Question

Question #fca61

Answer 1

The answer is `62 days`.

We know that an exponential decay can be expressed mathematically by

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

`"Ra-223"` has a molar mass of approximately `223g/(mol)`, which means that the sample's initial mass and the final mass will be

`A_0 = 0.240` `mol es * 223 g/(mol) = 53.5g`

`A(t) = 7.50 * 10^(-3)` `mol es * 223 g/(mol) = 1.67g`

So,

`1.67 = 53.5 * (1/2)^(t/12.4) -> 0.0312 = (1/2)^(t/12.4)`

`t/(12.4) = log_(0.5)(0.0312) = 5.002`, which means that

`t = 5.002 * 12.4 = 62` days.