Title:  Radioactive Decay and the Inverse-Square Relationship.
 

Audience: 11^th Grade Physics.
 

Duration:  Approx. 45 minutes.
 

References:  Physical Science, Hurd, etal. Ch. 11, pp. 256-262.
                        Physics: A General Introduction, 2^nd ed. Van Heuvelen, Alan. Ch. 34, pp. 720-726.
 

Specific Behavioral Objectives:

1. Students will be able to explain the concept of half-life and draw and label a general graph showing the number of radioactive nuclei remaining versus time.
2. Students will be able to perform calculations involving half-life, decay constant, and time for various situations, including problems involving radioactive dating.
3. Students will be able to determine distances and dose rates in various situations by using the inverse-square relationship.

Materials:

1. Laser pointer
2. Meterstick or ruler
3. Tape measure

Anticipatory Set:  Ask students to predict what will happen to the spot of light on a wall if the source is moved from some short distance away to a greater distance. Have students volunteer to come up and measure the diameter of the spot created by shining a laser light onto the wall, as well as the distance between the light source and the wall. Then move the source to a distance approximately twice the original distance and remeasure the diameter of the spot. The area of the spot should increase approximately by a factor of four. This illustrates the inverse-square relationship.
 

Main Body:

DEFINITIONS

• Half-life (T) - the time required for the number of radioactive nuclei in a given sample to be reduced by one half. After n half-lives, the fraction of radioactive nuclei that remain is

• Draw graph on the board depicting N vs. t
• Work out an example problem on the board

• Decay Rate - the number of radioactive nuclei that decay per unit time. (A.K.A. activity)
• Units: - becquerel (Bq) 1 Bq = 1 decay/second
                        - curie (Ci) 1 Ci = 3.70 x 10¹⁰ decays/second
• Decay rate is a function of an isotope's decay constant, l. The higher the decay constant, the faster that radioactive isotope decays.

• The number of remaining radioactive nuclei (N) depends on time (t) and the decay constant (l) of the isotope:

• Work out an example problem on the board.

• An isotope's decay rate (specified by its decay constant) is related to its half-life:

• Work out an example on the board.

• Radioactive dating is a method of determining the age of very old objects. It relies on the predictable nature of certain radioactive isotopes, such as carbon-14 and potassium-40.
• All living creatures, including plants, take-in carbon in various forms throughout their lives. Some of this carbon consists of trace amounts of the radioactive isotope, C-14. When a plant or animal dies, it no longer takes-in C-14. The C-14 that the plant or animal did have within it becomes trapped, with the only way for the C-14 to leave the organic material being via radioactive decay. So, if you can imagine, it is as if a clock begins ticking within the remains of the plant or animal at the time of its death. The "ticking of the clock" is actually the slow and predictable decay of C-14 atoms. Since the half-life of C-14 is about 5700 years, traces of the isotope remain present for a very long time. When scientists find the remains of organic matter, they can easily determine how long ago that particular plant or creature lived, using the equation below.
• Potassium-40 dating is similar to C-14 dating; however it is used to determine the age of certain rocks instead of the age of organic matter. K-40 has a half-life of over 1 billion years.

• With very little trouble, the exponential decay equation can be manipulated to solve for time (t):
t = -

• Using this equation, we can easily determine the time that has elapsed (t) since an object has ceased to have a particular radioactive isotope added to its make-up, as in when a living thing dies.
• Work out an example on the board.

• The radiation counts detected from a given source vary inversely with the square of the distance from that source:

                    -where C₁ and C₂ are the counts detected at distances D₁ and D₂ respectively

• Work out an example on the board.

Evaluation:  Students will receive verbal feedback on concept mastery by responding to questions and participating in a brief review session. Areas of confusion will be retaught or clarified.
 

PRACTICE PROBLEM

1. Students will work independently at their desks.
2. Given that a particular location on the earth's surface receives approximately 800 W/m² of radiation when the earth is 1.5 x 10¹¹ m from the sun, calculate the amount of radiation (in watts per square meter) incident upon the surface of the planet Mercury, which is 5.8 x 10¹⁰ m from the sun.

Assignments:

1. Do problems # 31-43 (odd only) inVan Heuvelen, pp.737.
2. Complete inverse-square law worksheet.
3. Read over your notes to prepare for laboratory experiment on the inverse-square law tomorrow.

Summary/Closing Statement: Today, we saw some of the relationships that are characteristic of the ways in which radiation and radioactive substances behave. We observed, at the beginning of class, that light becomes less intense as we increased our distance from the source. This was an example of which property? (Inverse-square law) We also talked about radioactive dating. Which method of radioactive dating do you think would be used to determine the age of an old piece of cloth? (C-14 dating) A piece of granite? (K-40 dating) As I said earlier, tomorrow we'll be doing a lab exercise which involves the inverse-square law, so be prepared!
 

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