The number of half-lives required for a radioactive atom to decay to a negligible amount depends on the half-life of the specific radioactive isotope. The half-life is the time it takes for half of the radioactive atoms in a sample to decay.
If there is only one radioactive atom remaining, it means that half of the original sample has already decayed, and we are left with one atom. In this case, we can consider it as the first half-life.
To determine the number of half-lives required for the remaining atom to decay, we need to consider the concept of exponential decay. Each half-life reduces the amount of radioactive material by half. So, if we start with one atom, after one half-life, we would expect half of that atom to decay, leaving us with 0.5 atoms. After the second half-life, half of that remaining atom would decay, leaving us with 0.25 atoms. After the third half-life, 0.125 atoms would remain, and so on.
However, it's important to note that when dealing with such small quantities, we reach a point where the concept of a "fractional" atom no longer applies. At the atomic scale, we cannot have a fraction of an atom. Therefore, after a certain number of half-lives, the remaining atom will decay, and no further decay can occur.
In practical terms, once the remaining quantity of radioactive material becomes extremely small (less than a few atoms), it effectively decays to zero within a few more half-lives. The exact number of half-lives required to reach this point depends on the specific radioactive isotope and its half-life.