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

1 Answer
Jun 10, 2017

Answer:

#"2 half-lives"#

Explanation:

The thing to remember about a radioactive nuclide's nuclear half-life, #t_"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 * 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 #t# passes, which means that you will have

#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 #color(red)(2)# 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