Radioactive Decay - Review and Examples Help
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.
- 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 .
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?
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 , we know r and want to find t.
- Suppose a radioactive substance decays at the rate of 2.5% per hour. What is its half-life?
Decay Rate and Half-Life Practice Problems
- Suppose a substance has a half-life of 45 days. Find its daily decay rate.
- The half-life of lead-210 is 22.3 years. Find its annual decay rate.
- Suppose the half-life 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 half-life?
- A radioactive substance decays at the rate of 0.02% per year. What is its half-life?
The substance decays at the rate of 46.2% per second.
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.
Carbon-14 decays at the rate of 0.012% per year.
- 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 .
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 .
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
- 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.
- How long would it take for an object to lose 25% of its carbon-14?
- Suppose a tree fell 400 years ago. How much of its carbon-14 remains?
The wood is about 2970 years old.
- An object has lost 25% of its carbon-14 when 75% of it remains.
After about 2400 years, an object will lose 25% of its carbon-14.
- 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.