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

##### 1 Answer

#### Answer:

#### Explanation:

The thing to remember about a radioactive nuclide's **nuclear half-life**, **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 **half-life** passes.

So, if you start with

#A_0 * 1/2 = A_0/2 = A_0/2^color(red)(1) -># after#color(red)(1)# half-life#A_0/2 * 1/2 = A_0/4 = A_0/2^color(red)(2) -># after#color(red)(2)# half-lives#A_0/4 * 1/2 = A_0/8 = A_0/2^color(red)(3) -># after#color(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

#A_t = A_0 * 1/4 = A_0/4 = A_0/2^color(red)(2)#

As you can see, this is exactly what you would expect to get after **half-lives** pass, so

#t = color(red)(2) * t_"1/2"#

and

#A_ (color(red)(2) xx t_"1/2") = A_0/2^color(red)(2) -># the initial sample is down to#25%# of its initial value after#color(red)(2)# half-lives