# The half-life or a radioactive isotope is the time takes tor a quantity for the isotope to be reduced to half its initial mass. Starting with 150 grams ot a radioactive isotope, how much will be left atter 6 half-lives?

Apr 5, 2017

After six half lives there will be $2.3 g$ of the isotope left.

#### Explanation:

If we start with $150 g$ of the isotope, after one half-life the quantity will diminish to $\frac{150 g}{2} = 75 g$ which is half the original quantity.

This can be written mathematically as: $150 g {\left(\frac{1}{2}\right)}^{1}$

After the second half life: $\frac{75 g}{2} = 37.5 g = \frac{150 g}{4} = \left(150 g\right) {\left(\frac{1}{2}\right)}^{2}$

After the third half life: $\frac{37.5 g}{2} = 18.75 g = \frac{150 g}{8} = \left(150 g\right) {\left(\frac{1}{2}\right)}^{3}$

But now we can see a pattern developing, because for each new half-life we are dividing the quantity by $2$ to a power that increases as the number of half-lives.

Then we can take the original quantity and quickly compute for $6$ half-lives:

$\left(150 g\right) {\left(\frac{1}{2}\right)}^{6} = \left(150 g\right) \left(\frac{1}{64}\right) = \frac{150 g}{64} = 2.3 g$

The time it takes for the isotope to do this will depend on the isotope in question. For some it will be microsecond, and for others it will be years and decades. The isotope will typically decay to become another element or isotope.