Half-Life Calculator

What is a half-life calculator?

A half-life calculator finds how much of a decaying substance is left after a given amount of time. You enter the starting amount, the half-life, and the elapsed time, and it returns the remaining amount, the percent remaining, and the number of half-lives that have passed. It is built for chemistry and physics students, lab workers, and anyone studying radioactive decay, medical isotopes, or carbon dating, where a quantity shrinks by half over each fixed interval.

How the half-life formula works

The calculator uses the exponential decay equation remaining = initial × (1/2)^(t / half-life). Decay is multiplicative: every time one half-life passes, whatever is left is cut in half again. Here is what each term means:

• initial — the starting amount (mass, atoms, activity, or concentration) in any unit you choose. The answer comes back in that same unit.
• t — the elapsed time since you started, in any time unit.
• half-life (T) — the time it takes for the amount to fall to half. It must use the same time unit as t.
• t / half-life — the number of half-lives that have elapsed. This does not have to be a whole number.
• (1/2)^(t / T) — the fraction remaining. Multiply by 100 to get the percent remaining.

Because the exponent is the ratio of two times, the units cancel: you only need t and the half-life to share a unit. The half-life itself never changes with starting amount, temperature, or pressure, which is why the same constant works for every calculation involving a given isotope.

Examples

Each example uses only the formula above, so you can reproduce every answer in the calculator.

Example 1 — three full half-lives

Start with 100 units, a half-life of 5, and an elapsed time of 15 (same units).

t / T = 15 / 5 = 3 half-lives (1/2)^3 = 0.125 remaining = 100 × 0.125 = 12.5 units

So 12.5 units remain, which is 12.5 percent of the original, after 3 half-lives.

Example 2 — one half-life

Start with 100 units, half-life 5, elapsed time 5.

t / T = 5 / 5 = 1 half-life (1/2)^1 = 0.5 remaining = 100 × 0.5 = 50 units

After exactly one half-life, 50 units (50 percent) remain. This is the definition of half-life.

Example 3 — two half-lives

Start with 100 units, half-life 5, elapsed time 10.

t / T = 10 / 5 = 2 half-lives (1/2)^2 = 0.25 remaining = 100 × 0.25 = 25 units

After two half-lives, 25 units (25 percent) remain. Notice the pattern: 50 percent, then 25 percent, then 12.5 percent.

Half-life decay reference table

This table shows the fraction and percent remaining for whole numbers of half-lives. It applies to any isotope or substance, because the math depends only on how many half-lives have passed (t / T).

Half-lives elapsed (t / T)	Fraction remaining (1/2)^(t/T)	Percent remaining
0	1	100%
1	1/2 = 0.5	50%
2	1/4 = 0.25	25%
3	1/8 = 0.125	12.5%
4	1/16 = 0.0625	6.25%
5	1/32 = 0.03125	3.125%
10	1/1024 ≈ 0.000977	0.0977%

How to find the elapsed time from the remaining amount

If you know how much is left and want the elapsed time instead, rearrange the same formula. Divide the remaining amount by the initial amount to get the fraction remaining, take its base-1/2 logarithm to find the number of half-lives, then multiply by the half-life. In other words, t = T × log(remaining / initial) / log(1/2). For example, with 100 units of a substance whose half-life is 5, finding when only 12.5 units are left gives 12.5 / 100 = 0.125, and log(0.125) / log(0.5) = 3, so t = 5 × 3 = 15 — exactly the elapsed time from Example 1, run in reverse. This inverse calculation is the heart of radiometric dating: measure the surviving fraction, then solve for age.

Why half-life is constant

The half-life of an isotope is fixed because radioactive decay is a first-order process — the rate at which atoms decay is proportional only to how many undecayed atoms remain, not to temperature, pressure, chemical state, or the size of the sample. That proportionality is what produces a constant fractional loss per unit time, and a constant fractional loss is exactly what gives an unchanging half-life. Doubling the sample doubles the number of decays per second, but the proportion lost in one half-life stays at one half. This is why the same constant works for a microgram or a kilogram of the same isotope, and why the percent-remaining column in the reference table above is universal rather than substance-specific.

Common uses

The half-life formula shows up wherever a quantity decays at a fixed proportional rate:

• Chemistry and physics homework — finding the remaining mass or number of atoms of a radioactive isotope after a set time.
• Carbon dating — estimating the age of organic material from how much carbon-14 (half-life about 5,730 years) is left.
• Nuclear medicine — calculating how much of a tracer such as technetium-99m (half-life about 6 hours) remains for imaging or dosing.
• Pharmacology — modeling how a drug concentration in the bloodstream drops by half over each elimination half-life.
• Environmental science — tracking how long radioactive contamination or other decaying substances persist.

Tips and common mistakes

• Use the same time unit for t and the half-life. If the half-life is in years, the elapsed time must be in years too. Mixing hours with days is the most common error.
• The number of half-lives can be a decimal. You do not need a whole number of half-lives; t / T = 2.5 is perfectly valid and gives (1/2)^2.5.
• Half-life does not depend on the starting amount. Doubling the initial amount doubles the remaining amount, but the percent remaining and the half-life stay exactly the same.
• Percent remaining never reaches zero. Each half-life halves what is left, so the amount approaches zero but never truly hits it.
• Do not confuse half-life with mean lifetime. The mean (average) lifetime is longer than the half-life; they are related but not equal.
• Check the answer against the table. If your elapsed time is close to a whole number of half-lives, the percent should be near 50, 25, or 12.5 percent.

Limitations and notes

This calculator assumes exponential, first-order decay with a constant half-life, which is an excellent model for radioactive isotopes and many drug-elimination processes. It treats the substance as a single decaying species and ignores any in-growth of decay products, branching decay chains, or replenishment of the substance. For very small numbers of atoms, real decay is random and statistical, so a continuous formula is only an average; with everyday sample sizes the approximation is essentially exact. The arithmetic is precise for the values you enter, but make sure the assumption of a fixed, single half-life truly fits your problem before relying on a number.

For related calculations, try the density calculator for mass and volume, the pH calculator for acid and base strength, or the combined gas law calculator for gas behavior — and browse more in the chemistry and physics category.