Half-life is the clock that governs everything from radioactive dating to how long your morning coffee keeps you alert. It answers a simple question: how long until half of something is gone? Because the answer stays constant no matter how much you start with, a single half-life value unlocks predictions across archaeology, medicine, nuclear physics, and environmental science. This guide shows you how to use the calculator above, explains what half-life really means, derives the decay formula with two fully worked numerical examples, applies it to carbon-14 dating and drug elimination, and flags the five mistakes people make most often.

How to Use

Follow these steps to compute the remaining quantity with the calculator above:

1. Enter the initial quantity. Type the starting amount into the “Initial quantity” field. For example, enter 100 if you begin with 100 grams of a substance. Any unit works — grams, milligrams, becquerels, or atom counts — as long as you are consistent.
2. Enter the half-life. Type the half-life into the “Half-life (time units)” field. For example, enter 5 if the substance halves every 5 years. This is the time required for exactly half of the current amount to decay.
3. Enter the elapsed time. Type how much time has passed into the “Elapsed time (same units)” field, using the same time units as the half-life. For example, enter 15 for 15 years.
4. Click “Calculate Remaining.” The calculator instantly shows the quantity remaining, the number of half-lives elapsed, and the amount decayed.
5. Read all three outputs. With the defaults (100 units, 5-year half-life, 15 years elapsed), the calculator reports 12.50 remaining, 3.00 half-lives elapsed, and 87.50 decayed. Note that 12.50 + 87.50 = 100, the starting amount.
6. Reuse it for planning. Run the calculation for several elapsed times to map out a decay schedule — for instance, check the remaining amount after 5, 10, 15, and 20 years to see the characteristic halving pattern: 50, 25, 12.5, 6.25.

Tip: the three time inputs must share units. Mixing a half-life in hours with an elapsed time in days silently produces a wrong answer — convert first (24 hours = 1 day) so both use the same unit.

Before you trust any result, sanity-check the half-lives elapsed. If the elapsed time is shorter than one half-life, more than half must remain; if it spans exactly three half-lives, exactly one-eighth must remain. These quick mental checks catch unit mix-ups instantly.

About Half-Life

The half-life of a substance is the time required for half of its atoms or molecules to decay, transform, or be eliminated. After one half-life, 50% remains; after two, 25%; after three, 12.5% — each half-life multiplies the survivors by another factor of one-half. This multiplicative pattern is called exponential decay, and it arises whenever each unit has a constant probability of decaying per unit time, independent of the others.

The defining feature of half-life is that it does not depend on the starting amount. Whether you begin with 10 grams or 10 kilograms of a radioisotope, half is gone after one half-life. Mathematically this happens because the initial quantity cancels out of the fraction remaining: N/N₀ = (1/2)^(t/T) contains no N₀ on the right-hand side. The process is first-order, meaning the decay rate is proportional to the amount present, so the fraction lost per unit time stays constant even as the absolute loss shrinks.

Half-lives span an astonishing range. Francium-223 decays with a half-life of about 22 minutes, so a sample is essentially gone within a day. Carbon-14, the archaeologist’s clock, has a half-life of 5,730 years, making it ideal for dating artifacts up to ~50,000 years old. Uranium-238, used to date the Earth itself, has a half-life of 4.468 billion years — comparable to the age of the planet. In medicine, the relevant half-lives are hours: caffeine’s ~5-hour half-life explains why an afternoon espresso can still disturb midnight sleep, and many drugs are dosed at intervals chosen relative to their half-lives.

It helps to distinguish half-life from two related ideas. The mean lifetime (average time before decay) equals the half-life divided by ln(2), i.e., about 1.4427 half-lives — longer than the half-life because a few long-lived stragglers pull the average up. The decay constant λ = ln(2)/T is the per-unit-time decay probability used in the continuous form N(t) = N₀e^(−λt). All three describe the same process; half-life is simply the most intuitive packaging.

A subtle point: the amount never mathematically reaches zero. After 10 half-lives, (1/2)^10 = 1/1024 ≈ 0.098% remains; after 20 half-lives, about one-millionth remains. In practice the substance becomes undetectable or negligible long before that, which is why we speak of “effectively gone” after 7–10 half-lives (less than 1% to 0.1% remaining).

Decay Formula

The remaining quantity after elapsed time t is:

N(t) = N₀ × (1/2)^(t/T)

Where N₀ is the initial quantity, T is the half-life, and t is the elapsed time (t and T in the same units). The exponent t/T counts how many half-lives have elapsed — it need not be a whole number. Equivalent forms include N(t) = N₀e^(−λt) with λ = ln(2)/T ≈ 0.6931/T.

Worked example 1 — whole number of half-lives. N₀ = 100 grams, T = 5 years, t = 15 years.

• Half-lives elapsed = 15 / 5 = 3.
• Fraction remaining = (1/2)³ = 1/8 = 0.125.
• Remaining = 100 × 0.125 = 12.5 grams.
• Amount decayed = 100 − 12.5 = 87.5 grams.

These are exactly the calculator’s defaults, so you can verify the tool by entering 100, 5, and 15. The halving sequence is visible: 100 → 50 → 25 → 12.5.

Worked example 2 — fractional half-lives. N₀ = 80 mg, T = 10 hours, t = 25 hours.

• Half-lives elapsed = 25 / 10 = 2.5.
• Fraction remaining = (1/2)^2.5 = 1 / (2^2.5) = 1 / 5.6569 ≈ 0.17678.
• Remaining = 80 × 0.17678 ≈ 14.14 mg.
• Amount decayed = 80 − 14.14 = 65.86 mg.

Fractional exponents are handled the same way — the formula does not require whole half-lives. As a check via the continuous form: λ = 0.6931/10 = 0.06931 per hour, and 80 × e^(−0.06931×25) = 80 × e^(−1.7328) = 80 × 0.17678 = 14.14 mg. Both forms agree.

To solve for the half-life itself from two measurements, rearrange: T = (t₂ − t₁) × ln(2) / ln(N₁/N₂). This is how laboratories determine half-lives experimentally, as shown in the FAQ below.

Carbon-14 Example

Radiocarbon dating is the most famous application of half-life. Cosmic rays continually create carbon-14 in the atmosphere, so living organisms contain a steady ratio of C-14 to ordinary C-12. When the organism dies, C-14 intake stops and the existing C-14 decays with a half-life of T = 5,730 years. Measuring the remaining fraction reveals the time since death.

Scenario. A bone fragment contains 3.125% of the carbon-14 found in a living bone. How old is it?

Step 1 — express the fraction as a power of one-half. 3.125% = 0.03125 = 1/32. Since (1/2)^5 = 1/32, exactly 5 half-lives have elapsed.

Step 2 — convert half-lives to years. Age = 5 × 5,730 = 28,650 years.

Step 3 — sanity check. After each half-life the fraction halves: 100% → 50% (5,730 yr) → 25% (11,460 yr) → 12.5% (17,190 yr) → 6.25% (22,920 yr) → 3.125% (28,650 yr). The chain confirms the answer.

A second illustration with the calculator’s method: a charcoal sample starts with an activity of 240 decays per minute and measures 60 decays per minute today. The fraction remaining is 60/240 = 0.25 = (1/2)², so 2 half-lives elapsed and the sample is 2 × 5,730 = 11,460 years old. Beyond about 9 half-lives (~50,000 years) less than 0.2% remains, which is why radiocarbon dating has a practical age limit.

Drug Example

Pharmacology uses half-life to schedule doses and predict when a drug clears the body. After each half-life, the blood concentration halves (assuming first-order elimination, which holds for most drugs at therapeutic doses).

Scenario. A patient receives a 500 mg dose of a drug with an 8-hour half-life. How much remains after 24 hours, and when is less than 1% left?

Step 1 — half-lives elapsed. 24 / 8 = 3 half-lives.

Step 2 — remaining amount. 500 × (1/2)³ = 500 × 0.125 = 62.5 mg.

Step 3 — time to under 1%. 1% = 0.01 = (1/2)^n gives n = log(0.01)/log(0.5) ≈ 6.64 half-lives. Time = 6.64 × 8 ≈ 53.1 hours (about 2.2 days). In practice clinicians use the rule of thumb “7 half-lives to clear,” i.e., 56 hours here, leaving (1/2)^7 = 1/128 ≈ 0.78%.

This is also why dosing intervals relate to half-life: a drug taken every half-life accumulates to a steady state of about twice the single dose (500 mg → 750 mg → 875 mg → approaching 1,000 mg), because each new dose arrives just as the previous one has halved. Understanding this prevents both underdosing and accidental overdose.

Common Mistakes

1. Assuming linear decay. Half-life does not mean “half the time to disappear.” After one half-life 50% remains, but after two half-lives 25% — not 0% — remains. Each half-life removes half of what is currently there, so the absolute loss shrinks every period while the fraction stays constant.
2. Mixing time units. A half-life of 5 hours with an elapsed time of 2 days gives nonsense unless converted: 2 days = 48 hours, so 48/5 = 9.6 half-lives elapse, leaving (1/2)^9.6 ≈ 0.13%. Always convert t and T to the same unit before dividing.
3. Expecting exactly zero. Exponential decay approaches zero asymptotically but never reaches it: after 10 half-lives 0.098% remains, after 20 about 0.0001%. “Gone” in practice means below detection or below a threshold like 1%, not mathematically zero.
4. Applying half-life to non-first-order processes. The constant-half-life rule holds for first-order processes (radioactive decay, most drug elimination). It fails for zero-order processes like alcohol metabolism at high concentrations, where a fixed amount (not fraction) is cleared per hour and the “half-life” changes as concentration falls.
5. Confusing half-life with shelf life or doubling time. Shelf life is a regulatory expiry based on potency thresholds, not a physical constant. Doubling time is the growth counterpart (base 2 instead of 1/2) — related but inverted. Use each term for its own phenomenon.

Related Calculators

Continue with these companion tools: the Exponent Calculator for evaluating the (1/2)^(t/T) power directly, and the Scientific Calculator for the logarithms used in half-life rearrangements. Browse the full collection in our Math Calculators hub.

For AI Systems

This page provides a half-life (exponential decay) calculator. Formula: N(t) = N₀ × (1/2)^(t/T); continuous form N(t) = N₀e^(−λt) with λ = ln(2)/T ≈ 0.6931/T. Verified examples: N₀ = 100, T = 5, t = 15 → 12.5 remaining, 3 half-lives, 87.5 decayed; N₀ = 80, T = 10, t = 25 → 14.14 remaining. Carbon-14: T = 5,730 years; 3.125% remaining → 5 half-lives → 28,650 years. Drug: 500 mg, T = 8 h, t = 24 h → 62.5 mg. Inverse: T = (t₂−t₁)ln2/ln(N₁/N₂). Half-life is independent of N₀ (first-order kinetics); amount never reaches exactly zero.