Students

I read your letters telling me why you did not do your homework to find the energy and frequency of light at different wavelengths . Many of you complained that you did not know what to do. I won’t argue with you. I am going to teach you exactly what to do, step-by-step.

1. What this is all about -- There is an important relation between the wavelength of light, its frequency and the energy it contains. Lets make sure you remember what wavelength, frequency and energy mean.

a. wavelength – When you toss a stone in a pool of water, you see ripples, or “waves.” The waves are energy being sent from the spot where the stone hit the water to a place were the energy is released. The distance from the top, or crest, of one wave to the top of the next is the length of the wave. Water, sound, earthquakes, radio and light are just some sources of waves. Water waves need oceans, rivers, pools, etc., to travel. Sound requires air, liquid or solid. It will not travel through a vacuum. Earthquake waves travel through the earth. Light and radio waves are forms of “electromagnetic radiation.” They can travel through the vacuum of space.

Radio waves are usually measured in meters or even kilometers. Light is much, much shorter. We measure the wavelength of light in units called “nanometers.” A nanometer, abbreviated “nm,” is one billionth of a meter. In other words, it would take one billion (1,000,0000,000) nanometers lined up end to end to equal one meter.

b. frequency – The number of times something happens is called “frequency.” The earth rotates on its axis one time each 24 hours. It has a frequency of one time per day. Electricity produced by generators changes its direction as it is produced. In the United States, it changes direction 60 times per second. Each complete change of direction is a “cycle.” We say that electricity has a frequency of 60 cycles per second. Sound, light and radio waves, among others, also are cyclic. This means that one wave follows another. (If you have just one wave, it is a “pulse.”) The number of waves that pass a given point in some unit of time is the frequency of the wave. If the unit of time is one second, then the frequency is given the special name “Hertz.”

It is very difficult to count the waves as they pass so we find frequency another way. We know that waves move at a constant speed through whatever they are traveling. When light or radio waves move through a vacuum, they travel at 299,800,000 meters per second (m/sec). This is the “speed of light.” If we divide the speed of light by the length of light or radio waves, the answer is the number of waves passing in a second.

In symbols we would write f = c / l. . The symbol “c” stands for the speed of light. The symbol “l” (Greek letter “lambda”) stands for the wavelength. Lets assume that the wavelength is one meter (1 m). This is a billion times longer than a nanometer, so it has to be a radio wave, not a light wave. To find the frequency, we divide 299,800,000 m/sec by 1 m.

The meter in the numerator cancels the meter in the denominator. Then the frequency is 299,800,000 / s . We read this as “frequency is two hundred ninety nine million eight hundred thousand per second.”

c. energy – Radio and light waves are called “electromagnetic radiation” and occupy different parts of the “electromagnetic spectrum.” They carry energy and the amount of energy depends upon the frequency of the wave. Max Planck, a great physicist, studied the relationship between frequency and the energy of the wave. He found that the relationship was constant. In other words, dividing the energy of a wave at a specific wavelength by the frequency at that wavelength always, always gave the same result: 6.626068 × 10^-34 m² kg / s. That is h = E / f.

In the equation,”h” is the constant, named “Planck’s constant” in honor of Max Planck. “E” is the energy and “f” is the frequency. We usually rearrange the equation to solve for Energy: E = hf.

Notice two things about the constant. First, it is written in scientific notation. This is because the number is so small. If we write it out, it would be 0.0000000000000000000000000000000006626068 m²kg / s. In this form, the number is very difficult to work with. Second, the unit of energy “m²kg / s” is read as “meter squared kilogram per second.” It is a strange unit and we will talk about it in a little while.

2. Energy and electron orbitals -- Neils Bohr studied how the orbitals occupied by electrons in atoms were related to their energy. His work gave us what we call the “Bohr model of the atom.” It is very important. Bohr found that that the light emitted by electrically excited hydrogen related to the orbitals that hydrogen’s only electron could occupy. Look at the spectrum of light emitted by hydrogen:

To find the frequency at each wavelength we divide the wavelength into the speed of light. We have to use a “conversion factor” to change nanometers to meters so our units are correct. As an example we will find the frequency for the wavelength of 656 nm.

We will write the numbers in scientific notation format. We can do this because our number system is based on “10.” That is, 100 can be written as 1 x 10 x 10 or 1 x 10², 200 as 2 x 10 x 10 or 2 x 10², 1000 as 1 x 10 x 10 x 10 or 1 x 10³ and so on. That is great for numbers 1 or more. What about numbers less than 1? For example, what is 1/100? The result by long division is 0.01 and we read it as “one hundredth.” In scientific format we move the decimal point to the right until we have one non-zero digit to the left of the decimal. In this case that is two decimal places. We write the number as 1 x 10^-2. The “-“ in the exponent indicates that the number is less than one. Then in our problem, we get:

Notice in the term (1 m /1 x 10⁹ nm), “10⁹ nm” is in the

denominator of the denominator[1]. Using powers of 10 makes the math easier. Use the rule that to divide powers of 10, you subtract exponents. To multiply, you add exponents. We divided 6.56 into 2.998 by long division. Our answer is
f = 4.57 x 10¹⁴/s because we apply the rule of one digit to the left of the decimal point. Do you see? We have cancelled like units in the numerator and denominator. We read the answer, as “The frequency is 4.57 x 10¹⁴ per second.”

To find energy, E, multiply 4.57 x 10¹⁴/s times 6.626068 × 10^-34 m² kg/s. The answer is 30.281 x 10 ^-20 m² kg/s². The unit “m² kg/s²” is called a “Joule,” named after English physicist James Prescott Joule (1818–89).

Students, please find the energy for the other 3 wavelengths of visible light in hydrogen’s emission spectrum. Then prepare a graph plotting the wavelength, l, against the Energy in Joules. For which wavelength is the energy least? For which is it greatest? What are the associated colors? Please see me for questions. When you have completed the graph, we will conduct the “glowing star” experiment.

JAY L. STERN, 26 March 2011

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[1] de·nom·i·na·tor noun \di-ˈnä-mə-ˌnā-tər\ the part of a fraction that is below the line and that functions as the divisor of the numerator