Half-Life Calculator
Calculate how much of a substance remains after a given time using exponential decay, plus the decay constant and elapsed half-lives.
How It's Calculated
Formula
N(t) = N_0 \times \left(\frac{1}{2}\right)^{t / T_{1/2}}Half-life describes how quickly a quantity decays exponentially: every time one half-life passes, exactly half of whatever remains decays away. The remaining quantity at any elapsed time t is the initial quantity N0 multiplied by one-half raised to the power of elapsed time divided by the half-life T. This calculator also reports the decay constant (lambda = ln(2) / T), which is the equivalent continuous decay rate used in the exponential form N(t) = N0 * e^(-lambda*t), and the number of half-lives that have elapsed (t / T), which does not need to be a whole number. Both the initial quantity and half-life must be greater than zero, and elapsed time must be zero or greater since this calculator only projects decay forward in time.
Worked Examples
Radioactive sample: 100g with a 10-day half-life, after 20 days
- Elapsed half-lives: 20 / 10 = 2
- Remaining: 100 x (1/2)^2 = 100 x 0.25 = 25g
- Decayed: 100 - 25 = 75g
Carbon-14 dating: 80g sample, half-life 5,730 years, after 5,730 years
- Elapsed half-lives: 5730 / 5730 = 1
- Remaining: 80 x (1/2)^1 = 40g
- Percent remaining: 40 / 80 x 100 = 50%
Frequently Asked Questions
What does half-life mean?
Half-life is the fixed amount of time it takes for exactly half of a decaying quantity to disappear, regardless of how much is present at the start. It applies to radioactive isotopes, drug concentrations in the body, and any process that decays exponentially.
What is the decay constant?
The decay constant (lambda) is an alternative way to express the same decay rate as half-life, related by lambda = ln(2) / half-life. It's used in the continuous exponential form N(t) = N0 * e^(-lambda*t), which is mathematically identical to the half-life formula.
Can elapsed time exceed many half-lives?
Yes. The remaining quantity shrinks toward zero but never becomes negative or reaches exactly zero mathematically — after 10 half-lives, less than 0.1% of the original quantity remains, and the calculator handles arbitrarily large elapsed times without producing invalid results.
Why must the half-life be greater than zero?
The half-life is used as a denominator when computing elapsed half-lives and the decay constant. A half-life of zero or less has no physical meaning and would make the calculation undefined, so this calculator rejects it.