# Question #40235

May 11, 2017

The half-life of radium-226 is 1600 years.

#### Explanation:

The half-life of an element is the time it takes for a sample to be reduced by a factor of $1 / 2$. To solve this problem we note that we were given the time for the sample to be reduced by a factor of $1 / 4$ which is a multiple of $1 / 2$:

$\frac{1}{4} = \frac{1}{2} \cdot \frac{1}{2}$

This implies that the sample has undergone 2 half-lives. To see this consider the amount of the sample remaining after one half-life:

${N}_{\text{one-halflife" = N_"beginning}} / 2$

The behaviour of the sample will be the same no matter how many atoms we started with - there will always be half of them at the end of one half life. So, at the end of the first half life we can start over and see how many we would have left at the end of a second half life:

${N}_{\text{two-halflives" = N_"one-halflife}} / 2$

Then substituting in from the first equation:

${N}_{\text{two-halflives" = (N_"beginning"/2)/2=N_"beginning}} / 4$

Therefore, it took 2 half lives to reduce the number of atoms by a factor of 4. This means that the half-life of the sample is half of the time provided:

${t}_{\text{Halflife" = "3200 years"/2 = 1600 " years}}$