Radioactive decay is the cleanest first-order process you’ll ever meet. There’s no concentration dependence, no temperature dependence, no catalyst that speeds it up — every nucleus has the same fixed probability per unit time of falling apart, and that probability is the decay constant λ. Half-life is the more intuitive way to talk about it (“how long until half is gone?”), but λ is the number you actually plug into equations, so converting between the two fluently is the first thing to learn.

The relationship that ties everything together

λ = ln(2) / t½ = 0.6931 / t½

That’s it. A faster-decaying isotope has a bigger λ and a shorter half-life. They’re inversely related through the natural log of 2, which appears because half-life is defined as the time for N to drop by a factor of 2 in an exponential, and ln(2) is the exponent that produces that factor.

A first calculation: carbon-14

Carbon-14 has a half-life of 5730 years. Its decay constant:

λ = 0.6931 / 5730 = 1.21 × 10⁻⁴ /year

Need it per second for an activity calculation? Convert:

λ = 1.21 × 10⁻⁴ / (365.25 × 24 × 3600) = 3.83 × 10⁻¹² /s

This is where most students slip — picking the wrong time unit for whatever they’re computing next. If you need activity in becquerel (decays per second), λ has to be in per-second. Mixing units silently gives wrong answers because the equation doesn’t complain.

The exponential decay equation

N(t) = N₀ × e^(−λt)

Same physics in two equivalent forms:

N(t) = N₀ × (1/2)^(t/t½)

Use whichever is convenient. The exponential form is what shows up in derivations and software; the half-life form is faster for back-of-envelope estimates (“three half-lives, so 1/8 left”).

Half-life from a measured decay constant

An isotope decays at λ = 0.0866 /hour. Its half-life:

t½ = 0.6931 / 0.0866 = 8.00 hours

That’s the iodine-131 thyroid clearance neighborhood (its physical half-life is 8.02 days; 8 hours is closer to certain biological clearance contexts).

Activity calculations

Activity A is just λ multiplied by the number of radioactive atoms present:

A = λN

A sample of 5.00 × 10²⁰ atoms of phosphorus-32 (t½ = 14.3 days). Activity in becquerel?

First convert λ to per-second:

λ = 0.6931 / (14.3 × 86400) = 5.61 × 10⁻⁷ /s

Then:

A = 5.61 × 10⁻⁷ × 5.00 × 10²⁰ = 2.81 × 10¹⁴ Bq

That’s 281 TBq, or roughly 7600 Ci. A very hot sample — well into “professional shielding required” territory.

Carbon-14 dating

A wooden artifact retains 62.5% of the C-14 activity of a living tree. How old?

Start from N/N₀ = 0.625 = e^(−λt). Take ln of both sides:

ln(0.625) = −λt

−0.4700 = −(1.21 × 10⁻⁴) × t

t = 0.4700 / 1.21 × 10⁻⁴ = 3884 years

Cross-check using the half-life form: 0.625 = (0.5)^(t/5730). Take log of both sides, divide, t = 3885 years. Same answer to rounding — the two forms are mathematically identical, so use whichever feels cleaner.

C-14 dating has a useful range of about 50,000 years (roughly 9 half-lives) before the remaining activity is too low to measure reliably above background.

Decay constant from experimental data

You don’t always know the isotope. Sometimes you measure activity twice and back out λ. A sample reads 800 Bq initially, then 200 Bq after 5.0 hours:

A = A₀ × e^(−λt)

200 = 800 × e^(−5.0λ)

0.25 = e^(−5.0λ)

ln(0.25) = −5.0λ

−1.386 = −5.0λ

λ = 0.277 /hour, t½ = 0.6931 / 0.277 = 2.50 hours

Sanity check: activity dropped by a factor of 4 in 5 hours, which is two half-lives, so t½ = 2.5 hours. Matches.

Common slip-ups

Using log instead of ln. Decay equations use the natural log because the exponential is base e. Plug in log₁₀ and you get an answer that’s wrong by a factor of ln(10) ≈ 2.303. The cleanest defense: never write log without specifying base.

Time-unit mismatches. If λ is per year and you ask for activity per second, your answer is off by a factor of 31.5 million. Convert λ first, then plug in.

Confusing decay constant with half-life. Bigger λ = faster decay = shorter half-life. They move in opposite directions. If you find yourself with a “decay constant” of 5730 years for C-14, you’ve grabbed the half-life by mistake.

Practice

Verify against the Half-Life Calculator:

Decay constant for radon-222 (t½ = 3.82 days), in /day and /s.
Cobalt-60 has λ = 0.1315 /year. Half-life?
Activity drops from 1000 Bq to 125 Bq in 24 hours. λ and t½?
K-40 (t½ = 1.25 × 10⁹ years), 2.0 × 10¹⁸ atoms in a rock. Activity in Bq?
Sr-90 (t½ = 28.8 years) was released in a 1962 nuclear test. Fraction remaining today?

How to Calculate the Decay Constant