Calculating how long it will take for the half-life of X amount to fall below Y amount I am trying to determine how long it will take for the half life of 500 amount of substance to fall below 100 (to be -- I am only concerned with integers) when it has a half life of 5. I have gotten this far in the calculation: So I guess my questions are as follows: 1.Have I set this up right? 2.Is there a better way to do this? 3.Can somebody point me in the direction of how to solve from here?
Answer & Explanation
Lilliana Rich
Beginner2022-03-17Added 3 answers
The half-life is the amount of time it takes for half of a given amount to disappear, or decay. After one half-life, there's half as much; two half-lives, a fourth as much; three half-lives, an eighth as much; etc. In other words, if is the original amount of substance, then the amount of substance left after n half-lives is In this situation, then, you want or equivalently, If we're dealing with integer numbers of half-lives, then half-lives must pass, so 15 units of time must pass (you didn't specify what units of time your half-life was in). If we aren't requiring integer numbers of half-lives, then we'll hit both sides with a logarithm to get so so more than units of time must pass. Edit in response to OP's edit I would note that there is a difference between "less than " and "less than or equal to ", but if you're only concerned with integer amounts, then you've set it up perfectly. If we divide by and multiply both sides by , we get the equivalent equation and hitting both sides with a logarithm gets us whence