Half-Life Calculator

Half-life describes a process that loses the same fraction during every equal time interval. Enter the values you know in the Half-Life Calculator to find the initial quantity, remaining quantity, half-life, elapsed time, decay constant, or mean lifetime. The result includes a step-by-step derivation so you can check both the equation and the units.

What the Half-Life Calculator Finds

The calculator connects six quantities in a single exponential-decay model:

• N(0): the quantity present at the starting time
• N(t): the quantity remaining after elapsed time t
• T: the half-life
• t: the total elapsed time
• λ: the decay constant
• τ: the mean lifetime

Quantities may represent mass, concentration, number of particles, activity, or any other consistently measured amount. Initial and remaining quantities must use the same unit. Half-life, elapsed time, and mean lifetime must also share a compatible time basis.

Half-Life Formula

The amount remaining after time t is:

N(t) = N(0) × (1/2)^t/T

The exponent t/T is the number of half-lives that have passed. The same relationship can be written with the decay constant:

N(t) = N(0) × e^−λt

Half-life, decay constant, and mean lifetime describe the same rate:

• λ = ln(2) / T
• T = ln(2) / λ
• τ = 1 / λ
• T = τ × ln(2)

To solve for elapsed time when the starting and remaining quantities are known, use t = ln[N(0) / N(t)] / λ. To recover the starting quantity, use N(0) = N(t) × 2^t/T.

How to Use the Calculator

1. Select the unit for half-life, elapsed time, and mean lifetime.
2. Enter enough known values to define one decay relationship. For example, enter the initial quantity, half-life, and elapsed time to calculate the amount remaining.
3. Click Calculate.
4. Review the completed fields and the step-by-step derivation. Check that the starting and remaining quantities use the same unit.

The live calculator displays λ in per second. If a published decay constant uses another reciprocal-time unit, convert it before entering the value. For example, a rate in day⁻¹ is divided by 86,400 to obtain s⁻¹.

Worked Example

Suppose a sample begins at 160 g, has a half-life of 8 days, and decays for 24 days.

First, find the number of half-lives:

t / T = 24 / 8 = 3 half-lives

Then calculate the amount remaining:

N(t) = 160 × (1/2)³ = 160 × 0.125 = 20 g

The decay constant is ln(2) / 8 = 0.08664 day⁻¹. Because the live field is per second, this is approximately 1.003 × 10⁻⁶ s⁻¹. After 24 days, 20 g—or 12.5% of the starting amount—remains.

How to Interpret Half-Life Results

Half-life is not the time required for a quantity to disappear. After one half-life, 50% remains; after two, 25% remains; after three, 12.5% remains. In an ideal exponential model, the value approaches zero without reaching it at a finite time.

A shorter half-life corresponds to a larger decay constant and faster decay. Mean lifetime is longer than half-life because τ = T / ln(2), or about 1.443T. These relationships apply to a single process with a constant fractional decay rate.

Assumptions and Limitations

• The process follows one first-order, single-exponential decay curve with a constant rate.
• No material is added, produced, replenished, or removed by another process.
• The model does not represent multi-stage decay chains, multi-compartment drug behavior, repeated dosing, or changing environmental conditions.
• Rounded output may differ slightly from a result calculated with full precision.
• This educational tool does not calculate radioactive activity, radiation dose, exposure, shielding, or safety limits.

For radioactive materials, use authoritative isotope data and required radiation-safety procedures. For medication questions, follow a doctor, pharmacist, or the official label; a simple half-life calculation cannot determine a safe dose or when a drug is fully cleared.

Common Mistakes to Avoid

• Subtracting half the initial amount repeatedly instead of halving the current remainder
• Mixing hours, days, and seconds without converting them
• Confusing the half-life T with the elapsed time t
• Entering a remaining quantity greater than the initial quantity in a decay-only model
• Applying a single exponential equation to a process with multiple decay rates

Half-Life Calculator FAQs

What percentage remains after one half-life?

Exactly 50% remains in the ideal exponential model. After two half-lives 25% remains, and after three half-lives 12.5% remains.

How do I calculate half-life from the decay constant?

Use T = ln(2) / λ. The half-life unit is the reciprocal of the decay constant’s time unit.

How do I find elapsed time from initial and remaining amounts?

Use t = ln[N(0) / N(t)] / λ. Make sure λ and t use matching reciprocal time units.

Is half-life decay linear?

No. The same fraction, not the same fixed amount, disappears during each half-life. That makes the decrease exponential.

Does a substance disappear after several half-lives?

No finite number of half-lives makes the modeled quantity exactly zero. It becomes progressively smaller and may eventually fall below a practical detection threshold.

Can this calculator determine medication timing or radiation safety?

No. It models only simple exponential decay and cannot account for dosing schedules, biological variability, radiation dose, exposure, shielding, or regulatory limits.

Related Calculators

For connected chemistry and science calculations, try the Atom Calculator, Molarity Calculator, pH Calculator, or Ideal Gas Law Calculator.

References

• OpenStax College Physics 2e: Half-Life and Activity
• OpenStax Chemistry 2e: Radioactive Decay
• International Atomic Energy Agency: Radiation in Everyday Life