Physics
How To Calculate Half Life
A half life is the time it takes for half of a sample to decay. Three quantities describe the whole process, and any two of them fix the third.
Quick Answer
N = N0 x (1/2)^(t / t1/2)
- N
- Amount remaining after the elapsed time
- N0
- Starting amount of the substance
- t
- Elapsed time
- t1/2
- Half life of the substance
- lambda
- Decay constant, equal to ln 2 divided by the half life
Divide the elapsed time by the half life to count how many half lives have passed, raise one half to that power, and multiply by the starting amount. After 24 days a 100 milligram sample with an 8 day half life falls to 12.5 milligrams, or 12.5 percent of the original, having passed through 3 half lives with a decay constant of 0.08664339756999316 per day.
What Is Half Life?
A half life is the time it takes for half of a radioactive sample to decay. It is a property of the substance rather than of the amount you happen to have, so a gram and a tonne of the same isotope share exactly the same half life. That is what makes the number useful: it describes the substance, not the sample.
The mathematics is exponential decay. After one half life half the sample remains, after two a quarter, after three an eighth. The pattern is not linear, and the amount never quite reaches zero on paper. Each half life removes half of whatever is left rather than half of the original, which is the distinction that separates exponential decay from a steady loss.
Counting half lives is the practical route through the calculation. Divide the elapsed time by the half life to get how many have passed, then raise one half to that power. After 24 days with an 8 day half life, three half lives have passed and the remaining fraction is one eighth, so a 100 milligram sample is down to 12.5 milligrams.
The decay constant is the other way to describe the same process. It is the natural logarithm of two divided by the half life, and it expresses the probability per unit time that any given atom decays. For an 8 day half life the decay constant is 0.08664339756999316 per day. A larger constant means faster decay.
The two descriptions are exactly equivalent. Using the decay constant, the remaining amount is the starting amount multiplied by e to the power of minus lambda t. Using half lives, it is the starting amount multiplied by one half to the power of t over the half life. Both formulas give the same answer at every time, and the identity linking them is that e to the power of minus lambda times the half life equals one half.
Real samples contain enormous numbers of atoms, and decay is a probabilistic process, so the smooth curve is a statistical description rather than a guarantee. The half life is the time by which, on average, half the atoms have decayed. For a sample of a few atoms the actual behaviour is visibly lumpy, and the formula describes the average rather than the individual case.
Half lives span an absurd range. Some isotopes decay in fractions of a second while others, such as tellurium-128, have half lives of around ten to the power of twenty-four years. This enormous spread is what makes some isotopes useful for dating and others hazardous for millennia, and it is why the number is always quoted with the isotope rather than on its own.
The concept is not restricted to radioactivity. Drugs have biological half lives describing how long the body takes to clear half a dose, pollutants have environmental half lives describing persistence in soil or water, and even a cooling cup of coffee can be described by a half life. The arithmetic is identical; only the mechanism differs.
A common source of confusion is that after several half lives the remaining activity can be negligibly small but the material is still present. Ten half lives leaves less than a tenth of a percent, which is often below detection, yet the atoms that have not decayed are still there. Half life describes the transformation, not the disappearance of matter.
Radiometric dating rests directly on this formula. If the starting amount is known or can be inferred from a stable daughter product, then measuring how much remains gives the elapsed time. Carbon-14 with its 5730 year half life suits archaeological timescales, while uranium-lead systems with half lives in the billions of years date rocks.
Formula
N = N0 x (1/2)^(t / t1/2)
Count the half lives that have passed, raise one half to that power, and scale the starting amount.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| N | Remaining amount | same as N0 | How much is left after the elapsed time. |
| N0 | Starting amount | mass or count | The amount at the beginning. |
| t | Elapsed time | same as half life | How long has passed. |
| t_1/2 | Half life | time | The time for half the sample to decay. |
lambda = ln(2) / t1/2
The probability per unit time that an atom decays. It is inversely proportional to the half life.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| lambda | Decay constant | per unit time | Fraction of the sample decaying per unit time. |
| ln 2 | Natural logarithm of two | dimensionless | The constant 0.6931471805599453. |
| t_1/2 | Half life | time | The time for half the sample to decay. |
N = N0 x e^(-lambda x t)
The same decay written with the decay constant instead of counting half lives. Equivalent at every time.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| N | Remaining amount | same as N0 | How much is left after the elapsed time. |
| N0 | Starting amount | mass or count | The amount at the beginning. |
| lambda | Decay constant | per unit time | Fraction of the sample decaying per unit time. |
| t | Elapsed time | time | How long has passed. |
How To Calculate Half Life
- 1
Write down the starting amount and the half life
Both must use consistent time units. An 8 day half life with a 24 day elapsed time is straightforward; mixing days and hours is not, and the conversion has to happen before anything else.
- 2
Divide the elapsed time by the half life
Twenty-four days divided by 8 days is 3, so three half lives have passed. The division removes the units, which is why the half life and the elapsed time must match.
- 3
Raise one half to that power
One half cubed is one eighth, or 0.125. Each whole half life halves the fraction, so the exponent is simply how many halvings have happened. A fractional exponent handles times that fall between whole half lives.
- 4
Multiply by the starting amount
One eighth of 100 milligrams is 12.5 milligrams. If you would rather have a percentage, the fraction on its own is the answer: 0.125, or 12.5 percent of the original.
- 5
Work out the decay constant if you need a rate
Divide the natural logarithm of two, which is 0.6931471805599453, by the half life. For 8 days that gives 0.08664339756999316 per day, the fraction of the sample that decays each day in the continuous model.
Examples
Example 1: A 100 milligram sample with an 8 day half life after 24 days
- Starting amount
- 100 mg
- Half life
- 8 days
- Elapsed time
- 24 days
| Step | Calculation | Result |
|---|---|---|
| Half lives elapsed | 24 / 8 | 3 |
| Fraction remaining | 0.5 x 0.5 x 0.5 | 0.125 |
| Decay constant | 0.6931471805599453 / 8 | 0.08664339756999316 |
| Amount remaining | 100 x 0.125 | 12.5 |
Result: After 24 days a 100 milligram sample with an 8 day half life has passed through 3 half lives, leaving 12.5 milligrams, which is 12.5 percent of the original, with a decay constant of 0.08664339756999316 per day.
Example 2: Carbon-14 dating after 11460 years
- Starting amount
- 100 percent
- Half life
- 5730 years
- Elapsed time
- 11460 years
| Step | Calculation | Result |
|---|---|---|
| Half lives elapsed | 11460 / 5730 | 2 |
| Fraction remaining | 0.5 x 0.5 | 0.25 |
| Decay constant | 0.6931471805599453 / 5730 | 0.00012096809433855938 |
| Amount remaining | 100 x 0.25 | 25 |
Result: A carbon-14 sample left for 11460 years has passed through 2 half lives, so 25 percent of the original remains, with a decay constant of 0.00012096809433855938 per year.
Example 3: An 80 milligram dose with a 3.5 hour biological half life
- Starting amount
- 80 mg
- Half life
- 3.5 hours
- Elapsed time
- 10.5 hours
| Step | Calculation | Result |
|---|---|---|
| Half lives elapsed | 10.5 / 3.5 | 3 |
| Fraction remaining | 0.5 x 0.5 x 0.5 | 0.125 |
| Decay constant | 0.6931471805599453 / 3.5 | 0.19804205158855578 |
| Amount remaining | 80 x 0.125 | 10 |
Result: An 80 milligram dose with a 3.5 hour biological half life is down to 10 milligrams after 10.5 hours, having passed through 3 half lives at a rate of 0.19804205158855578 per hour.
Calculator
Amount remaining
12.5
- Fraction of the original remaining
- 12.50%
- Half lives elapsed
- 3
- Decay constant
- 0.0866
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the Half Life calculator page.
Common Mistakes
Subtracting the half life instead of dividing
The half life is not an amount to take away from the elapsed time, it is a unit of measurement for it. Dividing 24 days by 8 days gives 3 half lives; subtracting would give 16 days, which means nothing at all.
Halving the original amount rather than what is left
Each half life removes half of the current amount, not half of the starting amount. After two half lives a quarter remains, not zero, and after three an eighth remains. Treating it as a straight line down to nothing is the single most common misunderstanding.
Mixing time units between the half life and the elapsed time
A half life given in hours and an elapsed time given in days must be converted before dividing. The division only makes sense if both are in the same unit, and the error is easy to miss because the numbers themselves still look reasonable.
Confusing the decay constant with the half life
The decay constant is a rate per unit time and the half life is a duration. They are inversely related through the natural logarithm of two, so a large constant means a short half life. Quoting one where the other is expected gives an answer that is wrong by a factor of about 0.69 even when the units look right.
Expecting the amount to reach exactly zero
Exponential decay approaches zero without arriving. After ten half lives about a tenth of a percent remains, which may be undetectable, but the atoms that have not decayed are still present. The model describes the transformation, not the removal of matter.
Assuming the half life depends on how much you start with
The half life is a property of the substance. A gram and a tonne decay with the same half life, because each atom decays independently of the others. Starting with more material means more atoms decay per second, but the time to lose half of them is unchanged.
FAQ
What is the half life formula?
The amount remaining equals the starting amount multiplied by one half raised to the power of the elapsed time divided by the half life. Dividing 24 days by an 8 day half life gives 3, so one half cubed is 0.125 and a 100 milligram sample is down to 12.5 milligrams.
What is the difference between half life and decay constant?
The half life is a duration and the decay constant is a rate per unit time. They are linked by the natural logarithm of two: the constant equals 0.6931471805599453 divided by the half life. A short half life means a large decay constant.
Does the half life depend on the starting amount?
No. The half life is a property of the substance, so a gram and a tonne of the same isotope decay with exactly the same half life. More material simply means more decays per second, not a different timescale.
How many half lives until the material is gone?
It never completely goes, because each half life removes half of whatever is left. After ten half lives about a tenth of a percent remains, which is often below the detection limit but is still physically present.
How is half life used in carbon dating?
Carbon-14 has a half life of 5730 years. Measuring how much of it remains in a sample, against the amount expected in living tissue, gives the number of half lives elapsed and therefore the age. After 11460 years, 25 percent remains.
Can half life apply to things other than radioactivity?
Yes. Drugs have biological half lives describing how long the body takes to clear half a dose, and pollutants have environmental half lives describing persistence in soil or water. The mathematics is the same exponential decay in every case.
References
- [1]Wikipedia, Half-life — https://en.wikipedia.org/wiki/Half-life
- [2]Wikipedia, Exponential decay — https://en.wikipedia.org/wiki/Exponential_decay
- [3]Wikipedia, Radioactive decay — https://en.wikipedia.org/wiki/Radioactive_decay
- [4]Wikipedia, Carbon-14 — https://en.wikipedia.org/wiki/Carbon-14