The half-life of strontium-90 is 28 years. How long will it take a 44 mg sample to decay to a mass of 11 mg?

1 Answer
Oct 20, 2015

Answer:

#"56 years"#

Explanation:

A radioactive isotope's nuclear half-life tells you how much time must pass until an initial sample is halved.

In your case, strontium-90 is known to have a half-life of #28# years. THis means that if you start with an initial mass of strontium-90, let's say #A_0#, you will have

  • #A_0 * 1/2 -># after one half-life passes;
  • #A_0/2 * 1/2 = A_0/4 -># after two half-lives pass;
  • #A_0/4 * 1/2 = A_0/8 -># after three half-lives pass;
  • #A_0/8 * 1/2 = A_0/16 -># after four half-lives pass;
    #vdots#

and so on.

Notice that you can write the remaining amount of an initial sample by using the number of half-lives that pass

#"remaining amount" = "initial amount"/2^n" "#, where

#n# - the number of half-lives that pass.

Now, you initial sample has a mass of #"44 mg"#. Notice that the remaining sample can be written as

#"11 mg" = "44 mg"/4 = "44 mg"/2^2#

This means that two half-lives must pass in order for the strontium-90 sample to decay to a quarter of its initial mass.

This implies that you have

#2color(red)(cancel(color(black)("half-lives"))) * "28 years"/(1color(red)(cancel(color(black)("half-life")))) = color(green)("56 years")#