Half-life describes radioactive decay, drug elimination and carbon dating. Enter the starting amount, the half-life and the time elapsed to see how much remains and the decay constant.
Enter the initial amount.
Enter the half-life and elapsed time in the same unit.
Read the remaining amount.
Remaining: N = N₀ × (½)^(t ÷ T½)
Decay constant: λ = ln 2 ÷ T½
A half-life is the time it takes for half of something to disappear through a steady, random process. This calculator takes an initial amount, the half-life and the time elapsed, and tells you how much is left, what percentage that is, how many half-lives have passed and the decay constant λ. Any unit of time works, as long as the half-life and the elapsed time use the same one.
The idea is central to Class 12 nuclear physics and to carbon dating in archaeology. It also describes medical isotopes used in Indian hospitals for scans and therapy, the fading of radioactive contamination, and, in pharmacology, how the level of many medicines in the blood falls over time. Whenever a quantity loses the same fraction in each equal interval, this calculator applies.
1. Write the initial amount N₀. It can be grams, number of atoms, activity in becquerels or a percentage.
2. Write the half-life T½ and the elapsed time t in the same time unit.
3. Find the number of half-lives: t ÷ T½.
4. Find the amount left: N = N₀ × (½)^(t ÷ T½).
5. Find the fraction remaining as N ÷ N₀ × 100%.
6. Find the decay constant λ = ln 2 ÷ T½ ≈ 0.693 ÷ T½, in units of per unit time.
Each radioactive nucleus has a fixed chance of decaying in any short time, independent of its age or of its neighbours. So the number decaying per second is proportional to the number present: dN/dt = −λN. The solution of this equation is N = N₀e^(−λt). Setting N = N₀ ÷ 2 gives e^(−λT½) = ½, so λT½ = ln 2. Substituting back gives N = N₀ × (½)^(t ÷ T½), the form the calculator uses. Both forms are the same curve.
After one half-life 50% remains, after two 25%, after three 12.5%, and after ten about 0.1%. The amount never mathematically reaches zero, but for a small sample the last few atoms do decay, one at a time, at random moments. That is the key point about a half-life: it is a statistical average for large numbers of nuclei. You cannot predict when a single atom will decay, only the fraction of a large group that will have decayed by a given time.
The mean life τ = 1 ÷ λ is the average lifetime of a nucleus and equals about 1.44 half-lives. In carbon dating, living things keep the same ratio of carbon-14 to carbon-12 as the air. After death no new carbon-14 comes in, so its fraction halves every 5,730 years or so. Measuring the remaining fraction and solving t = T½ × log₂(N₀ ÷ N) gives the age. Beyond about 50,000 years too little carbon-14 remains to measure reliably.
A hospital in Chennai receives 200 MBq of iodine-131, which has a half-life of about 8 days, and the physicist wants to know the activity left after 30 days in storage.
Number of half-lives: 30 ÷ 8 = 3.75
N = N₀ × (½)^(t ÷ T½): = 200 × 0.5^3.75 = 14.865089
Decay constant: λ = ln 2 ÷ T½ = 8.6643 × 10^-2 per unit time
Answer: Amount remaining 14.865089; Half-lives elapsed 3.75; Decay constant λ 8.6643 × 10^-2
Using different time units for the half-life and the elapsed time, such as days and hours.
Subtracting half the original amount each half-life instead of halving what is left.
Using the half-life where the decay constant is needed, or forgetting λ = 0.693 ÷ T½.
Using log₁₀ instead of ln when working with e^(−λt) by hand.
Expecting an exact prediction for a single atom rather than an average for many.
Class 12 nuclear physics numericals on radioactive decay.
Carbon dating of wood, bone and cloth in archaeology.
Planning storage and disposal of medical and laboratory radioisotopes.
Understanding how long a medicine stays in the body, as a teaching model only.
Modelling any quantity that falls by a fixed fraction per interval, such as charge on a leaking capacitor.
What is carbon-14's half-life?
About 5,730 years, which makes it useful for dating organic material up to about 50,000 years old.
Does a sample ever fully decay?
Mathematically it approaches zero; practically it becomes undetectable.