One-half of the nuclei of a given radioisotope decays during one half-life. Why doesn't the remaining half decay during the next half-life?

Half-life is the time it takes for half the nuclei of a given radioactive substance to decay. This means that after one half-life, half of the original nuclei will remain undecayed, while the other half will have decayed into a different element.

Radioactive decay is a random, probabilistic process on the atomic level. Each nucleus has a certain probability of decaying, and this probability remains constant throughout each half-life. The decay does not "know" when the half-life is over, nor does it "know" when it started; it is purely based on chance.

During the second half-life, the remaining half of the nuclei still have the same probability of decaying as they did in the first half-life. So, during the second half-life, half of the remaining nuclei (which is one-quarter of the original nuclei) will decay, and so on. The number of decaying nuclei in each half-life period decreases by half but never reaches zero.

Since radioactive decay is a continuous process, we can examine the fraction of undecayed nuclei at any point in time using the decay equation: N(t) = N_0 e^{-λt}. Here, N(t) represents the number of undecayed nuclei at time t, N_0 is the initial number of nuclei, λ is the decay constant, and t is the time elapsed. As t approaches infinity, N(t) approaches zero, but it will never actually reach zero. This indicates that there will always be some undecayed nuclei remaining, even though their number continuously decreases. In conclusion, the remaining half of the radioisotope nuclei do not decay during the next half-life because radioactive decay is a probabilistic process, and the number of decaying nuclei decreases by half with each half-life period. The remaining undecayed nuclei will continue to decay, but they will never entirely disappear.