Calculate radioactive decay, remaining mass, initial amount, and half-life instantly. Solve exponential decay problems with step-by-step formulas. Free online half-life calculator for chemistry, physics, and medicine.
Please provide any three of the following to calculate the fourth value.
Half-life is simply the time it takes for something to drop to half of what it started at. You'll hear it most often when talking about radioactive atoms breaking down, but it works for any kind of decay — exponential or not.
One of the of half-life is carbon-14 dating. Carbon-14 has a half-life of about 5,730 years, which makes it perfect for dating things up to around 50,000 years old. The whole idea was developed by William Libby. Here's how it works: carbon-14 is always being created in the atmosphere. Plants absorb it through photosynthesis, animals eat the plants, and it moves right up the food chain. But once a plant or animal dies, the carbon-14 starts decaying. By measuring how much is left in a sample, scientists can figure out when it died. Simple, brilliant, and it earned Libby a Nobel Prize.
Three equivalent formulas describing exponential decay:
Where:
N0 is the initial quantity
Nt is the remaining quantity after time t
t½ is the half-life
τ is the mean lifetime
λ is the decay constant
If an archaeologist found a fossil sample that contained 25% carbon-14 in comparison to a living sample, the time of the fossil sample's death could be determined since Nt, N0, and t½ are known.
Given: Nt/N0 = 0.25, t½ = 5,730 years
t = t½ × ln(N0/Nt) / ln(2)
t = 5,730 × ln(1/0.25) / 0.693147
t = 5,730 × 1.386294 / 0.693147
t = 11,460 years
This means that the fossil is 11,460 years old.
Using the above equations, it is possible to derive the relationship between t½, τ, and λ. This relationship enables the determination of all values, as long as at least one is known.
t½ = ln(2) / λ
Half-life from decay constant
τ = 1 / λ
Mean lifetime from decay constant
t½ = τ × ln(2)
Half-life from mean lifetime
Here's where it gets real. Half-life isn't just some abstract math concept. It affects your life in ways you probably never noticed.
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You have 100 grams of a radioactive substance with a half-life of 10 years. How much is left after 30 years? Solution: 30÷10=3 half-lives. 100→50→25→12.5 grams.
A medication has a half-life of 4 hours. You take 200 mg. How long until 25 mg remains? 200→100(4h)→50(8h)→25(12h). 12 hours. That's why your doctor says "take one every 12 hours."
A sample starts at 80 grams. After 6 days, only 10 grams remain. What's the half-life? 80→40→20→10 = 3 half-lives in 6 days. Each half-life = 6÷3 = 2 days.
I've seen students make the same mistakes over and over. Here's what to watch out for:
Mistake #1: Thinking half-life is linear. It's not. You don't lose the same amount each time. You lose half of what's left. Big difference.
Mistake #2: Forgetting to count from zero. If something has a half-life of 5 years and you wait 10 years, that's 2 half-lives, not 1. Count carefully.
Mistake #3: Mixing up units. If your half-life is in hours and your time is in days, convert first. Always use the same units.
Mistake #4: Thinking half-life only applies to radioactivity. Nope. It applies to anything that decays exponentially - medicine, population decline, even the brightness of a light bulb over time.
Decay Constant: Another way to describe how fast something decays. λ = ln(2) / t½. The higher it is, the faster things decay.
Exponential Decay vs. Half-Life: Half-life is just one way to talk about exponential decay. Scientists sometimes use "mean life" or "decay constant" instead. They all describe the same thing, just from different angles.
Why Half-Life Is Constant: Here's something that blows people's minds. The half-life of a substance doesn't change based on how much you have. A big chunk of uranium-238 decays at the same rate as a tiny speck. It's about probability, not quantity. Each atom has a certain chance of decaying, and that chance doesn't depend on how many other atoms are around.
The half-life of uranium-238 is 4.5 billion years. That's about the age of Earth.
Carbon-14's half-life (5,730 years) was discovered by Willard Libby in 1946. He won a Nobel Prize for it.
Some elements have half-lives measured in milliseconds. Others take longer than the universe has existed.
The term "half-life" was first used in 1907 by physicist Ernest Rutherford.
Your body has a natural half-life for many substances. Caffeine, for example, has a half-life of about 5 hours in most people.
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Linear decay: If something loses the same amount each time (like a car losing $2,000 in value every year), use a linear model instead.
Doubling time: If something is growing instead of shrinking, you want doubling time, not half-life.
Logistic decay: Some real-world systems don't follow perfect exponential decay. They might slow down or speed up over time.
When in doubt, ask yourself: "Is this thing losing half of what's left each time?" If yes, half-life works. If no, you need a different tool.
Half-life is one of those concepts that seems scary at first but is actually pretty simple once you get it. It's all about things cutting in half over and over again.
Our calculator is here to help you check your work and save time. But don't forget to understand the "why" behind the numbers. That's what makes you actually good at this stuff.
Got a specific problem you're stuck on? Try the calculator. And if you're still confused, come back to this guide. Sometimes reading it a second time makes everything click.
Half-life is the time it takes for half of something to disappear or decay. Think of it like a pizza where you eat half every hour. After one hour, half is gone. After two hours, half of what's left is gone. That's half-life - it's just repeated halving over time.
You can do it by hand if the numbers are nice. Count how many half-lives have passed, then divide your starting amount by 2 that many times. For example, if you start with 100 grams and 3 half-lives pass, you do 100 ÷ 2 = 50, then 50 ÷ 2 = 25, then 25 ÷ 2 = 12.5. For trickier numbers, use the formula N(t) = N₀ × (1/2)^(t / t½).
This confuses a lot of people. Half-life is about probability, not quantity. Each atom has a fixed chance of decaying, and that chance doesn't change based on how many other atoms are around. So a big pile decays at the same rate as a tiny pile - just with more atoms decaying each second.
They describe the same thing but in different ways. Half-life tells you how long it takes for half to decay. Decay constant tells you the probability of decay per unit time. You can convert between them using the formula λ = ln(2) / t½. Think of half-life as the "time" version and decay constant as the "rate" version.