# How many half-lives have elapsed when 25% of the parent nuclide is left?

Jun 10, 2017

$\text{2 half-lives}$

#### Explanation:

The thing to remember about a radioactive nuclide's nuclear half-life, ${t}_{\text{1/2}}$, is that it represents the time needed for half, hence the term half-life, of an initial sample of said nuclide to undergo radioactive decay.

In other words, the mass of a radioactive nuclide, regardless of its initial value, will always be halved after $1$ half-life passes.

So, if you start with ${A}_{0}$, you can say that you will be left with

• ${A}_{0} \cdot \frac{1}{2} = {A}_{0} / 2 = {A}_{0} / {2}^{\textcolor{red}{1}} \to$ after $\textcolor{red}{1}$ half-life
• ${A}_{0} / 2 \cdot \frac{1}{2} = {A}_{0} / 4 = {A}_{0} / {2}^{\textcolor{red}{2}} \to$ after $\textcolor{red}{2}$ half-lives
• ${A}_{0} / 4 \cdot \frac{1}{2} = {A}_{0} / 8 = {A}_{0} / {2}^{\textcolor{red}{3}} \to$ after $\textcolor{red}{3}$ half-lives
$\vdots$

and so on.

In your case, you know that

25% = 25/100 = 1/4

of the initial sample is left after a time $t$ passes, which means that you will have

${A}_{t} = {A}_{0} \cdot \frac{1}{4} = {A}_{0} / 4 = {A}_{0} / {2}^{\textcolor{red}{2}}$

As you can see, this is exactly what you would expect to get after $\textcolor{red}{2}$ half-lives pass, so

$t = \textcolor{red}{2} \cdot {t}_{\text{1/2}}$

and

${A}_{\textcolor{red}{2} \times {t}_{\text{1/2}}} = {A}_{0} / {2}^{\textcolor{red}{2}} \to$ the initial sample is down to 25% of its initial value after $\textcolor{red}{2}$ half-lives