First-Order Half-Life Calculator
In a first-order reaction the rate depends on a single reactant concentration, which gives the decay a fixed half-life that is independent of how much you start with. The half-life is the natural logarithm of 2 divided by the rate constant. This calculator finds the half-life from the rate constant, and also reports the fraction of reactant remaining after any elapsed time using the integrated first-order rate law. The same mathematics describes radioactive decay and many drug-clearance processes. Enter the rate constant in units that match your time scale.
First-order half-life formula
t-half = ln(2) / k = 0.693 / k
fraction remaining = e^(-k * t)
half-lives elapsed = t / t-half
percent remaining = 100 * fraction remaining
The rate constant must be greater than zero and time must be zero or greater. The half-life does not depend on the initial concentration for a first-order process.
Half-life context
- First-order half-life is constant and independent of starting concentration.
- After n half-lives the fraction remaining is (1/2) raised to the power n.
- Radioactive decay is a classic first-order process with a characteristic half-life.
- The rate constant and half-life are inversely related through the natural logarithm of 2.
- Use a time unit for k that matches the elapsed time you enter.
First-order half-life: frequently asked questions
What is the half-life of a first-order reaction?
The half-life of a first-order reaction is the time for the reactant concentration to fall to half its value. It equals the natural logarithm of 2 divided by the rate constant: t-half = 0.693 / k. It does not depend on the starting concentration.
Why is first-order half-life independent of concentration?
For a first-order reaction the rate is proportional to a single concentration, so the relative rate of decay is constant. This makes the time to halve always the same regardless of how much you start with, which is why radioactive decay has a fixed half-life.
How do I find the rate constant from the half-life?
Rearrange the formula: k = 0.693 / t-half. If you know the half-life you can find the rate constant, and vice versa. This calculator computes the half-life from k that you enter.
How much is left after several half-lives?
After n half-lives the fraction remaining is (1/2)^n. After 1 half-life half remains, after 2 a quarter, after 3 an eighth, and so on. This calculator reports the remaining fraction for any elapsed time you enter using the exponential decay law.
What is the integrated first-order rate law?
The integrated first-order rate law is ln([A]/[A]0) = -k t, which rearranges to [A] = [A]0 times e to the power minus k t. The remaining fraction after time t is therefore e to the power minus k t.
Official sources
- IUPAC Gold Book: half-life and rate constant terminology.
- NIST Chemistry WebBook: kinetics reference data.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.