# A radioactive element has a half life of 2,000 years. If a sample of this element begins with a mass of 100 grams, how long would you have to wait for the mass to decrease to 25 grams?

Mar 29, 2016

4000 years

#### Explanation:

You can solve this (and other) general half-life calculations with the general formula: $N = {N}_{0} {e}^{-} \left(\lambda t\right)$ where ${t}_{\frac{1}{2}} = \frac{0.693}{\lambda}$

In this case you can also just use the definition that a half-life is the time that it takes for ½ of the starting material to reduce to half of its amount.

So, one half-life of 2000 years means that in 2000 years 100 grams will decay to 50 grams. In another 2000 years that 50 grams will decay to 25 grams, which is the requested amount.

Thus, 2000 + 2000 = 4000 years for it to go from 100 to 25 grams.