# Half-life: why things fade by half, again and again, and never quite reach zero

> Half-life is the time it takes for half of something to disappear, and it applies to radioactive atoms, caffeine in your body and the age of ancient bones. This guide builds the formula from simple halving, with pictures and exact worked examples.

Published 2026-10-08 by Toolcloak. 4 minute read. Original: https://toolcloak.pages.dev/articles/half-life-why-things-fade-by-half-again-and-again-and-never-quite-reach-zero
Related free tool: [Half-Life Calculator (Radioactive Decay)](https://toolcloak.pages.dev/tools/half-life-calculator)

Imagine a cup of coffee with 200 mg of caffeine. Five hours later, about 100 mg is left. After another five hours, about 50 mg. The amount does not fall by the same number each time. It falls by the same **fraction**: half. That fixed time for halving is the half-life, and the pattern shows up in physics, medicine, chemistry and even archaeology.

> **In short**
> - After one half-life **50%** remains, after two **25%**, after three **12.5%**. After **7 half-lives** under 1% is left.
> - Formula: **N = N₀ × (½)^(t ÷ T)**, where T is the half-life and t the time passed.
> - Caffeine with a 5-hour half-life: a 200 mg coffee leaves about **29 mg** after 14 hours.
> - To find the **time** from a known fraction, use **t = T × log₂(N₀ ÷ N)**. 25% left means exactly 2 half-lives.

## Halving again and again

Start with 100 units of something that decays with a half-life of one hour. After the first hour 50 remain. After the second, half of those 50: 25. It never reaches exactly zero, though after enough halvings what is left is too small to matter.

**What is left after each half-life (% of the start)**

|  | Value |
| --- | --- |
| 1 half-life | 50% |
| 2 | 25% |
| 3 | 12.5% |
| 4 | 6.25% |
| 5 | 3.13% |
| 6 | 1.56% |
| 7 | 0.78% |

*Each bar is half of the one before it: 50, 25, 12.5, 6.25, 3.13, 1.56, 0.78.*

**Halving a pile of 64 coins**

| Step | Value |
| --- | --- |
| Start | 64 |
| After 1 half-life | 32 |
| After 2 | 16 |
| After 3 | 8 |
| After 4 | 4 |

*After 6 half-lives a single coin is left. Real atoms decay at random, so the half-life describes a large group, not any single atom.*

Try it: [Half-Life Calculator (Radioactive Decay)](https://toolcloak.pages.dev/tools/half-life-calculator). Enter the half-life and the time passed, or the fraction left, and get the missing value.

## The formula

> **N = N₀ × (1/2)^(t ÷ T):** N₀ is the starting amount, N the amount now, T the half-life and t the elapsed time, both in the same unit. The exponent t ÷ T counts how many half-lives have passed, and it need not be a whole number.

For a non-whole number of half-lives, use a calculator. After 1.5 half-lives, 0.5^1.5 = 0.354, so about 35% remains: more than the 25% after 2 and less than the 50% after 1.

**How much iodine-131 (half-life about 8 days) is left after 20 days?**

1. **Count the half-lives.** 20 ÷ 8 = 2.5 half-lives.
2. **Raise ½ to that power.** 0.5^2.5 = 0.177.
3. **Read the answer.** About 17.7% of the original amount remains.

**Iodine-131 over 64 days (% remaining, half-life 8 days)**

| Days | Remaining |
| --- | --- |
| 0 | 100 |
| 8 | 50 |
| 16 | 25 |
| 24 | 12.5 |
| 32 | 6.25 |
| 40 | 3.13 |
| 48 | 1.56 |
| 56 | 0.78 |
| 64 | 0.39 |

*Eight days halves the amount to 50%. By day 64 (eight half-lives) only about 0.4% is left.*

## Real half-lives, from seconds to billions of years

Half-lives span an enormous range. Some atoms last a fraction of a second, while others outlast the Earth. A shorter half-life means faster decay, and a more intense radiation level for the same amount.

| Substance | Half-life (approx.) | Where it matters |
| --- | --- | --- |
| Technetium-99m | 6 hours | Medical scans, used because it fades quickly |
| Iodine-131 | 8 days | Thyroid treatment, fallout monitoring |
| Caffeine (healthy adult) | ≈ 5 hours (3 to 7) | Why an evening coffee affects sleep |
| Caesium-137 | ≈ 30 years | Long-lived fallout, industrial sources |
| Carbon-14 | 5,730 years | Dating organic remains |
| Uranium-238 | 4.5 billion years | Age of rocks and the Earth |

*The values are standard rounded figures. For caffeine, the half-life varies with genes, pregnancy, smoking and medication.*

**How a half-life shapes what you do with a substance**

| When | What |
| --- | --- |
| Hours | Technetium-99m. Gone in a day or two, so a patient’s exposure is brief. |
| Days | Iodine-131. Weeks of caution, then it is mostly gone. |
| Decades | Caesium-137. Takes generations to fade, so sites stay monitored. |
| Millennia | Carbon-14. Useful as a clock for ancient objects. |

## A coffee at 8 a.m.

Assume one 200 mg coffee and a 5-hour half-life. Every five hours the amount in your body halves. By late evening there is still a measurable amount.

**Caffeine left in the body after a 200 mg coffee (mg)**

| Hours after drinking | Caffeine remaining |
| --- | --- |
| 0h | 200 |
| 2h | 151.57 |
| 4h | 114.87 |
| 6h | 87.06 |
| 8h | 65.98 |
| 10h | 50 |
| 12h | 37.89 |
| 14h | 28.72 |
| 16h | 21.76 |
| 18h | 16.49 |
| 20h | 12.5 |
| 22h | 9.47 |
| 24h | 7.18 |

*At 8 a.m. the cup starts at 200 mg; by 10 p.m. (14 hours later) about 29 mg remains. Taking a second cup in the afternoon adds to what is already there.*

> **Doses add up before they fade:** Because the earlier dose has not finished decaying, a second drink stacks on top. A 100 mg coffee at 3 p.m. still leaves about 50 mg at 8 p.m., and 25 mg at 1 a.m. For sleep, people often choose a cut-off several half-lives before bed.

## Dating ancient things with carbon-14

Living things keep a steady proportion of carbon-14 while they are alive. After death the proportion falls with a half-life of 5,730 years. Measure the fraction left and you can work backwards to the age.

**Age of a sample by the share of carbon-14 left (years)**

|  | Age |
| --- | --- |
| 50% left | 5,730 yr |
| 25% left | 11,460 yr |
| 12.5% left | 17,190 yr |
| 10% left | 19,035 yr |
| 6.25% left | 22,920 yr |

*Use t = T × log₂(N₀ ÷ N). For 10% left: 5,730 × log₂(10) ≈ 19,000 years. Beyond about 50,000 years too little carbon-14 remains to measure.*

> **Other ways to say the same thing:** A decay constant λ = ln 2 ÷ T ≈ 0.693 ÷ T gives N = N₀ × e^(−λt). The mean lifetime is T ÷ ln 2 ≈ 1.44 × T. All three describe the same curve. It is the mirror image of compound growth: see [the rule of 72](https://toolcloak.pages.dev/articles/the-rule-of-72-how-long-money-takes-to-double) for doubling.

## Quick practice

1. A sample has a half-life of 10 days. How much of 80 g is left after 30 days? Three half-lives: 80 ÷ 8 = **10 g**.
2. How many half-lives until under 1%? **7** (0.78% remain).
3. Caffeine at 5-hour half-life: 160 mg after 10 hours? 160 ÷ 4 = **40 mg**.
4. A bone has 25% of its carbon-14. Age? Two half-lives: **11,460 years**.

Whenever something shrinks by a fixed fraction per unit of time, the half-life is the natural way to talk about it: discounted values, drug levels, even the “half-life” of news. Try different numbers in the calculator to feel how fast halving adds up.

## Frequently asked questions

### What is half-life?

The time it takes for half of a quantity to decay or disappear. After one half-life 50% is left, after two 25%, after three 12.5%.

### What is the half-life formula?

N = N₀ × (1/2)^(t ÷ T), where T is the half-life and t the elapsed time in the same unit.

### Does a substance ever fully disappear?

Mathematically no, since halving never reaches zero. In practice the amount becomes too small to matter, usually within 7 to 10 half-lives.

### What is the half-life of caffeine?

About 5 hours in healthy adults, with a typical range of 3 to 7 hours. Pregnancy, medication and genes change it.

### How does carbon dating work?

It measures the share of carbon-14 left in organic material and uses the 5,730-year half-life to calculate how long ago the organism died.

### Is a shorter half-life more dangerous?

A shorter half-life means more decays per second for the same amount, so higher intensity but a shorter time of exposure. Both matter.