Half-life problem

$\frac{d[R]}{dt}=-k; [R]=-kt + [R]_{0}; t_{\frac{1}{2}}=\frac{[R]_{0}}{2k}$

Evelyn L 1H
Posts: 67
Joined: Fri Sep 29, 2017 7:05 am

Half-life problem

This is a question from a practice quiz:

A given compound decomposes with a half-life of 8.0 s and the half-life is independent of the concentration. How long does it take for the concentration to decrease to 1/9th of its initial value?

How would I solve this? I think it's zero order but there's no k given or initial concentration.
The answer should be 25 s.

Humza_Khan_2J
Posts: 56
Joined: Thu Jul 13, 2017 3:00 am

Re: Half-life problem

Well, if something takes 8 seconds for half of it to decompose, then we know how many half cycles it'll take for any amount to decompose. The formula for this would be (1/2)^n=(x), where x is the amount remaining(as a percent) and n is the number of cycles the system goes through to decompose to the amount.

Plugging this in, we get (1/2)^n=.1111. We can take the ln of both sides to get n(ln(.5))=ln(.1111). This yields n=3.17. Multiplying 3.17 cycles by 8 seconds per cycle yields ~25 seconds.

Evelyn L 1H
Posts: 67
Joined: Fri Sep 29, 2017 7:05 am

Re: Half-life problem

Thank you that was so helpful!!