# How can I calculate the half life of an element?

Dec 30, 2014

Nuclear half-life expresses the time required for half of a sample to undergo radioactive decay. Exponential decay can be expressed mathematically like this:

$A \left(t\right) = {A}_{0} \cdot {\left(\frac{1}{2}\right)}^{\frac{t}{t} _ \left(\text{1/2}\right)}$ (1), where

$A \left(t\right)$ - the amount left after t years;
${A}_{0}$ - the initial quantity of the substance that will undergo decay;
${t}_{\text{1/2}}$ - the half-life of the decaying quantity.

So, if a problem asks you to calculate an element's half-life, it must provide information about the initial mass, the quantity left after radioactive decay, and the time it took that sample to reach its post-decay value.

Let's say you have a radioactive isotope that undergoes radioactive decay. It started from a mass of 67.0 g and it took 98 years for it to reach 0.01 g. Here's how you would determine its half-life:

Starting from (1), we know that

$0.01 = 67.0 \cdot {\left(\frac{1}{2}\right)}^{\frac{98.0}{t} _ \left(\text{1/2")) -> 0.01/67.0 = 0.000149 = (1/2)^(98.0/(t_("1/2}\right)}$

$\frac{98.0}{t} _ \left(\text{1/2}\right) = {\log}_{0.5} \left(0.000149\right) = 12.7$

Therefore, its half-life is ${t}_{\text{1/2}} = \frac{98.0}{12.7} = 7.72$ $\text{years}$.

So, the initial mass gets halved every 7.72 years.

Sometimes, if the numbers allow it, you can work backwards to determine an element's half-life. Let's say you started with 100 g and ended up with 25 g after 1,000 years.

In this case, since 25 represents 1/4th of 100, two hal-life cycles must have passed in 1,000 years, since

$\frac{100.0}{2} = 50.0$ $\text{g}$ after the first ${t}_{\text{1/2}}$,

$\frac{50.0}{2} = 25.0$ $\text{g}$ after another ${t}_{\text{1/2}}$.

So, $2 \cdot {t}_{\text{1/2") = 1000 -> t_("1/2}} = \frac{1000}{2} = 500$ $\text{years}$.