Half-Life Calculator

Formulas & Assumptions

• Remaining amount: N(t) = N₀ × (1/2)^{t / T½}
• Decay constant: λ = ln(2) / T½
• Elapsed time: t = T½ × log₂(N₀ / N)
• Half-life from data: T½ = t / log₂(N₀ / N)

These relations are valid for first-order exponential decay with a constant half-life. That model is standard for radioactive decay and many idealized first-order processes.

The important idea behind half-life is proportional decay: the system loses the same fraction in each equal interval, not the same absolute amount. That is why a short-lived isotope, a first-order drug clearance model, and a simple population-decay example can all use the same mathematics even when their units and physical interpretations differ. Once the half-life is known, the decay constant, the fraction remaining, and the elapsed-time relationship all follow from the same exponential model.

• Initial amount must be greater than zero.
• Remaining amount must be greater than zero and cannot exceed the initial amount.
• Half-life and elapsed time must be positive when solving for finite decay.
• If the remaining amount equals the initial amount after positive time, the implied half-life is infinite.

How to Use It

1. Select what you want to solve for.
2. Enter the known amount values using the same amount unit.
3. Enter the known time value and choose its unit.
4. Calculate to get the result, fraction remaining, and λ.
5. Use copy or download if you need a quick lab note or calculation record.

In practice, the safest workflow is to keep the amount units consistent and focus on whether the time relationship is sensible. If one half-life passes, half the amount should remain. If two half-lives pass, one quarter should remain. If three pass, one eighth should remain. Those checkpoints are useful for spotting input mistakes quickly before you rely on the detailed result. The calculator exposes the decay constant and milestone table for that reason: they give you an immediate sanity check on whether the entered half-life matches the story told by the remaining amount.