Half-Life Calculator
Solve any of N₀, N(t), t, or T from the other three using the standard decay equation N(t) = N₀ × (½)^(t/T). Works for radioactive isotopes, first-order chemical reactions, and drug elimination — any process with a constant half-life.
Remaining amount N(t)
25
- Half-lives elapsed (t / T)
- 2
- Fraction remaining
- 0.25
- Percent decayed
- 75%
- Decay constant λ = ln(2) / T
- 1.209681e-4
- Mean lifetime τ = 1 / λ
- 8,266.6426
N(t) = N₀ × (1/2)^(t/T). This is the closed-form solution of first-order decay (dN/dt = −λN) with λ = ln(2) / T. Use any consistent time unit — the formula is scale-free as long as t and T share units.
How to use this calculator
Pick which quantity you want to solve for. The calculator needs three of the four values: initial amount (N₀), remaining amount (N(t)), elapsed time (t), and half-life (T). Enter values in any units you like — the formula is scale-free, but the elapsed time and the half-life must share the same unit. If you put the half-life in years, put the elapsed time in years too. The breakdown shows the decay constant λ, the mean lifetime τ, the fraction of the original sample still present, and how many half-lives have elapsed.
How the calculation works
Half-life describes any process where a quantity decreases by half over a fixed interval, regardless of the starting amount. The closed-form equation is N(t) = N₀ × (½)^(t/T), which is the solution to the first-order rate law dN/dt = −λN with decay constant λ = ln(2) / T. After one half-life, half remains; after two, a quarter; after n half-lives, (½)ⁿ. The decay constant gives the instantaneous rate (units of inverse time), and the mean lifetime τ = 1/λ = T / ln(2) is the average time an individual particle survives — always about 1.443 times the half-life. Although the term comes from nuclear physics, the same maths applies to any first-order process: drug elimination, capacitor discharge, fluorescence decay, and many enzyme-catalysed reactions.
Worked example
A 100-gram sample of carbon-14 has a half-life of 5,730 years. How much remains after 11,460 years? Set N₀ = 100, T = 5,730, t = 11,460. The exponent t / T = 11,460 / 5,730 = 2, so N(t) = 100 × (½)² = 25 grams. The decay constant is λ = ln(2) / 5,730 ≈ 1.210 × 10⁻⁴ per year and the mean lifetime is τ = 5,730 / ln(2) ≈ 8,267 years. Solve in reverse: if a fossil sample contains 25 % of its original C-14, t = T × log₂(N₀ / N(t)) = 5,730 × log₂(4) = 5,730 × 2 = 11,460 years.
Frequently asked questions
What is the half-life formula?
The standard form is N(t) = N₀ × (½)^(t / T), where N₀ is the initial amount, N(t) is the amount remaining after time t, and T is the half-life. Equivalently, you can write it with the natural exponential as N(t) = N₀ × e^(−λt), where the decay constant λ = ln(2) / T. The two forms are mathematically identical — the first is easier to reason about in half-lives, the second is what falls out of the differential equation dN/dt = −λN.
Does the formula assume anything about the substance?
Only that the process is first-order — the rate of decay is proportional to the amount present. This is exactly true for individual radioactive isotopes because each nucleus decays independently, but it is also a very good approximation for many other processes: most drug elimination in the body, RC capacitor discharge, the fluorescence decay of an excited atom, and pseudo-first-order chemical reactions. If the rate depends on concentration in a more complex way (second-order, zero-order, autocatalytic), the half-life is not constant and the formula does not apply.
How is the decay constant λ related to the half-life T?
They are reciprocally linked through ln(2): λ = ln(2) / T, equivalently T = ln(2) / λ. The decay constant is the fraction of the sample that decays per unit time at any given instant — it has units of inverse time (s⁻¹, yr⁻¹). The mean lifetime τ = 1 / λ = T / ln(2) is the average time before any given particle decays; it is always about 1.443 × T, never equal to the half-life itself.
Can I use this for drug half-lives or pharmacokinetics?
Yes — for drugs that follow first-order elimination, the same equation describes plasma concentration over time. After one half-life, 50 % of the dose remains; after roughly 4 to 5 half-lives, more than 95 % has been eliminated, which is why clinicians often quote "five half-lives" as a rule of thumb for clearance. The result is a plasma concentration, not blood-level safety guidance — always defer to dosing tables and clinical judgement for real medication decisions.
How do I solve for the half-life if I know how much has decayed?
Rearrange the formula: T = t × ln(2) / ln(N₀ / N(t)). Equivalently, T = t / log₂(N₀ / N(t)). Pick the "Half-life T" mode and enter the initial amount, the remaining amount, and the time that has elapsed. This is the basis of radiocarbon dating — measure the ratio of C-14 to C-12 in a sample and back-calculate the age from the known C-14 half-life.
What if my units do not match?
They have to. The formula is scale-free in time, which means it works in any unit, but t and T have to be in the same unit. If your half-life is given in seconds and your elapsed time in minutes, convert one of them first. The amounts N₀ and N(t) just need to be in the same unit as each other — grams, atoms, moles, counts per minute — the ratio is what matters.