Last updated: July 3, 2026
Half-Life Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
The half-life calculator solves general first-order decay using N(t) = N0 × (1/2)^(t/t½) or N(t) = N0 × e^(−λt). It can calculate remaining amount, elapsed time, half-life, decay constant λ, mean lifetime τ, percent remaining, and half-lives elapsed. For N0 = 100 and t½ = 10, t = 10 gives 50, t = 20 gives 25, and t = 30 gives 12.5.
Half-life decay means the amount is cut in half every half-life. The remaining amount equals the initial amount times one half raised to elapsed time divided by half-life. The decay constant equals natural log of two divided by the half-life.
Key Takeaways
- Half-life decay follows N(t) = N0 × (1/2)^(t/t½).
- The equivalent exponential form is N(t) = N0 × e^(−λt), with λ = ln2/t½.
- One, two, and three half-lives leave 50%, 25%, and 12.5% of the initial amount.
- Use t = [ln(N0/N)/ln2] × t½ to solve elapsed time from a measured remaining amount.
- Mean lifetime is τ = t½/ln2, about 1.443 times the half-life.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
N(t) = N0 × (1/2)^(t/t½) = N0 × e^(−λt); λ = ln2/t½; τ = t½/ln2
Where:
- N(t)=Remaining quantity after elapsed time(same as N0)
- N0=Initial quantity(amount, mass, atoms, or activity)
- t=Elapsed time(same time unit as t½)
- t½=Half-life(time)
- λ=Decay constant(time⁻¹)
- τ=Mean lifetime(time)
Worked Examples
One half-life leaves 50 g
A 100 g sample with a 10 time-unit half-life is measured after 10 time units.
- 1Compute half-lives elapsed: t/t½ = 10/10 = 1.
- 2Apply N = N0 × (1/2)^(t/t½) = 100 × (1/2)^1.
- 3N = 50 g, so 50% remains.
Two half-lives leave 25 g
The same 100 g sample measured after 20 time units has gone through two halvings.
- 1t/t½ = 20/10 = 2 half-lives.
- 2N = 100 × (1/2)^2 = 100 × 0.25.
- 3N = 25 g, or 25% of the initial amount.
Three half-lives leave 12.5 g
After 30 time units, a 10-unit half-life sample has completed three half-lives.
- 1t/t½ = 30/10 = 3.
- 2N = 100 × (1/2)^3 = 100 × 0.125.
- 3N = 12.5 g, or 12.5% remaining.
Carbon-14 decay constant from half-life
The general half-life equation gives the C-14 decay constant when t½ = 5730 years.
- 1Use λ = ln2/t½.
- 2λ = 0.693147/5730 yr.
- 3λ ≈ 0.000121 yr⁻¹.
Solve half-life from two amounts
A process drops from 100 to 25 units in 20 time units, so the half-life is 10 time units.
- 1Find half-lives elapsed: log2(100/25) = log2(4) = 2.
- 2Half-life = elapsed time / half-lives = 20/2.
- 3t½ = 10 time units.
Introduction
The half-life calculator applies the general first-order decay law to any isotope, radioactive sample, drug elimination step, or process that loses a constant fraction per unit time. It is deliberately broader than the radiocarbon dating calculator: radiocarbon dating is C-14-specific dating, while this tool solves general N(t), elapsed time, half-life, decay constant, and mean lifetime problems. If your work also needs isotope masses or formula mass checks, see the atomic mass calculator and molar mass calculator.
Half-life formula for first-order decay
For first-order decay, the amount remaining after time t is N(t) = N0 × (1/2)^(t/t½). The exponent t/t½ is the number of half-lives elapsed. The same law is often written as N(t) = N0 × e^(−λt), where λ is the decay constant. These forms are identical because λ = ln2/t½.
N0 is the initial amount, activity, concentration, or atom count.
N(t) is the amount remaining after elapsed time t.
t and t½ must use the same time unit.
Each additional half-life halves whatever amount remains, not the original amount.
What the calculator can solve
The strategy logic chooses the correct rearrangement from the values you provide. Enter initial amount, elapsed time, and half-life to solve remaining amount. Enter initial amount, remaining amount, and half-life to solve time. Enter initial amount, remaining amount, and elapsed time to solve half-life. Enter a half-life by itself to compute λ and τ.
If all values are provided, the default primary answer remains N(t), matching the most common classroom and lab use.
Decay constant and mean lifetime
The decay constant λ = ln2/t½ is the fractional first-order rate constant in reciprocal time units. A half-life of 5730 years gives λ ≈ 1.21 × 10⁻⁴ yr⁻¹. The mean lifetime τ = 1/λ = t½/ln2 is longer than the half-life by a factor of about 1.443. NIST summarizes radionuclide half-life measurement practices at NIST PML.
Half-life benchmark table
The powers of one-half make quick checks easy. These benchmarks are useful for radioactive decay, kinetics, and pharmacokinetic elimination.
| Half-lives elapsed | Fraction remaining | Percent remaining | From 100 units |
|---|---|---|---|
| 0 | 1 | 100% | 100 |
| 1 | 1/2 | 50% | 50 |
| 2 | 1/4 | 25% | 25 |
| 3 | 1/8 | 12.5% | 12.5 |
| 4 | 1/16 | 6.25% | 6.25 |
Connections to other chemistry calculators
Radioactive decay and many chemical kinetics problems share exponential mathematics. The Nernst equation calculator and pKa calculator use logarithms to turn ratios into linear quantities, while the Beer-Lambert law calculator uses exponential attenuation of light through matter. LibreTexts gives a teaching overview of radioactive decay at Chemistry LibreTexts/Nuclear_Chemistry/Radioactive_Decay_Rates).
Assumptions and limitations
Use this calculator for ideal first-order decay where the half-life is constant and the daughter products do not affect the measured quantity. It does not model production, branching decay chains, changing environmental exchange, or calibrated calendar ages. For radiocarbon sample ages, reservoir corrections and calibration curves remain separate from this general equation.
Always label time units. A decay constant from a half-life in years has units yr⁻¹; from days it has units day⁻¹.
Quick Reference Card
Half-Life Decay — Quick Reference
Quick reference • Half-Life Calculator
N = N0(1/2)^(t/t½); t = log2(N0/N)t½; λ = ln2/t½; τ = t½/ln2Valid range: N0 > 0, t ≥ 0, t½ > 0, and 0 < N ≤ N0 for inverse solutions
Common Values
⚠ Watch Out
- •Use consistent time units for t, t½, λ, and τ.
- •The inverse half-life formula requires N to be less than N0 and greater than zero.
- •This model assumes ideal first-order decay with a constant half-life.
- •Do not use the general decay result as a calibrated radiocarbon calendar date.
Pro Tips
- →Count half-lives first: t/t½ often gives the fastest mental estimate.
- →Use logarithms when the remaining amount is not an exact power of one-half.
- →Report λ with reciprocal units, such as yr⁻¹ or day⁻¹.
- →For lab data, fit ln(N/N0) versus time; the slope is −λ.
FAQs
How do I calculate the amount remaining after a half-life?
Use N(t) = N0 × (1/2)^(t/t½). For N0 = 100, t = 10, and t½ = 10, the exponent is 1 and N = 100 × 1/2 = 50.
How do I find elapsed time from the amount remaining?
Rearrange the decay law: t = [ln(N0/N)/ln2] × t½. The logarithmic factor is the number of half-lives elapsed.
How do I find half-life from experimental data?
If you know N0, N, and elapsed time t, compute t½ = t / [ln(N0/N)/ln2]. For 100 dropping to 25 in 20 time units, two half-lives elapsed, so t½ = 10.
What is the difference between half-life and decay constant?
Half-life is the time for a quantity to halve. The decay constant λ is the first-order rate constant in reciprocal time units. They are linked by λ = ln2/t½.
Is this calculator only for radioactive substances?
No. It works for radioactive decay and any ideal first-order process with a constant half-life, such as simple pharmacokinetic elimination or first-order decomposition.
Why is mean lifetime longer than half-life?
Mean lifetime is τ = 1/λ = t½/ln2 ≈ 1.443 × t½. It averages the survival time of individual atoms or molecules in an exponential distribution, rather than marking the median time to 50% remaining.