Question

How can I calculate the half life of an element?

Answer 1

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(t) = A_0 * (1/2)^(t/t_("1/2"))` (1), where

`A(t)` - the amount left after t years;
`A_0` - the initial quantity of the substance that will undergo decay;
`t_("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 * (1/2)^(98.0/t_("1/2")) -> 0.01/67.0 = 0.000149 = (1/2)^(98.0/(t_("1/2"))`

`98.0/t_("1/2") = log_(0.5)(0.000149) = 12.7`

Therefore, its half-life is `t_("1/2") = 98.0/(12.7) = 7.72` `"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

`100.0/2 = 50.0` `"g"` after the first `t_("1/2")`,

`50.0/2 = 25.0` `"g"` after another `t_("1/2")`.

So, `2 * t_("1/2") = 1000 -> t_("1/2") = 1000/2 = 500` `"years"`.