Question

The activity of a sample of radioactive material is measured,and found to be 880 Bq. After 160 minutes the activity has fallen to 55 Bq. What is the half-life of Bq?

Answer 1

`"40 minutes"`

Explanation:

Before doing anything else, make sure that you understand what's going on here.

You're dealing with a sample of radioactive material that has an activity equal to `"880 Bq"`. Now, what does that mean?

A becquerel, `"Bq"`, is a unit used to measure of the radioactivity of an isotope in which one nucleus decays per second.

`color(blue)("1 becquerel" = "1 decay"/"1 s")`

This means that you have

`"1 Bq" = "1 s"^(-1)`

So, an activity of `"880 Bq"` means that `880` nuclei are decaying every second.

Now, after `160` minutes pass, the activity falls to `"55 Bq"`, which is equivalent to saying that at this point only `55` nuclei are decaying per second.

As you know, nuclear half-life is simply the time needed for a sample of radioactive material to decay to half of its initial size.

The equation that establishes a relationship between the amount left undecayed, `A`, the initial amount, `A_0`, and the number of half-lives that pass in a period of time `t` looks like this

`color(blue)(A = A_0 * 1/2^n)" "`, where

`n` - the number of half-lives

Plug in your values to get

`55 color(red)(cancel(color(black)("Bq"))) = 880 color(red)(cancel(color(black)("Bq"))) * 1/2^n`

This is equivalent to

`55/880 = 1/2^n`

`1/16 = 1/2^n implies 2^n = 16`

You will thus have

`2^n = 16 implies n = 4`

Therefore, four half-lives must pass in order for the activity of the sample to go from `"880 Bq"` to `"55 Bq"`.

Since the number of half-lives can be thought of as

`color(blue)(n = "given period of time"/"half-life" = t/t_"1/2")`

you can say that

`t_"1/2" = t/n`

This will get you

`t_"1/2" = "160 minutes"/4 = color(green)("40. minutes")`