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 halflife of a radioactive substance to describe how fast its radioactivity decays. For example, bismuth210 has a halflife of 5 days. After 5 days, 16 grams of bismuth210 decays to 8 grams of bismuth210 (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 halflife 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 bismuth210.

Because its halflife is 5 days, at t = 5, onehalf of n _{0} remains, so .
Bismuth210 decays at the rate of 13.86% per day.
 The halflife of radium226 is 1600 years. What is its annual decay rate?
The decay rate for radium226 is about 0.0433% per year.
Finding the HalfLife from the Decay Rate
In the same way we found the decay rate from the halflife, we can find the halflife from the decay rate. In the formula , 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 halflife?
Decay Rate and HalfLife Practice Problems
Practice
 Suppose a substance has a halflife of 45 days. Find its daily decay rate.
 The halflife of lead210 is 22.3 years. Find its annual decay rate.
 Suppose the halflife for a substance is 1.5 seconds. What is its decay rate per second?
 Suppose a radioactive substance decays at the rate of 0.1 % per day. What is its halflife?
 A radioactive substance decays at the rate of 0.02% per year. What is its halflife?
Solutions
The substance decays at the rate of 46.2% per second.
Carbon14 and Decay Rate
All living things have carbon14 in them. Once they die, the carbon14 is not replaced and begins to decay. The halflife of carbon14 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 carbon14 then will answer some typical carbon14 dating questions.
Carbon14 decays at the rate of 0.012% per year.
Examples
 How long will it take for 80% of the carbon14 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} .
After about 13,400 years, 80% of the carbon14 will have decayed.
 Suppose a bone is discovered and has 60% of its carbon14. How old is the bone? 60% of its carbon14 is 0.60 n _{0} .
The bone is about 4260 years old.
 Suppose an animal dies today. How much of its carbon14 will remain after 250 years?
n (250) = n _{0} e ^{−0.00012(250)} ≈ 0.97 n _{0}
About 97% of its carbon14 will remain after 250 years.
Carbon14 and Decay Rate Practice Problems
Practice
 Suppose a piece of wood from an archeological dig is being carbon14 dated, and found to have 70% of its carbon14 remaining. Estimate the age of the piece of wood.
 How long would it take for an object to lose 25% of its carbon14?
 Suppose a tree fell 400 years ago. How much of its carbon14 remains?
Solutions
The wood is about 2970 years old.
 An object has lost 25% of its carbon14 when 75% of it remains.
After about 2400 years, an object will lose 25% of its carbon14.
 n (400) = n _{0} e ^{−0.00012(400)} ≈ 0.953 n _{0}
About 95% of its carbon14 remains after 400 years.
Find practice problems and solutions for these concepts at: Exponents and Logarithms Practice Test.
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