# Question 72e48

Nov 30, 2015

$\text{40 days}$

#### Explanation:

As you know, nuclear half-life expresses the time needed for an initial sample of a radioactive isotope to get halved.

Now, you know that your radioactive isotope's activity decreases from $1200$ disintegrations per minute to $150$ disintegrations per minute in $120$ days.

The number of disintegrations per minute (dpm) is simply the number of atoms that undergo radioactive decay in one minute. The important thing to realize here is that the number of disintegrations per minute will also be halved with the passing of one half-life.

In other words, you can use disintegrations per minute as a substitute for the mass of the sample.

Your goal now is to figure out how many half-lives must have passed in order for the dpm count to reach $150$.

You can say that

• $\text{1200 dpm"/2 = "600 dpm } \to$ after the passing of one half-lfie

• $\text{600 dpm"/2 = "300 dpm } \to$ after the passing of two half-lives

• $\text{300 dpm"/2 = "150 dpm } \to$ after the passing of three half-lives

So, three half-lives must pass in order for your sample to go from $\text{1200 dpm}$ to $\text{150 dpm}$. And remember, this happened in a total fo $120$ days. This means that one half-life will be equivalent to

1 color(red)(cancel(color(black)("half-life"))) * "120 days"/(3 color(red)(cancel(color(black)("half-lives")))) = color(green)("40 days")#