Quick Answer

100 grams of a substance with a half-life of 5 years has 25.00 grams remaining after 10 years (two half-lives elapsed, 75.00 grams decayed).

Calculate Remaining Quantity

Initial Amount (N₀)

Half-Life

Any time unit (years, hours, seconds, etc.)

Elapsed Time

Same time unit as half-life

Find Half-Life from Decay Data

Initial Amount (N₀)

Remaining Amount (N)

Elapsed Time

Common Examples

Input	Result
N0 = 100 g, t_half = 5 years, t = 10 years	Remaining: 25.00 g (25.00%)
N0 = 500 g, t_half = 8 hours, t = 24 hours	Remaining: 62.50 g (12.50%)
N0 = 200 g, t_half = 30 days, t = 90 days	Remaining: 25.00 g (12.50%)
N0 = 1000 g, N = 250 g, t = 10 years	Half-life: 5.00 years
N0 = 400 g, N = 100 g, t = 6 hours	Half-life: 3.00 hours

How It Works

The Decay Formula

N(t) = N₀ x (1/2)^(t / t_half)

Where:

• N(t) = quantity remaining at time t
• N₀ = initial quantity at time zero
• t = elapsed time
• t_half = half-life (time for half the substance to decay)

After each half-life period, exactly half of the remaining substance has decayed. After 1 half-life, 50% remains. After 2 half-lives, 25%. After 3 half-lives, 12.5%. After 10 half-lives, about 0.1% remains.

Solving for Half-Life

Given measured initial and remaining quantities plus elapsed time:

t_half = t x ln(2) / ln(N₀ / N)

This is derived by taking the natural logarithm of both sides of the decay equation and solving for the half-life.

Decay Constant

The decay constant (lambda) relates to half-life as: lambda = ln(2) / t_half. The decay formula can also be written as N(t) = N₀ x e^(-lambda x t).

Applications

Half-life calculations apply to radioactive decay (nuclear physics), drug metabolism (pharmacology), chemical reaction kinetics, and any process that follows first-order exponential decay. Carbon-14 dating, for example, uses the 5,730-year half-life of C-14 to estimate the age of organic materials.

Notable Half-Lives

• Carbon-14: 5,730 years
• Uranium-238: 4.47 billion years
• Iodine-131: 8.02 days
• Radon-222: 3.82 days
• Caffeine in the body: approximately 5 hours

Worked Example

A lab starts with 500 mg of Iodine-131 (half-life = 8.02 days). After 24.06 days: halvings = 24.06 / 8.02 = 3.00. Remaining = 500 x (0.5)^3 = 500 x 0.125 = 62.50 mg. So 437.50 mg has decayed, and 12.50% remains.

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Frequently Asked Questions

What is a half-life?

A half-life is the time required for exactly half of a substance to decay or be eliminated. After one half-life, 50% remains. After two half-lives, 25% remains. The concept applies to radioactive isotopes, drug metabolism, and any exponential decay process. Each half-life reduces the remaining quantity by half, regardless of the starting amount.

Does the substance ever fully decay to zero?

Mathematically, the exponential decay formula never reaches exactly zero; it approaches zero asymptotically. In practice, after about 10 half-lives, less than 0.1% of the original substance remains, which is often considered negligible. For radioactive materials, individual atoms decay randomly, so eventually all atoms will have decayed.

What time units should I use?

The half-life and elapsed time must use the same units (both in years, both in hours, both in seconds, etc.). The calculator does not convert between units. Common choices depend on the substance: years for geological isotopes, days for medical isotopes, hours for drugs, and seconds for very short-lived particles.

Can this be used for drug half-life calculations?

Yes. Drug elimination from the body often follows first-order kinetics, which is modeled by the same exponential decay formula. For example, caffeine has a half-life of about 5 hours. After consuming 200 mg of caffeine, approximately 50 mg would remain in the body after 10 hours (two half-lives).

What is carbon-14 dating?

Carbon-14 dating uses the known half-life of Carbon-14 (5,730 years) to estimate the age of organic materials. Living organisms maintain a constant ratio of C-14 to C-12. After death, C-14 decays without replacement. By measuring how much C-14 remains, scientists calculate how many half-lives have passed and estimate the age of the sample.