g if 2 half-lives elapsed in a rock containing Carbon-14 and Nitrogen-14, what is the ratio of parent to daughter isotopes

The ratio of parent to daughter isotopes is .

Explanation:

We know that Carbon-14 decays in time and transforms into Nitrogen-14, being the latter the "daughter" of the first one. The decay of any isotope is represented by the following ordinary linear differential equation:

(Eq. 1)

Where:

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

- Time constant, measured in years.

- Current mass of the isotope, measured in grams.

The solution of this differential equation is:

(Eq. 2)

Where:

- Time, measured in years.

- Initial mass of the isotope, measured in grams.

Time constant can be found as a function of half life. Please notice that half-life of Carbon-14 is 5760 years. The equation of time constant is:

(Eq. 3)

Where is the half-life of the isotope, measured in years.

If we know that and , then we have that:

Which means that 75 % of the original mass of Carbon-14 became Nitrogen-14. The parent-to-daughter ratio is:

The ratio of parent to daughter isotopes is .