Question

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?

Answer 1

`"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")`