Final answer:
The half-life of a radioactive element is the time it takes for 50% of a sample to decay. After three half-lives, the amount of radioactive substance remaining is one-eighth of the original amount. The decay constant shows how much of a sample decays in one second, not the total time it will take to decay.
Step-by-step explanation:
The half-life of a radioactive element is the period needed for half of a given quantity of a substance to undergo decay. For example, if the half-life of fluorine-20 is 11.0 seconds, this means that it takes 11.0 seconds for 50% of a sample of fluorine-20 to decay. After one half-life (11.0 s), a 100.0 mCi sample would become 50.0 mCi. After two half-lives (22.0 s), the sample would be reduced to 25.0 mCi. After three half-lives (33.0 s), it would be 12.5 mCi. Eventually, following this pattern, we can determine it takes a little more than three half-lives for a sample to decay from 100.0 mCi to 10.0 mCi.
What is often misunderstood is that the decay constant, which is a percentage, does not directly indicate the time it will take for the entire sample to decay. Instead, it tells us the fraction of the sample that decays in one second. Therefore, it is incorrect to think that if the decay constant is 0.05 s⁻¹ the sample will completely decay in 20 seconds.
Moreover, after three half-lives have passed, the amount of a radioactive sample remaining is nether N/3, N/6, nor N/27, but instead N/8, since with each half-life, the quantity of atoms is reduced by half, leading to a reduction to a eighth of its initial number after three half-lives. For instance, a sample with two different isotopes both starting with equal amounts, but with different half-lives, will change in ratio over time. After a duration equal to the half-life of the shorter-lived isotope, the ratio of the isotopes will shift significantly favoring the longer-lived isotope.