Download Article
Download Article
The half-life of a substance undergoing decay is the time it takes for the amount of the substance to decrease by half. It was originally used to describe the decay of radioactive elements like uranium or plutonium, but it can be used for any substance which undergoes decay along a set, or exponential, rate. You can calculate the half-life of any substance, given the rate of decay, which is the initial quantity of the substance and the quantity remaining after a measured period of time. [1] X Research source
Steps
-
What is half-life? The term “half-life” refers to the amount of time that half of the starting substance takes to decay or change. It’s most often used in radioactive decay to figure out when a substance is no longer harmful to humans. [2] X Research source
- Elements like uranium and plutonium are most often studied with half-life in mind.
-
Does temperature or concentration affect the half-life? The short answer is no. While chemical changes are sometimes affected by their environment or concentration, each radioactive isotope has its own unique half-life that isn’t affected by these changes. [3] X Research source
- Therefore, you can calculate the half-life for a particular element and know for certain how quickly it will break down no matter what.
Advertisement -
Can half-life be used in carbon dating? Yes! Carbon dating, or figuring out how old something is based on how much carbon it has, is a very practical way to use half-life. Every living thing intakes carbon while it’s alive, so when it dies, it has a certain amount of carbon in its body. The longer it decays, the less carbon is present, which can be used to date the organism based on carbon’s half-life. [4] X Research source
- Technically, there are 2 types of carbon: carbon-14, which decays, and carbon-12, which stays constant.
Advertisement
-
Understand exponential decay. Exponential decay occurs in a general exponential function where
- In other words, as increases, decreases and approaches zero. This is exactly the type of relationship we want to describe half-life. In this case, we want so that we have the relationship
-
Rewrite the function in terms of half-life. Of course, our function does not depend on generic variable but time [5] X Research source
- Simply replacing the variable doesn't tell us everything, though. We still have to account for the actual half-life, which is, for our purposes, a constant.
- We could then add the half-life into the exponent, but we need to be careful about how we do this. Another property of exponential functions in physics is that the exponent must be dimensionless. Since we know that the amount of substance depends on time, we must then divide by the half-life, which is measured in units of time as well, to obtain a dimensionless quantity.
- Doing so also implies that and be measured in the same units as well. As such, we obtain the function below.
-
Incorporate the initial amount. Of course, our function as it stands is only a relative function that measures the amount of substance left after a given time as a percentage of the initial amount. All we need to do is to add the initial quantity Now, we have the formula for the half-life of a substance. [6] X Research source
-
Solve for the half-life. In principle, the above formula describes all the variables we need. But suppose we encountered an unknown radioactive substance. It is easy to directly measure the mass before and after an elapsed time, but not its half-life. So, let's express half-life in terms of the other measured (known) variables. Nothing new is being expressed by doing this; rather, it is a matter of convenience. Below, we walk through the process one step at a time. [7] X Research source
- Divide both sides by the initial amount
- Take the logarithm, base
of both sides. This brings down the exponent.
- Multiply both sides by
and divide both sides by the entire left side to solve for half-life. Since there are logarithms in the final expression, you'll probably need a calculator to solve half-life problems.
- Divide both sides by the initial amount
Advertisement
-
Read the original count rate at 0 days. Take a look at your graph and find the starting point, or the 0 day mark, on the x-axis. The 0 day mark is right before the material starts decaying, so it’s at its original point. [8] X Research source
- On half-life graphs, the x-axis will usually show the timeline, while the y-axis usually shows the rate of decay.
-
Go down half the original count rate and mark it on the graph. Starting from the top of the curve, note the count rate on the y-axis. Then, divide that number by 2 to get the number at the halfway point. Mark that point on the graph with a horizontal line. [9] X Research source
- For example, if the starting point is 1,640, divide 1,640 / 2 to get 820.
- If you are working with a semi log plot, meaning the count rate is not evenly spaced, you’ll have to take the logarithm of any number from the vertical axis. [10] X Research source
-
Draw a vertical line down from the curve. Starting from the halfway point that you just marked on the graph, draw a second line going downward until it touches the x-axis. Hopefully, the line will touch an easy-to-read number that you can identify. [11] X Research source
-
Read the half-life where the line crosses the time axis. Take a look at the point that your line touched and read where on the timeline it hits. Once you identify the point on your timeline, you’ve found your half-life. [12] X Research source
Advertisement
-
Determine 3 of the 4 relevant values. If you’re solving for half-life, you’ll need to know the initial quantity, the quantity that remains, and the time that has passed. Then, you can use any half-life calculator online to determine the half-life. [13] X Research source
- If you know the half-life but you don’t know the initial quantity, you can input the half-life, the quantity that remains, and the time that has passed. As long as you know 3 of the 4 values, you’ll be able to use a half-life calculator.
-
Calculate the decay constant with a half-life calculator. If you want to calculate how old an organism is, you can input the half-life and the mean lifetime to get the decay constant. This is a great tool to use for carbon dating or figuring out the lifespan of an organism. [14] X Research source
- If you don’t know the half-life but you do know the decay constant and the mean lifetime, you can input those instead. Just like the initial equation, you only need to know 2 of the 3 values to get the third one.
-
Plot your half-life equation on a graphing calculator. If you know your half-life equation and you want to graph it, open up your Y-plots and input the equation into Y-1. Then, hit “graph” to open up your graph and adjust the window until you can see the whole curve. Finally, move your cursor above and below the midpoint of the graph to get your half-life. [15] X Research source
- This is a helpful visual, and it can be useful if you don’t want to do all of the equation work.
Advertisement
-
Problem 1. 300 g of an unknown radioactive substance decays to 112 g after 180 seconds. What is the half-life of this substance?
- Solution: we know the initial amount final amount and elapsed time
- Recall the half-life formula
Half-life is already isolated, so simply substitute the appropriate variables and evaluate.
- Check to see if the solution makes sense. Since 112 g is less than half of 300 g, at least one half-life must have elapsed. Our answer checks out.
-
Problem 2. A nuclear reactor produces 20 kg of uranium-232. If the half-life of uranium-232 is about 70 years, how long will it take to decay to 0.1 kg?
- Solution: We know the initial amount final amount and the half-life of uranium-232
- Rewrite the half-life formula to solve for time.
- Substitute and evaluate.
- Remember to check your solution intuitively to see if it makes sense.
-
Problem 3. Os-182 has a half-life of 21.5 hours. How many grams of a 10.0 gram sample would have decayed after exactly 3 half-lives? [16] X Research source
- Solution: (the amount remaining after 3 half-lives)
- remain
- have decayed
- For this particular equation, the actual length of the half-life did not play a role.
-
Problem 4. A radioactive isotope decayed to 17/32 of its original mass after 60 minutes. Find the half-life of this radioisotope. [17] X Research source
- Solution: (this is the decimal amount that remains)
- (this is how many half-lives have elapsed)
- (to 2 sig figs)
Advertisement
Calculator, Practice Problems, and Answers
Expert Q&A
Search
-
QuestionIf a sample contains 100 g of a radioactive isotope that has a half-life of 2 days, how much of the isotope remains after 6 days?Meredith Juncker is a PhD candidate in Biochemistry and Molecular Biology at Louisiana State University Health Sciences Center. Her studies are focused on proteins and neurodegenerative diseases.One quick way to do this would be to figure out how many half-lives we have in the time given. 6 days/2 days = 3 half lives 100/2 = 50 (1 half life) 50/2 = 25 (2 half lives) 25/2 = 12.5 (3 half lives) So 12.5g of the isotope would remain after 6 days.
-
QuestionIf the half-life of a material is 6 hours, how much material remains in 36 hours?Meredith Juncker is a PhD candidate in Biochemistry and Molecular Biology at Louisiana State University Health Sciences Center. Her studies are focused on proteins and neurodegenerative diseases.We would need to know how much material you start with in order to solve this problem.
-
QuestionWhat is the half-life of an isotope that decays to 25% of its original activity in 26.7 hours?Community AnswerSince the whole is 100%, the first half-life would drop to 50% and then to 25%. Because it takes the isotope 26.7 hours to reach 25%, and there are only 2 halves from 100 to 25%, divide 26.7/2, and you'll get 13.35 hours as the half life.
Ask a Question
200 characters left
Include your email address to get a message when this question is answered.
Submit
Advertisement
Video
Tips
- An alternative formulation for half-life makes use of an integer base. Note that this flips the and in the logarithm expression.Thanks
Submit a Tip
All tip submissions are carefully reviewed before being published
Name
Please provide your name and last initial
Thanks for submitting a tip for review!
Advertisement
References
- ↑ https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Kinetics/Reaction_Rates/Half-lives_and_Pharmacokinetics
- ↑ https://chem.libretexts.org/Courses/Furman_University/CHM101%3A_Chemistry_and_Global_Awareness_(Gordon)/05%3A_Basics_of_Nuclear_Science/5.07%3A_Calculating_Half-Life
- ↑ https://atomic.lindahall.org/what-is-meant-by-half-life.html
- ↑ https://chem.libretexts.org/Courses/Furman_University/CHM101%3A_Chemistry_and_Global_Awareness_(Gordon)/05%3A_Basics_of_Nuclear_Science/5.07%3A_Calculating_Half-Life
- ↑ https://www.ausetute.com.au/halflife.html
- ↑ https://socratic.org/chemistry/nuclear-chemistry/nuclear-half-life-calculations
- ↑ https://socratic.org/chemistry/nuclear-chemistry/nuclear-half-life-calculations
- ↑ https://www.khanacademy.org/test-prep/mcat/physical-processes/atomic-nucleus/a/decay-graphs-and-half-lives-article
- ↑ https://www.gcsescience.com/prad17-measuring-half-life.htm
- ↑ https://www.khanacademy.org/test-prep/mcat/physical-processes/atomic-nucleus/a/decay-graphs-and-half-lives-article
- ↑ https://www.gcsescience.com/prad17-measuring-half-life.htm
- ↑ https://www.khanacademy.org/test-prep/mcat/physical-processes/atomic-nucleus/a/decay-graphs-and-half-lives-article
- ↑ https://www.calculator.net/half-life-calculator.html
- ↑ https://www.calculator.net/half-life-calculator.html
- ↑ https://www.youtube.com/watch?v=5mKrIv1lo1E&feature=youtu.be&t=163
- ↑ https://www.chemteam.info/Radioactivity/Radioactivity-Half-Life-probs1-10.html
- ↑ https://www.chemteam.info/Radioactivity/Radioactivity-Half-Life-probs1-10.html
About This Article
Article Summary
X
To find the half life of a substance, or the time it takes for a substance to decrease by half, you’ll be using a variation of the exponential decay formula. Plug in ½ for a, use the time for x, and multiply the left side by the initial quantity of the substance. Rearrange the equation so that you’re solving for what the problem asks for, whether that’s half life, mass, or another value. Plug in the values you have and solve, writing the answer in seconds, days, or years. To see the half life equation and look at examples, read on!
Did this summary help you?
Thanks to all authors for creating a page that has been read 1,171,934 times.
Reader Success Stories
- "I am an IB student and am in the process of completing my Math IA. I needed an example of application of number e. This helped a lot! Thank you very much." ..." more
Advertisement
"I am an IB student and am in the process of completing my Math IA. I needed an example of application of number e. This helped a lot! Thank you very much."
..."
more
"I am a student, and my whole class is using this site for part of our "Science Geological Time Study Guide." This site is very useful for our purpose!"
..."
more
"I'm just hooked on studying. Well, everyone needs a hobby. wikiHow helps a ton, because things are explained in steps that are easy to follow."
..."
more
"I was given a very well detailed explanation. The steps were well organized and not complex at all."
"Taught me a whole new easy way of finding half life so quickly and simply, thanks."
