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Radioactive Decay - Review and Examples Help

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By — McGraw-Hill Professional
Updated on Oct 4, 2011

Introduction to Radioactive Decay

Some radioactive substances decay at the rate of nearly 100% per year and others at nearly 0% per year. For this reason, we use the half-life of a radioactive substance to describe how fast its radioactivity decays. For example, bismuth-210 has a half-life of 5 days. After 5 days, 16 grams of bismuth-210 decays to 8 grams of bismuth-210 (and 8 grams of another substance); after 10 days, 4 grams remain, and after 15 days, only 2 grams remains. We can use logarithms and the half-life to find the rate of decay. We will use the decay formula n ( t ) = n 0 e rt in the following problems.

Examples

  • Find the daily decay rate of bismuth-210.
  • Because its half-life is 5 days, at t = 5, one-half of n 0 remains, so Exponents and Logarithms Radioactive Decay .

Exponents and Logarithms Radioactive Decay

Bismuth-210 decays at the rate of 13.86% per day.

  • The half-life of radium-226 is 1600 years. What is its annual decay rate?

Exponents and Logarithms Radioactive Decay

The decay rate for radium-226 is about 0.0433% per year.

Finding the Half-Life from the Decay Rate

In the same way we found the decay rate from the half-life, we can find the half-life from the decay rate. In the formula Exponents and Logarithms Radioactive Decay, we know r and want to find t.

Example

  • Suppose a radioactive substance decays at the rate of 2.5% per hour. What is its half-life?

Exponents and Logarithms Radioactive Decay

Decay Rate and Half-Life Practice Problems

Practice

  1. Suppose a substance has a half-life of 45 days. Find its daily decay rate.
  2. The half-life of lead-210 is 22.3 years. Find its annual decay rate.
  3. Suppose the half-life for a substance is 1.5 seconds. What is its decay rate per second?
  4. Suppose a radioactive substance decays at the rate of 0.1 % per day. What is its half-life?
  5. A radioactive substance decays at the rate of 0.02% per year. What is its half-life?

Solutions

Exponents and Logarithms Radioactive Decay

Exponents and Logarithms Radioactive Decay

Exponents and Logarithms Radioactive Decay

Exponents and Logarithms Radioactive Decay

The substance decays at the rate of 46.2% per second.

Exponents and Logarithms Radioactive Decay

Exponents and Logarithms Radioactive Decay

Carbon-14 and Decay Rate

All living things have carbon-14 in them. Once they die, the carbon-14 is not replaced and begins to decay. The half-life of carbon-14 is approximately 5700 years. This information is used to find the age of many archeological finds. We will first find the annual decay rate for carbon-14 then will answer some typical carbon-14 dating questions.

Exponents and Logarithms Radioactive Decay

Exponents and Logarithms Radioactive Decay

Carbon-14 decays at the rate of 0.012% per year.

Examples

  • How long will it take for 80% of the carbon-14 to decay in an animal after it has died?
  • If 80% of the initial amount has decayed, then 20% remains, or 0.20 n 0 .

Exponents and Logarithms Radioactive Decay

After about 13,400 years, 80% of the carbon-14 will have decayed.

  • Suppose a bone is discovered and has 60% of its carbon-14. How old is the bone? 60% of its carbon-14 is 0.60 n 0 .

Exponents and Logarithms Radioactive Decay

The bone is about 4260 years old.

  • Suppose an animal dies today. How much of its carbon-14 will remain after 250 years?

n (250) = n 0 e −0.00012(250) ≈ 0.97 n 0

About 97% of its carbon-14 will remain after 250 years.

Carbon-14 and Decay Rate Practice Problems

Practice

  1. Suppose a piece of wood from an archeological dig is being carbon-14 dated, and found to have 70% of its carbon-14 remaining. Estimate the age of the piece of wood.
  2. How long would it take for an object to lose 25% of its carbon-14?
  3. Suppose a tree fell 400 years ago. How much of its carbon-14 remains?

Solutions

Exponents and Logarithms Radioactive Decay

The wood is about 2970 years old.

  1. An object has lost 25% of its carbon-14 when 75% of it remains.

Exponents and Logarithms Radioactive Decay

After about 2400 years, an object will lose 25% of its carbon-14.

  1. n (400) = n 0 e −0.00012(400) ≈ 0.953 n 0

About 95% of its carbon-14 remains after 400 years.

Find practice problems and solutions for these concepts at: Exponents and Logarithms Practice Test.

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