A certain radioactive isotope decays at a rate of 0.2% annually. Determine the half-life of this isotope, to the nearest year.

The half-life of the radioactive isotope is 346 years.

Explanation:

The decay rate of the isotope is modelled after the following first-order linear ordinary differential equation:

Where:

- Current isotope mass, measured in kilograms.

- Time, measured in years.

- Time constant, measured in years.

The solution of this differential equation is:

Where is the initial mass of the isotope. It is known that radioactive isotope decays at a yearly rate of 0.2 % annually, then, the following relationship is obtained:

The time constant associated to the decay is:

Finally, the half-life of the isotope as a function of time constant is given by the expression described below:

If , the half-life of the isotope is:

The half-life of the radioactive isotope is 346 years.

that's what i got you can count the number of atoms