How to Calculate Half Life

The Formula

Amount remaining equals the initial amount multiplied by a half raised to the power of the time elapsed divided by the half life. Equivalently, the fraction remaining is a half to the power of the number of half lives elapsed.

Why the Pattern Repeats

Each halving removes half of whatever is currently present, not half of the original. That is what makes the curve exponential rather than linear. After one half life 50 percent remains, after two half lives 25 percent, after three 12.5 percent, and after ten roughly a thousandth.

A Carbon 14 Example

With an initial 100 percent and a half life of 5,730 years, after 11,460 years two half lives have passed and 25 percent remains. After 17,190 years it is 12.5 percent. Each of those figures is exactly half the one before.

The Decay Constant Form

The same relationship is often written N equals N0 multiplied by e raised to minus lambda t, where lambda is ln(2) divided by the half life. If you are ever given lambda instead of a half life, convert with half life = ln(2) / lambda.

Units Are the Main Trap

The time elapsed and the half life must be in the same unit. A half life of 5,730 years compared against a time of 30 days produces a number, but a meaningless one. Convert first, then calculate.

It Never Reaches Zero

The remaining amount is always initial multiplied by a positive fraction, so it approaches zero without ever arriving. Any question asking when the sample hits zero has the answer never, which is a real result rather than a limitation of the arithmetic.