Half-Life Calculator

What this calculator does

The Half-Life Calculator solves the four quantities in exponential decay: the remaining amount, the initial amount, the elapsed time, and the half-life. You fill in three values and leave one field empty; the calculator solves for the empty field.

Half-life is the time required for half of a substance to decay. The idea is used in radiometric dating, nuclear physics, pharmacokinetics, and any process that decreases by a constant fraction over equal time intervals.

How to use this calculator

1. Fill in three of the four fields.
2. Leave exactly one field empty.
3. Click Calculate.
4. Read the solved value, the mean lifetime, and the decay constant.

The default example uses an initial amount of 100, a remaining amount of 10, and an elapsed time of 50. The half-life is about 15.0515 time units.

Half-life formulas

Nt = N0 × (1/2)^(t / t½)
N0 = Nt × 2^(t / t½)
t  = t½ × ln(N0 / Nt) ÷ ln(2)
t½ = t × ln(2) ÷ ln(N0 / Nt)

mean lifetime = t½ ÷ ln(2)
decay constant = ln(2) ÷ t½

The remaining amount decreases by half after each half-life. After two half-lives, one quarter remains; after three, one eighth remains.

What the results mean

The result panel names the solved quantity and shows its value. It also reports the mean lifetime and the decay constant, which describe the same decay process in different ways. A larger decay constant means faster decay.

For the default example, the half-life is 15.0515, the mean lifetime is about 21.7147, and the decay constant is about 0.04605 per time unit.

Assumptions and limitations

• Exactly one of the four fields must be empty.
• Amounts must be positive; time must be zero or greater; half-life must be positive.
• When solving for half-life or elapsed time, the initial amount must be greater than the remaining amount, because the model describes decay.
• The calculator assumes exponential decay with a constant half-life. It does not model a changing decay rate.
• Units are not converted. Keep time and half-life in the same unit.
• The calculator is a mathematical model, not a substitute for laboratory measurement or radiometric dating uncertainty analysis.

Frequently asked questions

What does it mean to leave one field empty?

The empty field is the unknown. If you know the initial amount, remaining amount, and elapsed time, leave the half-life field empty and the calculator will solve for it.

Why must the initial amount be larger than the remaining amount?

The half-life model describes decay, so the amount can only decrease. If the remaining amount were larger, the calculation would imply a negative elapsed time or a negative half-life, which is not meaningful here.

What is the difference between half-life and mean lifetime?

The half-life is the time for half of the substance to decay. The mean lifetime is the average lifetime of an individual particle, equal to the half-life divided by ln(2). The mean lifetime is longer than the half-life.

Can I use this for radioactive decay?

Yes, as a mathematical model. For real radiometric dating, you also need to account for measurement uncertainty, background contamination, and the specific decay chain.

Reference table

Formula variables

The decay constant λ = ln(2) ÷ t½. Mean lifetime = 1 ÷ λ. The calculator accepts any consistent time unit as long as time and half-life use the same unit.

Source

Reference: Calculator.net — Half-Life Calculator.