What fraction of the nuclei of a radioactive element decays in four half-lives.

If the half-life of a radioactive element is T, then the number of nuclei remaining after time t will decrease from No to N:

N = N0 * 2 ^ (- t / T);

In our case, the decay time is equal to 4 half-lives:

t = 4T;

N = N0 * 2 ^ (- 4T / T) = N0 * 2 ^ (- 4) = N0 / 16.

The number of decayed nuclei:

N0 – N = N0 – N0 / 16 = (15/16) N0.

Answer: 15/16 nuclei of a radioactive element will decay in 4 half-lives.