What is the formula for half-life?

The half-life formula is N(t) = N₀ × (1/2)^(t/t½), where N(t) is the remaining quantity, N₀ is the initial quantity, t is elapsed time, and t½ is the half-life.

How do you find the half-life from the decay constant?

Half-life equals ln(2) divided by the decay constant: t½ = 0.6931 / λ. If the decay constant is 0.000121 per year, the half-life is 0.6931 / 0.000121 = 5730 years.

How many half-lives does it take for a substance to fully decay?

Technically a substance never fully decays — it approaches zero asymptotically. In practice, after about 10 half-lives, less than 0.1% remains, which is considered negligible for most purposes.

How to Calculate Half-Life — Step-by-Step With Examples

Feb 18, 2026 · 8 min read

Half-life shows up constantly in physics, chemistry, geology, medicine — anywhere radioactive decay matters. The idea is simple: it's the time it takes for exactly half of a radioactive substance to decay. But actually doing the calculation trips people up, especially when the numbers get large or the units don't match. So here's the full breakdown, from the basic formula to worked problems.

The core formula

Everything starts from one equation. If you know the initial quantity of a substance, its half-life, and how much time has passed, you can find how much remains:

N(t) = N₀ × (½)^(t / t½)

N(t) = quantity remaining after time t
N₀ = initial quantity (at t = 0)
t = elapsed time
t½ = half-life

That's it. The exponent t / t½ tells you how many half-lives have passed, and raising ½ to that power gives you the fraction still left. If two half-lives have passed, the fraction remaining is (½)² = ¼. Three half-lives: (½)³ = ⅛. And so on.

A simple example first

Problem: You start with 80 grams of Iodine-131 (half-life = 8 days). How much remains after 24 days?

Step 1: Count half-lives → n = 24 / 8 = 3 half-lives

Step 2: Apply formula → N = 80 × (½)³ = 80 × 0.125

Step 3: Result → N = 10 grams remaining

After 24 days (3 half-lives), 10 grams remain and 70 grams have decayed. You can verify: 80 → 40 → 20 → 10. Each step cuts it in half.

That was clean because the elapsed time was an exact multiple of the half-life. But real problems are rarely that neat.

When the numbers aren't clean

Problem: A lab has 500 mg of Cesium-137 (half-life = 30.17 years). How much remains after 100 years?

Step 1: Number of half-lives → n = 100 / 30.17 = 3.315 half-lives

Step 2: Apply formula → N = 500 × (½)^3.315

Step 3: Calculate (½)^3.315 = 0.1005

Step 4: Result → N = 500 × 0.1005 = 50.24 mg remaining

About 90% of the original Cesium-137 has decayed after a century. This makes sense — 3.3 half-lives means more than three halvings but less than four.

The key here is that the formula works with any value of t / t½, not just whole numbers. Your calculator handles the fractional exponent. On most scientific calculators, you'd compute 0.5^3.315 directly.

Finding the decay constant and mean lifetime

The half-life connects to two other important quantities. The decay constant λ (lambda) represents the probability that any single atom will decay per unit time. The mean lifetime τ (tau) is the average time an atom survives before decaying.

λ = ln(2) / t½ = 0.6931 / t½
τ = 1 / λ = t½ / 0.6931 ≈ 1.443 × t½

The mean lifetime is always about 44% longer than the half-life. This sometimes confuses people, but it makes sense if you think about it: the atoms that survive past the first half-life have a disproportionate effect on the average, pulling it above the median.

For Carbon-14 (t½ = 5730 years):

λ = 0.6931 / 5730 = 1.21 × 10⁻⁴ per year

τ = 1 / (1.21 × 10⁻⁴) = 8267 years

So any individual C-14 atom survives on average about 8,267 years — but half of a large sample will have decayed after 5,730 years.

Working backwards: finding elapsed time

Sometimes you know the initial and final quantities and need to find how much time has passed. This is exactly what radiocarbon dating does. Rearrange the formula:

t = −(t½ / ln2) × ln(N / N₀)

or equivalently:
t = t½ × log₂(N₀ / N)

Problem: An archaeological sample has 12.5% of its original C-14. How old is it?

Step 1: Fraction remaining = 0.125

Step 2: t = 5730 × log₂(1 / 0.125) = 5730 × log₂(8)

Step 3: log₂(8) = 3

Step 4: t = 5730 × 3 = 17,190 years old

The sample has been through exactly 3 half-lives. In practice, radiocarbon ages are adjusted using calibration curves because atmospheric C-14 levels haven't been perfectly constant over time.

Working backwards: finding the half-life itself

If you measure how much of a substance remains after a known time, you can determine its half-life:

t½ = −t × ln(2) / ln(N / N₀)

Problem: You measure that 1000 atoms of an unknown isotope decay to 700 atoms in 15 minutes. What's the half-life?

Step 1: N/N₀ = 700/1000 = 0.7

Step 2: t½ = −15 × 0.6931 / ln(0.7)

Step 3: ln(0.7) = −0.3567

Step 4: t½ = −15 × 0.6931 / (−0.3567) = 29.15 minutes

The half-life is about 29 minutes. You can verify: after 15 minutes (roughly half a half-life), you'd expect somewhat more than half remaining — and 70% checks out.

Common mistakes to watch for

The most frequent error is mixing up time units. If the half-life is in years and your elapsed time is in days, the formula gives nonsense. Always convert to the same unit before calculating. The second common mistake is confusing "remaining" with "decayed" — if 25% remains, then 75% has decayed, not the other way around.

Another subtle issue: the formula assumes pure exponential decay from a single isotope. If a sample contains multiple radioactive species (as real samples often do), or if the daughter products are themselves radioactive (as in decay chains), the simple formula doesn't apply directly. You'd need to account for each species separately.

Don't want to do the math by hand? Use our free calculator — it handles all units, shows every step, and draws the decay curve.

Open Half-Life Calculator

Quick reference: half-lives of common isotopes

Some values worth knowing by heart, or at least having handy. Technetium-99m at 6 hours is the workhorse of nuclear medicine. Iodine-131 at 8 days is used in thyroid treatments. Carbon-14 at 5,730 years is the basis of radiocarbon dating. Uranium-238 at 4.47 billion years is used for geological timescales. The range spans over 15 orders of magnitude, from microseconds to billions of years, yet the same formula governs all of them.

If you need values for specific isotopes, our half-life calculator includes 16 built-in presets covering the most commonly encountered radioactive isotopes, from Tritium to Americium-241. For activity calculations (Becquerels and Curies), the decay and activity calculator handles that directly.