Suppose you found a material in which 12.5% of the original radioactive atoms were present. If the half-life is 10years, how old is the material?

The material is 30 years old.

Explanation:

The decay of radioactive isotopes can be modelled by the following ordinary differential equation:

(Eq. 1)

Where:

- Current mass of the isotope, measured in grams.

- Time constant, measured in years.

- Rate of change of the isotope in time, measured in grams per year.

After some handling, we find that the solution of the differential equation is:

(Eq. 2)

Where:

- Initial mass of the isotope, measured in grams.

- Time, measured in years.

Now we clear time within the expression above:

Besides, we determine the time constant in terms of the half-life (), measured in years:

(Eq. 3)

If we know that and , then the age of the material is:

The material is 30 years old.

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