Half-Life Calculator
Radioactive & Exponential Decay Tool
Solve the exponential decay equation N = N₀ × (1/2)^(t / half-life) for any one of its four variables.
Enter t and the half-life in the same time units. N and N₀ must use the same quantity units.
Understanding Half-Life
Half-life describes how quickly a quantity shrinks under exponential decay — a fixed amount of time in which exactly half of whatever remains disappears, no matter how much you started with.
Half-life is the constant time interval in which a decaying quantity is reduced to exactly half of its current amount. It's a property of the decay process itself, not of the starting quantity — a huge sample and a tiny sample of the same substance both lose half their mass in the same amount of time.
Radioactive isotopes decay at a fixed half-life, which is the basis of carbon-14 dating. In pharmacokinetics, a drug's half-life determines how often a dose must be repeated. The same exponential model also describes signal decay in circuits and certain financial depreciation curves.
The Formula
Where N is the remaining quantity, N₀ is the initial quantity, t is elapsed time, and t₁ᐟ₂ is the half-life. Solving the same equation for t or t₁ᐟ₂ uses logarithms: t = t₁ᐟ₂ × log₂(N₀ / N) and t₁ᐟ₂ = t / log₂(N₀ / N).
Working With the Four Variables
Because the decay equation relates four quantities, knowing any three lets you solve for the fourth. If you know how much material you started with, its half-life, and how long it's been decaying, you can find how much remains. If instead you know a starting and remaining amount along with the half-life, you can work out how much time has passed. And if you know a starting amount, a remaining amount, and the elapsed time, you can back out the substance's half-life itself — this is exactly how radiometric dating techniques like carbon-14 dating work.
Every half-life period cuts the remaining quantity exactly in half.
The approximate half-life of carbon-14, used to date organic remains.
N, N₀, t, and half-life — solve for any one given the other three.
Key Takeaways
- Half-life is constant — it doesn't depend on how much material you start with, only on the substance or process itself.
- The remaining quantity can never exceed the initial quantity under pure decay, which is why solving for time or half-life requires N ≤ N₀.
- Time and half-life must share the same units, and the two quantity values (N and N₀) must share the same units, or the result will be meaningless.
- The same equation powers carbon dating, drug dosing schedules, and radioactive safety planning — anywhere a quantity decays exponentially over time.
Frequently Asked Questions
- Choose which value you want to solve for from the Solve For dropdown: Remaining Quantity (N), Initial Quantity (N₀), Elapsed Time (t), or Half-Life.
- The field matching your selection is disabled — fill in the other three fields instead.
- Click "Calculate" to see the solved value.
Half-life is the amount of time it takes for exactly half of a quantity to decay or be eliminated. It's a constant property of an exponential decay process — no matter how much material you start with, the same half-life always cuts the remaining amount in half. After two half-lives, a quarter remains; after three, an eighth; and so on.
The formula is N = N₀ × (1/2)^(t / t₁ᐟ₂), where N is the remaining quantity, N₀ is the initial quantity, t is the elapsed time, and t₁ᐟ₂ is the half-life. This calculator can rearrange the same equation to solve for any one of the four variables given the other three.
Pure exponential decay only ever reduces a quantity — it never increases it, so the remaining amount N can never exceed the starting amount N₀. When solving for elapsed time or half-life, the calculator takes a logarithm of the ratio N₀/N, which only produces a valid (non-negative) time or half-life when N ≤ N₀. If N equals N₀, no decay has happened yet, so time must be exactly zero and the half-life itself can't be determined from the inputs.
Half-life shows up anywhere a quantity decays exponentially. Radioactive isotopes decay at a fixed half-life — carbon-14 dating uses its ~5,730-year half-life to estimate the age of organic remains. In pharmacology, a drug's biological half-life describes how long it takes the body to eliminate half of a dose, which determines dosing schedules. The same math also models depreciation of certain assets and the decay of a signal or charge in some electrical circuits.