Calculating how long it will take for the

Damian Hanna

Damian Hanna

Answered question

2022-03-16

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 (t) for the half life of 500 amount of substance to fall below 100 (to be 99 -- I am only concerned with integers) when it has a half life of 5.
I have gotten this far in the calculation:
99=500(12)t5
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

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 A0 is the original amount of substance, then the amount of substance left after n half-lives is
A=A02n.
In this situation, then, you want
5002n<100,
or equivalently,
5<2n.
If we're dealing with integer numbers of half-lives, then n=3 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
ln5<nln2,
so
n>ln5ln2,
so more than 5ln5ln2 units of time must pass.
Edit in response to OP's edit
I would note that there is a difference between "less than 100" and "less than or equal to 99", but if you're only concerned with integer amounts, then you've set it up perfectly. If we divide by 99 and multiply both sides by 2t5, we get the equivalent equation
2t5=50099,
and hitting both sides with a logarithm gets us
t5ln2=ln(50099),
whence
t=c5ln(50099)ln2.

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