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

Today is 6 February 2008 -- Please complete the following "Basic Science Skills" assessment.

Download the assessment or copy to a word processor. You may turn in your work on-line or in class.

1. What is 5/8 of 16/30?

2. Divide 27/32 by 3/8.

3. The concentration of a chemical in a mixture is 0.023%. How much of the chemical is in 78 grams?

4. 3/8 of a pound of iron is added to 2/3 of a pound of carbon. What is the total weight?

5. 3/26 of an ounce of magnesium oxide is taken from 29/13 ounces of it. How much is left?

6. How many feet in a yard?

7. How many inches in a yard?

8. How many centimeters in an inch?

9. How many centimeters in a yard?

10. 200 milligrams of a substance is added to 2.79 grams of the same substance. What percentage has been added?

11. A mixture is analyzed. It contains 4.5 grams of “A,” 17.06 grams of “B,” 0.06 grams of “C” and the balance is “D.” The overall weight is 30.19 grams. How much “D” is present? What is the percentage each of A, B, C and D in the mixture?

12. Express 0.0011 in scientific notation.

13. What is “standard format?” Give an example.

14. Express 3,259,609 in scientific notation.

15. How many significant figures in each of the following (NOTE: Not multiple choice!):

a. 3.104?

b. 1.0400?

c. 4000?

d. 0.0023

e. 1.9 x 10-3

16. The formula for a certain chemical compound is HxSyOz where "x, y and z" are subscripts that tell how many of a particular atom is present. For example, H2O -- water contains two atoms of hydrogen and one atom of oxygen for a total of three atoms in the molecule. Now, for the problem, if z = 4, x = 1/2 z and y = 1/2 x, how many atoms are present?

17. Carbon-14 is a radioactive isotope of carbon that decays to nitrogen at a predictable rate. The decay rate is used as a "clock" to determine the date of archelogical specimens. The "half-life" is the time it takes for one half of the radioactive carbon in a sample to decay. The half life for carbon-14 is 5730 years. In other words, if you have 10 grams of carbon-14 to begin with, then after one half life, you have 5 grams left. (Make sure you understand the question. Explain to yourself what "half-life" means. )

a. How many half lives will have passed if the sample has 2.5 grams of carbon-14 left?

b. How many years will have passed in two half-lives for carbon-14?

18. The atomic weight of methane gas (CH4) is 16. The atomic weight of air is about 29. This means that methane gas is less dense than air so it will rise. We can demonstrate this by blowing a soap bubble with gas. As it rises, we can touch a flame to the bubble and watch it ignite. It's really fun to watch. But here is my question: if the bubble is 4 inches in diameter, how much gas is in it? (In other words, what is the volume of a sphere whose diameter is 4 inches? The formula is V = (4/3) x Pi (the value is 3.14, if you have forgotten) x radius cubed.

19. In biology and chemistry we often use equations of the form A = B divided by C. (A=B/C). If you know any two of these variables, you can solve for the third. For example, the volume (V) of a solid is 5.75 cubic inches and its mass (m) is 11.50 lbs. What is the density? (D = m / V)

20. This is the last question and it contains a trick. Think carefully before you answer.

a. What is the order of the numbers 1, 3, 5, 7, 9 ?

b. What is the order of the numbers 2, 4, 8, 16, 32, 64 ?

c. What is the order of the numbers eight, five, four, nine, one, seven, six, three, two, zero?

Today is 14 March 2010 -- The Hollow Penny exam

Pennies minted after 1982 were made of zinc with a thin layer of copper plated on top.  The zinc can be dissolved out leaving a copper shell.  We conducted an experiment to explore this and then were given a test.  Here is the test, with answers and the logic to understand HOW to answer the questions:

1.  Pennies minted from 1962 to 1982 were an alloy of 95% copper and 5% zinc.

a. (5) If one of the pennies weighs 3.1 grams, how many grams of copper is in it? 

b. (5) How many grams of zinc is in the penny?

Ans:  An "alloy" is a usually considered a mixture of two or more metals.  Steel is a mixture of iron and carbon.  Brass is a mixture of copper and zinc.  Bronze is a mixture of copper and tin. 

You are told that copper is 95% of the coin and zinc is the remaining 5%.  The percentages must add up to 100%.  This means that 95% of 3.1 grams is the amount of copper in the coin, and 5% of 3.1 grams is the amount of zinc in the coin.  This means that the whole coin -- 3.1 grams -- represents 100% of the mass of it.  To find the amount of copper, set up a proportion like this:  100%/95% = 3.1 grams/x.  Cross multiply to obtain

(100%)(x) = (3.1 grams)(95%).  

Then divide both sides by 100% to get "x" by itself:

x = (3.1 grams)(95%)/100%

Performing the arithmetic you find x = 2.945 grams of copper.  Remember, the "%" signs cancel out and the units that are left are grams.  Since 95% is a part of 100% -- it is LESS than 100% -- the number of grams you get have to be less than 3.1.  If you get MORE than 3.1, you have done something wrong.

To find the amount of zinc in the penny, subtract 2.945 grams from 3.1 grams:

3.1 grams - 2.945 grams = 0.155 grams.  Since there are two digits in the least accurate factor (3.1), we round off the answer to 016 grams of zinc.

2.  Pennies minted from 1983 to the present consist of a core of zinc and a copper cladding.  The penny weighs 2.4 grams.  Zinc is supposed to be 97.5% of the weight of the penny.  Copper is supposed to be the balance.

a.  (5) What percent of copper is present in these pennies?

b.  (5) What is the difference between an "alloy" penny and a "clad" penny?

Ans:  You are told that zinc is 97.5% of the mass of the coin.  To find the percent of copper in this penny, subtract 97.5% from 100% and get 2.5%.  Notice that the question does NOT ask for the number of grams of copper; just the PERCENT.

In class you were told that an alloy is a mixture of metals.  "Clad" means a coating or covering over something.  For example, "When the house caught on fire, Jenny ran out CLAD only in her pajamas."

3.  When zinc is placed in an acid, it reacts to form a zinc salt and a gas. The formula for the reaction of zinc with hydrogen chloride (which is found in hydrochloric acid) is:

                              Zn +  2HCl --> ZnCl2 + H2

a.  (5) What is the name of the salt formed?

b.  (5) What is the name of the gas formed?

ANS:  You were told in class that the most common salt is table salt, or sodium chloride.  It is composed of a metal (sodium) and a non metal (chlorine).  You were told that any metal and non metal are called "salts."  This is the name of a class of compound.  So the salt formed from zinc and chlorine is named "zinc chloride."  There is no sodium in the compound.  You can see this by looking at the chemical equation.  You were also taught that hydrogen has the symbol "H" and that two atoms of hydrogen form a molecule of hydrogen gas, (H2).  

 4.  After soaking the penny in question 2 in hydrochloric acid for two days, Raquel found that the zinc had all disappeared.  She carefully washed and dried what remained of the penny; it was a thin, copper shell.  When she weighed it, she discovered that it was 0.07 gram.  (Raquel had to estimate the answer because the balance was only calibrated in tenths of grams.) 

a.  (5) What percent of the coin remained?

b.  (5) What percent of zinc reacted?

c.  (5) What percent of the coin was zinc?

d.  (5) How do these percentages (of zinc and of copper) compare to the percentages stated (and calculated) in question 2?  ("The 1983 - current penny is said to be ______% Zn and ______% Cu.  Raquel's results indicated the penny was _____% Zn and _____% Cu.  Based on these results Raquel's penny contained _________ (more/less) copper than was predicted from the data.")

ANS:  (part a) You are told that 0.07 grams copper remains.  To find the percent, set up a proportion:

100%/2.4 grams = x/0.07 grams. 

Cross multipy:  (100%)(0.07 grams) = (2.4 grams)(x)

Divide both sides by 2.4 grams to get the unknown "x" by itself.

(100%)(0.07 grams)/(2.4 grams) = x

The grams units cancel.  Divide 0.07 by 2.4 and multiply by 100% to find 2.92% copper.

(part b)  Since ALL of the zinc reacted, 100% of the zinc is gone.

(part c) Since 2.92% of the coin was copper, the zinc must be 100% - 2.92% or 97.08%.  You could also set up another proportion:  100%/2.4 grams = x/(2.4 grams - 0.07 grams).  Then (100%)(2.33 grams) = (2.4 grams)(x) and

(100%)(2.33 grams)/(2.4 grams) = x = 97.08%

(part d) All you had to do was to substitute the numbers you found in the spaces provided and compare the results:  "The 1983 - current penny is said to be _97.5_____% Zn and __2.5____% Cu.  Raquel's results indicated the penny was _97.08____% Zn and _2.92____% Cu.  Based on these results Raquel's penny contained ___more______ (more/less) copper than was predicted from the data.")

5.  After soaking in the HCl, some of the pennies looked silvery.  Why?

ANS:  Because some of the zinc that was inside the penny had to have formed zinc chloride.  The zinc chloride was in solution.  The amount of hydrochloric acid must have been very low, some of the water or HCl gas evaporated over the weekend, and the zinc precipitated out of solution.  [All you needed to write was that the zinc from inside the penny somehow deposited on the outside.]