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Half-Life — Definition, Formula & Examples

Half-Life

For a substance decaying exponentially, the amount of time it takes for the amount of the substance to diminish by half.

 

Table showing Carbon-14 decay: 100g halves every 5700 years (0y=100g, 5700y=50g, 11400y=25g, 17100y=12.5g).

 

 

See also

Doubling time

Key Formula

A(t)=A0(12)t/hA(t) = A_0 \left(\frac{1}{2}\right)^{t/h}
Where:
  • A(t)A(t) = Amount remaining after time t
  • A0A_0 = Initial amount at time t = 0
  • tt = Elapsed time
  • hh = Half-life (the time for the quantity to halve)

Worked Example

Problem: A radioactive sample starts with 800 grams and has a half-life of 3 years. How much remains after 9 years?
Step 1: Determine how many half-lives have passed by dividing the elapsed time by the half-life.
th=93=3\frac{t}{h} = \frac{9}{3} = 3
Step 2: Apply the half-life formula.
A(9)=800(12)3=800×18A(9) = 800 \left(\frac{1}{2}\right)^3 = 800 \times \frac{1}{8}
Step 3: Compute the final amount.
A(9)=100 gramsA(9) = 100 \text{ grams}
Answer: After 9 years (3 half-lives), 100 grams of the sample remain.

Why It Matters

Half-life is central to nuclear physics, where it describes how quickly radioactive isotopes decay — carbon-14's half-life of about 5,730 years is the basis of radiocarbon dating. It also appears in pharmacology (how fast a drug leaves the body) and in any mathematical model involving exponential decay.

Common Mistakes

Mistake: Assuming that after two half-lives the substance is completely gone.
Correction: After two half-lives, one-quarter of the original amount remains. The quantity never fully reaches zero; it just keeps halving.

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