# 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?

Jan 13, 2016

$\text{40 minutes}$

#### 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 $\text{880 Bq}$. Now, what does that mean?

A becquerel, $\text{Bq}$, is a unit used to measure of the radioactivity of an isotope in which one nucleus decays per second.

$\textcolor{b l u e}{\text{1 becquerel" = "1 decay"/"1 s}}$

This means that you have

${\text{1 Bq" = "1 s}}^{- 1}$

So, an activity of $\text{880 Bq}$ means that $880$ nuclei are decaying every second.

Now, after $160$ minutes pass, the activity falls to $\text{55 Bq}$, 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

$\textcolor{b l u e}{A = {A}_{0} \cdot \frac{1}{2} ^ n} \text{ }$, where

$n$ - the number of half-lives

Plug in your values to get

$55 \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{Bq"))) = 880 color(red)(cancel(color(black)("Bq}}}} \cdot \frac{1}{2} ^ n$

This is equivalent to

$\frac{55}{880} = \frac{1}{2} ^ n$

$\frac{1}{16} = \frac{1}{2} ^ n \implies {2}^{n} = 16$

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 $\text{880 Bq}$ to $\text{55 Bq}$.

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

$\textcolor{b l u e}{n = \text{given period of time"/"half-life" = t/t_"1/2}}$

you can say that

${t}_{\text{1/2}} = \frac{t}{n}$

This will get you

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