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Summary

  • In the decay of a radioactive substance, if the decay constant ( λ ) is large, the half-life is small, and vice versa.
  • The radioactive decay law, N = N 0 e λ t , uses the properties of radioactive substances to estimate the age of a substance.
  • Radioactive carbon has the same chemistry as stable carbon, so it mixes into the ecosphere and eventually becomes part of every living organism. By comparing the abundance of 14 C in an artifact with the normal abundance in living tissue, it is possible to determine the artifact’s age.

Conceptual questions

How is the initial activity rate of a radioactive substance related to its half-life?

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For the carbon dating described in this chapter, what important assumption is made about the time variation in the intensity of cosmic rays?

That it is constant.

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Problems

A sample of radioactive material is obtained from a very old rock. A plot ln A verses t yields a slope value of 10 −9 s −1 (see [link] (b)). What is the half-life of this material?

The decay constant is equal to the negative value of the slope or 10 −9 s −1 . The half-life of the nuclei, and thus the material, is T 1 / 2 = 693 million years .

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Show that: T = 1 λ .

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The half-life of strontium-91, 38 91 S r is 9.70 h. Find (a) its decay constant and (b) for an initial 1.00-g sample, the activity after 15 hours.

a. The decay constant is λ = 1.99 × 10 −5 s 1 . b. Since strontium-91 has an atomic mass of 90.90 g, the number of nuclei in a 1.00-g sample is initially
N 0 = 6.63 × 10 21 nuclei .
The initial activity for strontium-91 is
A 0 = λ N 0 = 1.32 × 10 17 decays/s
The activity at t = 15.0 h = 5.40 × 10 4 s is
A = 4.51 × 10 16 decays/s .

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A sample of pure carbon-14 ( T 1 / 2 = 5730 y ) has an activity of 1.0 μ Ci . What is the mass of the sample?

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A radioactive sample initially contains 2.40 × 10 2 mol of a radioactive material whose half-life is 6.00 h. How many moles of the radioactive material remain after 6.00 h? After 12.0 h? After 36.0 h?

1.20 × 10 2 mol ; 6.00 × 10 3 mol ; 3.75 × 10 4 mol

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An old campfire is uncovered during an archaeological dig. Its charcoal is found to contain less than 1/1000 the normal amount of 14 C . Estimate the minimum age of the charcoal, noting that 2 10 = 1024 .

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Calculate the activity R , in curies of 1.00 g of 226 R a . (b) Explain why your answer is not exactly 1.00 Ci, given that the curie was originally supposed to be exactly the activity of a gram of radium.

a. 0.988 Ci; b. The half-life of 226 R a is more precisely known than it was when the Ci unit was established.

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Natural uranium consists of 235 U ( percent abundance = 0.7200 % , λ = 3.12 × 10 −17 / s ) and 238 U ( percent abundance = 99.27 % , λ = 4.92 × 10 −18 / s ) . What were the values for percent abundance of 235 U and 238 U when Earth formed 4.5 × 10 9 years ago?

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World War II aircraft had instruments with glowing radium-painted dials. The activity of one such instrument was 1.0 × 10 5 Bq when new. (a) What mass of 226 R a was present? (b) After some years, the phosphors on the dials deteriorated chemically, but the radium did not escape. What is the activity of this instrument 57.0 years after it was made?

a. 2.73 μ g ; b. 9.76 × 10 4 Bq

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The 210 P o source used in a physics laboratory is labeled as having an activity of 1.0 μCi on the date it was prepared. A student measures the radioactivity of this source with a Geiger counter and observes 1500 counts per minute. She notices that the source was prepared 120 days before her lab. What fraction of the decays is she observing with her apparatus?

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Armor-piercing shells with depleted uranium cores are fired by aircraft at tanks. (The high density of the uranium makes them effective.) The uranium is called depleted because it has had its 235 U removed for reactor use and is nearly pure 238 U . Depleted uranium has been erroneously called nonradioactive. To demonstrate that this is wrong: (a) Calculate the activity of 60.0 g of pure 238 U . (b) Calculate the activity of 60.0 g of natural uranium, neglecting the 234 U and all daughter nuclides.

a. 7.46 × 10 5 Bq ; b. 7.75 × 10 5 Bq

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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