Half-life

**n8thatsme** · October 2nd 2008, 12:15 PM

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 $r=ln(2)/30$ but what about the 85%? Any help is appreciated.

**Chris L T521** · October 2nd 2008, 12:24 PM

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 $r=ln(2)/30$ but what about the 85%? Any help is appreciated.

Let $N_0$ be the initial amount of Radium-221, and let $N$ 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 $N=.15N_0\implies \frac{N}{N_0}=.15$

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

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

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

Can you take it from here?

--Chris

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