Remaining Percentage After 5 Half-Lives
Starting from 100 units. Use the calculator below to try your own values.
Remaining Quantity
3.125
Example
Starting with 100 units and a half-life of 1, after 5 time units (5 half-lives), about 3.125 units remain — 3.13% of the original.
Percent Remaining
3.13%
Half-Lives Elapsed
5
Decayed Amount
96.875
Decay Constant (λ)
0.693147
What Is Half-Life?
Half-life is the time it takes for a quantity undergoing exponential decay — most famously radioactive isotopes, but also drug concentrations in the body, capacitor discharge, and other decay processes — to reduce to exactly half its starting value. This holds true at every point in the decay: it takes the same amount of time to go from 100% to 50% as it does to go from 50% to 25%.
Because the decay rate is proportional to the amount remaining rather than a fixed quantity per unit time, half-life decay never technically reaches zero — the remaining quantity gets smaller and smaller but only approaches zero in the mathematical limit.
Decay curve
Quantity Remaining at Each Half-Life
| Half-Lives | Quantity Remaining | % Remaining |
|---|---|---|
| 0 | 100 | 100% |
| 1 | 50 | 50% |
| 2 | 25 | 25% |
| 3 | 12.5 | 12.5% |
| 4 | 6.25 | 6.25% |
| 5 | 3.125 | 3.125% |
| 6 | 1.5625 | 1.5625% |
| 7 | 0.7812 | 0.7812% |
| 8 | 0.3906 | 0.3906% |
| 9 | 0.1953 | 0.1953% |
| 10 | 0.0977 | 0.0977% |
How Is Half-Life Calculated?
The exponential decay formula relates the remaining quantity to the initial quantity, the elapsed time, and the half-life. Given any three of these values (plus the decay constant, which is derived from half-life), the fourth can always be solved for algebraically.
- N(t) — remaining quantity at time t
- N₀ — initial quantity
- t — elapsed time
- T — half-life
The Decay Constant (λ)
The decay constant λ = ln(2) / T describes the same decay process as an instantaneous rate, used in the alternative form N(t) = N₀ × e^(−λt). Both formulas describe identical decay — half-life is simply a more intuitive way to express the same underlying rate.
Half-Life in Radioactive Decay
Radioactive isotopes decay at wildly different rates — some, like Carbon-14 (half-life ~5,730 years), decay slowly enough to be useful for dating ancient organic material, while others decay in fractions of a second. This is why radiocarbon dating works for artifacts up to roughly 50,000 years old but not much older — beyond that, too little Carbon-14 remains to measure reliably.
Half-Life Beyond Physics
The same math describes how drugs clear from the bloodstream (pharmacokinetic half-life), how a charged capacitor discharges, and how many other real-world quantities decay proportionally to their current amount rather than at a constant rate — any process following this pattern can be analyzed with the same half-life formula.
Example — Your Current Inputs
Starting with 100 units and a half-life of 1, after 5 time units (5 half-lives), about 3.125 units remain — 3.13% of the original.
Additional Example — Carbon-14 Dating
A sample with 25% of its original Carbon-14 remaining has gone through 2 half-lives (since 0.5² = 0.25). With Carbon-14's ~5,730-year half-life, that sample is roughly 11,460 years old — a real calculation used in archaeological radiocarbon dating.
About These Parameters
- Solve For
- Choose which of the four related values you want calculated. The other three fields become your known inputs — for example, choose "Half-Life" if you know the initial and remaining quantities plus how much time has passed.
- Initial & Remaining Quantity
- Any consistent unit works — grams, becquerels, milligrams of a drug, percentage of a whole — as long as both quantities use the same unit.
- Half-Life & Elapsed Time
- Must use the same time unit as each other (both in years, both in days, both in hours, etc.) for the calculation to be meaningful.
Frequently Asked Questions
Does a substance ever fully decay to zero?
Mathematically, no — exponential decay only approaches zero asymptotically. In practice, after enough half-lives the remaining quantity becomes negligible or undetectable, which is treated as effectively zero.
How is half-life different from average lifetime?
Half-life is the time for half a population to decay; mean (average) lifetime — equal to 1/λ — is somewhat longer, about 1.44 times the half-life, since it accounts for the full decay curve rather than just the halfway point.
Can I mix units, like years for half-life and days for elapsed time?
No — both must be in the same unit for the ratio t/T in the decay formula to be meaningful. Convert one to match the other before entering values.