How do you calculate half-life?

If you know the decay constant λ, the half-life is t½ = ln(2)/λ ≈ 0.693/λ. If you instead know how much of a sample remains after some time, count the number of halvings: a sample that has dropped to 25% has passed 2 half-lives, so divide the elapsed time by 2. In general, t½ = t / log₂(N₀/N), where N₀ is the starting amount and N is the amount remaining after time t.

How long until a radioactive substance is completely gone?

Mathematically, never — exponential decay halves what remains but never reaches exactly zero. In practice, a sample is considered negligible after about 10 half-lives, when less than 0.1% (1/1024) of the original remains. This is why nuclear medicine uses short-lived isotopes like technetium-99m (6 hours): the radioactivity is effectively gone from the patient within a few days.

What is the difference between half-life and decay constant?

They express the same physics in different units. The decay constant λ is the probability per unit time that any single nucleus decays (units: s⁻¹), while the half-life t½ is the time for half of a large sample to decay. They are related by t½ = ln(2)/λ, so a large decay constant means a short half-life and an intensely radioactive sample.

Why is carbon dating not accurate after 50,000 years?

After 50,000 years — nearly 9 half-lives of carbon-14 — only about 0.2% of the original carbon-14 remains. The tiny signal becomes indistinguishable from background radiation and contamination, so uncertainties explode. For older samples, scientists use isotopes with longer half-lives, such as potassium-40 (1.25 billion years) or uranium-238 (4.47 billion years), which date rocks rather than organic material.

Does temperature or pressure affect radioactive decay?

No — radioactive decay is a nuclear process, and chemical conditions like temperature, pressure, or bonding only affect the electron cloud, not the nucleus. Heating, freezing, or compressing a sample leaves its half-life unchanged. This insensitivity is exactly why radiometric dating is trustworthy: a rock's clock ticks at the same rate whether it spent a billion years buried, frozen, or baked.

What is activity and what is a becquerel?

Activity is the number of nuclear decays per second in a sample, A = λN. The SI unit is the becquerel (Bq): 1 Bq = 1 decay per second. The older unit, the curie (Ci), equals 3.7 × 10¹⁰ Bq. Because activity is proportional to the number of remaining nuclei, it decays with exactly the same half-life as the sample itself.

Which isotope has the longest half-life?

Tellurium-128 holds the record, with a half-life of about 2.2 × 10²⁴ years — roughly 160 trillion times the age of the universe. Its decay is so slow it was only detected indirectly. Among more familiar nuclides, bismuth-209 (2 × 10¹⁹ years) was long believed to be stable until its decay was finally observed in 2003.

Half-Life Calculator and Radioactive Decay Simulator

Q: What is the formula for radioactive decay?

The exponential decay law is N(t) = N₀e^(−λt), where N₀ is the initial number of nuclei and λ is the decay constant. An equivalent and often easier form uses half-lives directly: N = N₀ × (1/2)^(t/t½). Both describe the same curve — the second form is usually faster for exam problems where the elapsed time is a whole number of half-lives.

Right now, roughly 5,000 atoms of potassium-40 are decaying inside your body every second. You cannot predict which atom will go next — nobody can, not even in principle — and yet physicists can tell you with astonishing precision that half of any potassium-40 sample will be gone in 1.25 billion years. That strange combination of individual randomness and collective certainty is captured by a single number: the half-life.

This page is both a half-life calculator and an interactive radioactive decay simulator. You can watch 400 simulated nuclei decay one random atom at a time, see the exponential decay curve emerge from pure chance, and use the decay formula to solve the exact problems that appear in AP Physics, A-Level, JEE and NEET exams — including how carbon dating reveals the age of an 11,460-year-old bone.

What Is Half-Life in Physics?

The half-life ( $t_{1/2}$ ) of a radioactive isotope is the time it takes for half of the nuclei in a sample to decay. Start with 1,000 atoms of carbon-14, wait 5,730 years, and about 500 remain. Wait another 5,730 years and about 250 remain — not zero. Every half-life halves whatever is left:

Half-lives elapsed	0	1	2	3	4	5
Fraction remaining	100%	50%	25%	12.5%	6.25%	3.125%

Think of it like flipping coins. Give every atom in a sample a coin, and once per half-life every atom flips: heads it survives, tails it decays. You can never say which specific coin will land tails — but you can say with near-certainty that about half of a large pile will. Half-life is not a property of any individual atom; it is a statistical property of the ensemble, which is why the same idea shows up in our post on statistical distributions and sampling.

Why Is Radioactive Decay Random but Predictable?

Radioactive decay is a quantum process. A nucleus does not age, weaken, or "wear out" — a carbon-14 atom created this morning has exactly the same probability of decaying in the next hour as one that has already survived 50,000 years. Physicists call this being memoryless.

A good everyday analogy is popcorn. Watch a single kernel and you have no idea when it will pop — it might go in the first second or never. But put 400 kernels in the pan and the pattern of popping is so reproducible that microwave manufacturers print the time on the button. Individual randomness plus large numbers equals collective predictability.

This is exactly what the simulator below shows. Each square is one nucleus assigned a random decay moment. Press New random sample and every individual square changes — but the decay curve underneath barely moves.

Interactive Half-Life Simulator

Choose an isotope, then drag the Elapsed Time slider to watch the sample decay. The top panel shows 400 individual nuclei (purple = original parent isotope, amber = decayed daughter product). The bottom panel plots the decay curve: the x-axis shows elapsed time measured in half-lives (0 to 6 — using half-lives instead of years or days means the same curve describes every isotope, whether its half-life is six hours or 4.47 billion years), and the y-axis shows the fraction of the original sample remaining, falling from 1 toward 0. Diamonds mark each whole half-life. The status box doubles as a calculator readout: it reports the elapsed time in real units, the percentage remaining, and the decay constant λ for the selected isotope.

Loading chart...

Isotope

Elapsed Time

1 half-lives

Random Sample

Each nucleus decays at a random moment — but the curve barely changes. That is the statistical heart of half-life.

After 5,730 years (1.00 half-lives), 50.0% of the original Carbon-14 remains. λ ≈ 1.210e-4 per year. Used for radiocarbon dating of organic remains.

What Is the Formula for Radioactive Decay?

The number of parent nuclei $N$ remaining after time $t$ follows the exponential decay law:

$N(t) = N_0 \, e^{-\lambda t}$

where $N_0$ is the initial number of nuclei and $\lambda$ is the decay constant — the probability per unit time that any single nucleus decays. The half-life and the decay constant are two ways of stating the same fact:

$t_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$

For quick exam work there is an equivalent form that skips $e$ entirely. If $n = t / t_{1/2}$ is the number of half-lives elapsed:

$N(t) = N_0 \left(\frac{1}{2}\right)^{n}$

Two related quantities appear constantly in problems:

Activity $A = \lambda N$ — the number of decays per second, measured in becquerels (Bq). Activity decays with exactly the same half-life as $N$ itself.
Mean lifetime $\tau = 1/\lambda = t_{1/2} / \ln 2 \approx 1.44 \, t_{1/2}$ — the average time a nucleus survives, slightly longer than the half-life because a few lucky nuclei last many half-lives.

How Do You Calculate the Amount Remaining Step by Step?

Find the number of half-lives elapsed: $n = t / t_{1/2}$
Halve the starting amount $n$ times: $N = N_0 \cdot (1/2)^n$
If $n$ is not a whole number, use $N = N_0 \, e^{-0.693\, t / t_{1/2}}$ or $N = N_0 \cdot 2^{-n}$ directly.

Example: An 80 g sample of iodine-131 ( $t_{1/2} = 8.02$ days) is stored for 24.06 days. Then $n = 24.06 / 8.02 = 3$ half-lives, so $N = 80 \cdot (1/2)^3 = 10$ g remains.

How Does Carbon Dating Work?

Cosmic rays striking the upper atmosphere constantly create carbon-14, which mixes into the CO₂ that plants absorb and animals eat. While an organism is alive, its carbon-14 level stays in equilibrium with the atmosphere. The moment it dies, intake stops — and the carbon-14 clock starts ticking down with a half-life of 5,730 years.

To date a sample, measure what fraction of the original carbon-14 remains, then invert the decay law:

$t = t_{1/2} \cdot \log_2\!\left(\frac{N_0}{N}\right)$

Example: A bone fragment retains 25% of its original carbon-14. Since $25\% = (1/2)^2$ , exactly 2 half-lives have passed: $t = 2 \times 5{,}730 = 11{,}460$ years.

After about 50,000 years (nearly 9 half-lives) only about 0.2% of the carbon-14 remains — too little to measure reliably against background radiation. That is why geologists switch to slower clocks like uranium-238 ( $t_{1/2} = 4.47$ billion years) for dating rocks, which is how we know the Earth is about 4.54 billion years old.

Half-Lives of Common Isotopes

Isotope	Half-life	Primary use
Technetium-99m	6.01 hours	Medical diagnostic imaging
Radon-222	3.82 days	Environmental radiation monitoring
Iodine-131	8.02 days	Thyroid treatment
Cobalt-60	5.27 years	Radiotherapy, sterilisation
Carbon-14	5,730 years	Radiocarbon dating
Plutonium-239	24,100 years	Nuclear fuel / weapons
Uranium-235	704 million years	Nuclear reactors
Uranium-238	4.47 billion years	Dating rocks, age of Earth

Notice the enormous range — from hours to billions of years. Short half-lives mean intense activity that fades fast (ideal for medical imaging); long half-lives mean feeble activity that persists essentially forever (ideal for geological clocks). The energy scales involved connect directly to the quantum physics we explore in the photoelectric effect.

Worked Examples for Physics Exams

Example 1: Medical dose decay (Iodine-131)

A hospital receives 64 mg of iodine-131 ( $t_{1/2} = 8.02$ days). How much remains after 32.08 days?

$n = 32.08 / 8.02 = 4$ half-lives → $N = 64 \cdot (1/2)^4 = 4$ mg. Verify in the simulator: select Iodine-131 and drag the slider to 4 half-lives — the readout shows 6.25% remaining.

Example 2: Finding a decay constant (Radon-222)

What is the decay constant of radon-222 ( $t_{1/2} = 3.82$ days)?

$\lambda = \ln 2 / t_{1/2} = 0.693 / 3.82 = 0.1814$ per day. In SI units: $0.1814 / 86{,}400 \text{ s} = 2.10 \times 10^{-6}\ \text{s}^{-1}$ .

Example 3: Dating a wooden tool (Carbon-14)

A wooden tool shows carbon-14 activity at 12.5% of the level found in living wood. How old is it?

$12.5\% = (1/2)^3$ , so 3 half-lives have passed: $t = 3 \times 5{,}730 = 17{,}190$ years.

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