Half-Life Calculator Formula

Understand the math behind the half-life calculator. Each variable explained with a worked example.

Use the Half-Life Calculator

Formulas Used

Remaining

remaining = initial * pow(0.5, time / half_life)

Decayed

decayed = initial - initial * pow(0.5, time / half_life)

Num Halves

num_halves = time / half_life

Pct Remaining

pct_remaining = pow(0.5, time / half_life) * 100

Variables

VariableDescriptionDefault
initialInitial Quantity100
half_lifeHalf-Life5
timeElapsed Time15

How It Works

Half-Life Formula

Formula

N(t) = N₀ × (1/2)^(t/t½)

Where:

  • N₀ = initial quantity
  • t = elapsed time
  • t½ = half-life

    • Properties

  • After 1 half-life: 50% remains
  • After 2 half-lives: 25% remains
  • After 3 half-lives: 12.5% remains
  • After 10 half-lives: ~0.1% remains

    • Worked Example

    100 units with a half-life of 5 after 15 time units.

    initial = 100half_life = 5time = 15
    1. 01Number of half-lives = 15/5 = 3
    2. 02Remaining = 100 × (0.5)³
    3. 03= 100 × 0.125
    4. 04= 12.5

    Frequently Asked Questions

    What is half-life?

    Half-life is the time required for a quantity to reduce to half of its initial value through decay. It is constant regardless of the starting amount.

    Does half-life ever reach zero?

    Mathematically, no. The exponential decay function asymptotically approaches zero but never reaches it. In practice, the quantity becomes negligibly small.

    What are some real half-lives?

    Carbon-14: 5,730 years. Uranium-238: 4.5 billion years. Iodine-131: 8 days. Caffeine in the human body: about 5 hours.

    Ready to run the numbers?

    Open Half-Life Calculator