# Half-life

• Oct 2nd 2008, 12:15 PM
n8thatsme
Half-life
Radium-221 has a half-life of 30 seconds. How long will it take for 85% of a sample to decay?

Ok so I really don't know what to do here. I know I can find r because its $\displaystyle r=ln(2)/30$ but what about the 85%? Any help is appreciated.
• Oct 2nd 2008, 12:24 PM
Chris L T521
Quote:

Originally Posted by n8thatsme
Radium-221 has a half-life of 30 seconds. How long will it take for 85% of a sample to decay?

Ok so I really don't know what to do here. I know I can find r because its $\displaystyle r=ln(2)/30$ but what about the 85%? Any help is appreciated.

Let $\displaystyle N_0$ be the initial amount of Radium-221, and let $\displaystyle N$ be the current amount of Radium.

Since 85% decayed, that means that the current amount is 15% of the original amount.

Thus, this implies that $\displaystyle N=.15N_0\implies \frac{N}{N_0}=.15$

Since the decay is modeled by $\displaystyle N=N_0e^{-kt}$, we see now that $\displaystyle .15 = e^{-kt}$

Now, k can be found using the half-life formula: $\displaystyle \lambda=\frac{\ln2}{k}\implies k=\frac{\ln 2}{\lambda}$, where $\displaystyle \lambda$ is the half-life.

So our equation now has the form $\displaystyle 0.15=e^{-\frac{\ln2}{\lambda}t}$

Can you take it from here?

--Chris