Computing Decay from a Known Half-Life

Using a Known Half-Life

A half-life tells us how long it takes a quantity undergoing exponential decay to fall to half its starting value. Once the half-life is known, we can calculate the amount remaining after any elapsed time. The starting amount does not need to be one unit, and the elapsed time does not need to be a whole number of half-lives.

The key is to count elapsed time in units of the half-life. If 2.5 half-lives have passed, for example, the amount has been multiplied by \(1/2\) two and a half times in the exponential model. It is not correct to treat the last half of a half-life as a linear loss.

The ratio \(t/T\) is the number of half-lives that have elapsed. When \(t/T\) is an integer, the calculation amounts to repeated halving. When it is not an integer, the exponent is fractional; the same formula still applies.

Why the Formula Works

The formula follows from the exponential decay model. In that model, an amount with initial value \(Q_0\) has the form \(Q(t)=Q_0e^{kt}\), where \(k\) is a constant. The half-life condition says that after time \(T\), the amount is \(Q_0/2\). Substituting \(t=T\) gives \(Q_0e^{kT}=Q_0/2\), so \(e^{kT}=1/2\). For any elapsed time \(t\), write \(t=(t/T)T\). Then

This also explains why fractional half-lives use fractional exponents: the exponential model gives a consistent amount at every elapsed time, not only at integer multiples of \(T\). As a check on the formula, setting \(t=T\) gives \(Q(T)=Q_0(1/2)^1=Q_0/2\), as the definition of half-life requires.

A Reliable Calculation Process

Before substituting numbers, identify three quantities: the initial amount \(Q_0\), the half-life \(T\), and the elapsed time \(t\). Confirm that \(t\) and \(T\) use matching units. Then calculate the exponent \(t/T\), evaluate the power of \(1/2\), and multiply by \(Q_0\). Keep exact values through the calculation when possible, and round only the final result.

Worked Examples

Common Mistake: Treating Decay as Linear

A frequent error is to assume that half of a half-life produces half of the total loss associated with a full half-life. The exponential model does not reduce the amount by a fixed number of units per unit of time. Instead, it applies a time-dependent factor to the amount present.

Reading and Reporting the Result

The starting amount \(Q_0\) is a scale factor: doubling \(Q_0\) doubles the predicted amount at every elapsed time, while leaving the half-life unchanged. The time ratio \(t/T\) determines the decay factor. This separation is useful when comparing calculations with different initial quantities but the same half-life.

Keep the units attached to the amount. If the starting amount is measured in milligrams, the calculated amount is also in milligrams; if it is measured in litres, the result is in litres. The ratio \(t/T\) has no units, provided both times are expressed in the same units. Do not insert a time in days and a half-life in hours into the ratio without first making their units consistent.

When the elapsed time is a whole number of half-lives, repeated division by two is a quick check. For a fractional number, retain the power notation or an exact radical when available, then calculate a decimal approximation if the problem calls for one. State the precision of that approximation, and avoid rounding intermediate values because small rounding errors can carry into the final amount.

The half-life formula is intended for a quantity that follows a constant half-life exponential model over the elapsed interval. A given half-life alone does not guarantee that every real process follows that model indefinitely. When the model is specified or appropriate, however, the formula handles whole and fractional elapsed half-lives in the same way.