Half-Life Calculator: Solve Exponential Decay

A half-life calculator solves one of the most common problems in chemistry, physics, and biology: how much of a substance remains after a certain time, when it shrinks by half at a steady rate. The same math describes radioactive atoms, carbon dating, and how long medicines stay in your body.

This guide explains the formula, shows how to solve for any unknown value, and gives quick mental shortcuts to check your answers.

What Is Half-Life?

Half-life is the time it takes for half of a substance to decay or be removed. After one half-life, 50% remains. After two, 25% remains. After three, 12.5%, and so on.

This pattern is called exponential decay: the amount drops by the same fraction in each time period, not by the same amount.

The Half-Life Formula

N = N₀ × (1/2)^(t ÷ T)

• N = amount remaining
• N₀ = starting amount
• t = time that has passed
• T = half-life (in the same time units as t)

A half-life calculator lets you enter any three of these values to find the fourth.

Solving for the Amount Remaining

Example: Iodine-131 has a half-life of about 8 days. How much of a 100 mg sample remains after 24 days?

1. Number of half-lives: 24 ÷ 8 = 3
2. Remaining fraction: (1/2)³ = 1/8
3. Remaining amount: 100 × 1/8 = 12.5 mg

When the time is an exact multiple of the half-life, you can solve it in your head. For other times, the formula or a half-life calculator gives the exact answer.

Example: What remains after 10 days?

100 × (1/2)^(10 ÷ 8) = 100 × (1/2)^1.25 ≈ 42 mg

Solving for Elapsed Time

Rearranging the formula with logarithms:

t = T × log(N₀ ÷ N) ÷ log(2)

Example: Carbon-14 dating. Carbon-14 has a half-life of about 5,730 years. A sample has 25% of its original carbon-14. How old is it?

1. N₀ ÷ N = 100 ÷ 25 = 4
2. log(4) ÷ log(2) = 2 half-lives
3. t = 5,730 × 2 = 11,460 years

If only 30% remains:

t = 5,730 × log(100 ÷ 30) ÷ log(2) ≈ 5,730 × 1.737 ≈ 9,950 years

Solving for the Half-Life

T = t × log(2) ÷ log(N₀ ÷ N)

Example: A sample falls from 80 g to 20 g in 12 hours.

• 80 ÷ 20 = 4, which is 2 half-lives
• T = 12 ÷ 2 = 6 hours

The Decay Constant

Scientists often use the decay constant (λ):

λ = ln(2) ÷ T ≈ 0.693 ÷ T

The formula then becomes N = N₀ × e^(−λt). Both versions give the same result; a half-life calculator may show either.

Pro tip: Always check that t and T use the same units. Mixing days and hours is the most common mistake in decay problems.

Half-Lives at a Glance

Half-Life in Medicine

Drugs are removed from the body at rates that are often described by half-life. For example, caffeine commonly has a half-life of around 5 hours in healthy adults, though it varies from person to person.

If you drink 200 mg of caffeine at 8 a.m., about 100 mg may remain at 1 p.m. and about 50 mg at 6 p.m. – which helps explain why an afternoon coffee can affect sleep.

Medication half-lives are affected by age, liver and kidney function, other medicines, and pregnancy. Always follow your doctor's or pharmacist's dosing instructions rather than calculating doses yourself.

The Unique Value Section: The "5 Half-Lives" Rule of Thumb

In pharmacology and lab work, a widely used rule says that after about 5 half-lives, roughly 97% of a substance is gone. That's why:

• A drug is often considered mostly cleared after about 5 half-lives.
• A medicine taken regularly is often said to reach a steady level after about 4–5 half-lives.

You can use this rule to sanity-check any half-life calculator result:

• If a problem asks how long until "almost all" is gone, the answer should be close to 5 × T.
• If your calculator says 99% is gone after just 2 half-lives, something is wrong (it should be 75%).

Quick Mental Estimates

• Time to fall to 10%: about 3.3 half-lives
• Time to fall to 1%: about 6.6 half-lives
• Time to fall to 0.1%: about 10 half-lives

These anchors make it easy to check answers on tests without redoing the full calculation.

Common Mistakes

• Subtracting half the original amount each period (linear thinking) instead of halving the current amount
• Mixing time units
• Using log instead of ln inconsistently in the same formula (either works if you use the same one top and bottom)
• Forgetting that decay never reaches exactly zero mathematically

Practice Problems With Answers

Test yourself, then confirm with a half-life calculator:

1. A 64 g sample, half-life 5 days. Amount after 20 days? → 4 half-lives → 64 ÷ 16 → 4 g
2. A sample falls to 1/8 in 9 hours. Half-life? → 3 half-lives → 3 hours
3. Half-life 10 years. Time to reach 25%? → 2 half-lives → 20 years
4. 200 mg, half-life 6 hours. Amount after 15 hours? → 200 × (1/2)^2.5 → about 35 mg
5. Carbon-14 sample with 50% remaining. Age? → 1 half-life → about 5,730 years

Problems 1–3 and 5 use whole numbers of half-lives, so mental math is enough. Problem 4 needs the full formula – a good reminder to estimate first: 15 hours is 2.5 half-lives, so the answer must fall between 25% (50 mg) and 12.5% (25 mg).

Expert Tip and Final Takeaway

A half-life calculator turns exponential decay into a simple input-output problem. Learn the core formula, use logarithms to find time or half-life, and keep units consistent.

Expert tip: Before calculating, count roughly how many half-lives have passed. If 3 half-lives passed, you know about 12.5% should remain. That one-second estimate catches most errors before they reach your final answer.