photon with a particular amount of energy it can be made to hop up to a higher shell. If an electron is in a higher shell it can spontaneously drop to a lower one. In the process it emits a photon with a particular amount of energy. These particular amounts of energy correspond exactly to the difference in potential energy of the two shells. This is the reason why light energy emitted or absorbed by atoms can only have certain values.
Often the shells of an atom are depicted as simple levels, like shelves. Such a diagram is shown on the next page. These energy-level diagrams show different distances between levels, which correspond to different amounts of potential energy. In a hydrogen atom, which only has one electron, the diagram would have one electron in the lowest level (level 1). When the electron is in its lowest level we say that the hydrogen atom is in the ground state. Other atoms have 2 or more electrons. In order for them to be in the ground state all of their electrons must be in the lowest available levels. Level 1 holds just two electrons but level 2 can hold up to eight. If an electron absorbs energy in the form of a photon with the right amount of energy then it enters into a higher level. Whenever any electron in an atom is in a higher level than it would be in the ground state then we say that the atom is in an excited state.
c = λf
(relates frequency and wavelength of light)
E = hf
(relates energy of one photon and frequency of light)
electron volts (eV) and joules (J):
1 eV = 1.602 × 10–19 J
|micrometers (μm): 1 m = 1 × 106 μm|
|nanometers (nm): 1 m = 1 × 109 nm|
|gigahertz (GHz): 1 GHz = 1,000 MHz = 1 × 109 Hz|
|megahertz (MHz): 1 MHz = 1,000 kHz = 1 × 106 Hz|
|kilohertz (kHz): 1 kHz = 1,000 Hz|
Atoms emit and absorb light. When they absorb light an electron moves from a lower energy level to a higher energy level. When they emit light an electron has moved from a higher energy level to a lower energy level. An excited state is when one or more electrons is in a higher energy level than it would be in the ground state. The ground state is the special condition in which all of the electrons in an atom are in the lowest possible energy level.
|Energy Level||Energy (eV)|
The first six energy levels of a hydrogen atom are shown at right. The potential energy an electron has in each level is shown in a table at left. The energies in the table are given in electron volts (eV). This is a special unit used in atomic physics and chemistry and it is valuable because it saves the user from a lot of very small numbers. One electron volt (eV) equals 1.602 × 10-19 J. The energies are negative because they represent potential energy and when the electron is infinitely far away it has zero potential energy. As it gets closer to the nucleus the electron must lose energy (think of this as slowing down so it can be part of an atom). The energy values are negative because in order for the electron to get to that energy level it had to give up the amount of energy shown.
When an electron moves between energy levels it either absorbs or emits energy in the form of a single photon. The energy of the photon matches the difference in the energies of two energy levels that the electron transitions between. To find the difference in energies just subtract the energy of the lower energy level from the energy of the higher level. For example, say the electron transitions between level 5 and level 1. The difference in energies is then (-0.544 eV) – (-13.606 eV) = 13.062 eV. This energy belongs to one particular kind of photon with a specific frequency and wavelength. In order to calculate f and λ the energy in eV must be converted to joules (J):
1.602 × 10-19 J 13.062 eV · ---------------- = 2.093 × 10-18 J 1 eV
Using Planck’s Formula you can find the frequency of the photon that has this amount of energy.
E hf E 2.093 × 10-19 J E = hf - = -- so f = - = ---------------- = 3.158 × 1015 1/s (or Hz) h h h 6.626 × 10-34 J· s
The Speed Formula can be used to find the wavelength of this frequency of light:
c λf c 3.00 × 108 m/s c = λf - = -- so λ = - = ---------------- = 9.50 × 10-8 m f f f 3.158 × 1015 1/s This is a wavelength in the ultraviolet portion of the electromagnetic spectrum.
The above examples can be run in reverse to find the energy change between two energy levels in an atom. In fact, that is really how scientists figure out the internal structure of the energy levels of atoms. Data are collected about what wavelengths of light are produced by atoms of an element in the form of an emission spectrum. The emission spectrum of an element is a unique signature for each element which shows the wavelengths of light emitted by the atoms when electrons in the atoms move from higher energy levels to lower energy levels. The spectrum of hydrogen atoms in the visible range is shown below. The lines you see are specific wavelengths of light that correspond to the energy changes from higher levels down to energy level 2.
Answer the following questions using one or more complete sentences. Everyone in the group must write down complete answers. Discuss among your group members what the best way to answer the question is and then write it down. All members must write down the answer.
|1/s (s–1), Hz, kHz, MHz|
|Energy per photon|
Here are the formulas important to this lesson solved for each variable that you will need to be able to find in this activity. Study them and learn how to rearrange the formula yourself to solve for each variable.
Planck’s Formula Use the first to find energy in J from frequency in Hz. E hf E E = hf - = -- so f = - h h h Use the second to find frequency in Hz from energy in J. The letter h stands for Planck’s Constant and h = 6.626 × 10-34 J·s
Speed Formula c λf c c = λf - = -- so λ = - f f f This is how to find wavelength in m from frequency in Hz. c λf c c = λf - = -- so f = - λ λ λ This is how to find frequency in Hz from wavelength in m. The letter c stands for the speed of light and c = 3.00 × 108 m/s
Use the Speed Formula to find wavelength (λ) given frequency or to find frequency (f) given wavelength. For each problem identify what part of the electromagnetic spectrum each problem refers to by using the spectrum reference sheet you have received.
Sometimes wavelengths of light are given in nanometers (nm, 1 m = 1 × 109 nm) or micrometers (μm, 1 m = 1 × 106 μm). Visible and ultraviolet light are measured using nanometers (nm) and infrared and some microwave wavelengths are are measured using micrometers (μm).
For each of the following problems find the wavelength in meters and then decide whether the light is in the ultraviolet (UV) (10 nm – 400 nm), Visible (400 nm – 700 nm) or the infrared (IR) range (0.7 μm – 1000 μm). If it is in the UV or visible range convert meters to nanometers. If it is in the IR range convert meters to micrometers.
Frequencies, too, are sometimes given in units other than Hz (1/s). The only proper unit to use in c = λf is 1/s (which is the same as Hz). So when frequencies are given in kHz, MHz, or GHz it is necessary to convert them first. 1 GHz = 1,000 MHz = 1 × 109 Hz 1 MHz = 1,000 kHz = 1 × 106 Hz 1 kHz = 1,000 Hz.
Find the wavelength in meters for each of the following frequencies.
Use Planck’s Formula (E = hf) to find energy per photon (E) given frequency (f) or to find frequency given energy per photon. For each problem identify what part of the electromagnetic spectrum each problem refers to by using the spectrum reference sheet you have received.
Often it is important to be able to use a wavelength (λ) to find the energy per photon (E). This is a two-step calculation in which you must first use c = λf to find frequency (f) and second use E = hf to find energy per photon (E). For each of the following, perform the indicated calculation. For each problem identify what part of the electromagnetic spectrum each problem refers to by using the spectrum reference sheet you have received.
The energy in joules (J) for a single photon is extremely small. A better unit for discussing individual photons and energy transitions in atoms is the electron-volt (eV). The conversion between them is 1 eV = 1.602 × 10–19 J. For each energy in joules below, convert to electron-volts. For each energy in eV, convert to J. For each problem identify what part of the electromagnetic spectrum each problem refers to by using the spectrum reference sheet you have received. You will need to calculate the frequency or wavelength in order to do so.
In order to related measurements of wavelengths to energy transitions in atoms it is useful to calculate the energy in eV for a wavelength in nm, or vice versa. For the a given wavelength, calculate frequency in Hz using f = c/λ. Then calculate energy in J using E = hf. Finally, convert the answer in joules to eV. For a given energy in eV, convert to J. Then use f = E/h to calculate the frequency. Then calculate wavelength in meters using λ = c/f. Finish by converting to nm. For each problem identify what part of the electromagnetic spectrum each problem refers to by using the spectrum reference sheet you have received.
|Frequency (Hz)||Energy (J)||Energy (eV)||Transition|
|ultraviolet||121.6||1.216 × 10–7||2.467 × 1015||1.64 × 10–18||10.204||2 —> 1|