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.

Example: Carbon-14 has a half-life of about 5700 years. This means that a 100 g. sample will decay as shown below.

Time	Grams of ${}^{14}\text{C}$
0 years	100
5700 years	50
11,400 years	25
17,100 years	12.5

See also

Doubling time

Key Formula

A(t) = A_0 \left(\frac{1}{2}\right)^{t/h}

Where:

$A(t)$ = Amount remaining after time t
$A_0$ = Initial amount at time t = 0
$t$ = Elapsed time
$h$ = 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.

\frac{t}{h} = \frac{9}{3} = 3

Step 2: Apply the half-life formula.

A(9) = 800 \left(\frac{1}{2}\right)^3 = 800 \times \frac{1}{8}

Step 3: Compute the final amount.

A(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.

Related Terms

Exponential Decay — The broader process that half-life measures
Doubling Time — The growth counterpart of half-life
Decay Factor — The constant multiplier per time period in decay