# Finding half life given decay rate

• Apr 26th 2012, 12:54 AM
jkort13
Finding half life given decay rate
Hey guys, I'm working on a problem right now and I'm having some trouble.
The decay of a certain chemical is 9.3% per year. Using the exponential decay model P(t) = P0-kt d where k is the decay rate, and P0 is the original amount of chemical find the half-life.

Thanks!
• Apr 26th 2012, 02:39 AM
biffboy
Re: Finding half life given decay rate
Just out of interest did you get the answer 7.1 years?
• Apr 26th 2012, 02:48 PM
jkort13
Re: Finding half life given decay rate
Yes that is the correct answer.
• Apr 26th 2012, 06:00 PM
HallsofIvy
Re: Finding half life given decay rate
I'm a bit surprized- that formula doesn't make sense. If $P(t)= P_0^{kt d}$, then the "initial value" is $P(0)= 1$, not $P_0$. Did you mean $P(t)= P_0e^{kt d}$? And if $P_0$ is the initial value, t is the time, and k is the decay rate, what is "d"?

More common is the formula $P(t)= P_0e^{kt}$ for decay. In one year, we will have $P(1)= P_0e^k$. If the chemical decays 9.3% per year, P(1) should equal $(1- .093)P_0= .907P_0$ so $P_0e^k= .907P_0$. That is, k must satisfy $e^k= .907$ so k= ln(.907)= -.0976.

Once you know that, the "half life" is the time until $P_0$ becomes $P_0/2$. Solve $P_0e^{-.0976t}= P_0/2$ which is the same as $e^{-.0976t}= 1/2$. Solve that for t.