A sample of a radioactive substance decayed to 95.5% of its original amount after a year. What is the half-life of the substance in years, and how long would it take the sample to decay to 25% of its original amount in years.

2. Originally Posted by acg716
A sample of a radioactive substance decayed to 95.5% of its original amount after a year. What is the half-life of the substance in years, and how long would it take the sample to decay to 25% of its original amount in years.

Let $\displaystyle N_0$ be the original amount of the substance and $\displaystyle N$ be the current amount of the substance.

Now, radioactive decay takes on the form $\displaystyle N=N_0e^{-kt}$, where $\displaystyle k$ is the decay constant. We need to know what $\displaystyle k$ is, because the half life of this substance is defined by $\displaystyle \lambda=\frac{\ln 2}{k}$, where $\displaystyle \lambda$ is the half life.

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The first bit of information gives us enough info to find k.

It takes a year for the substance to decay to 95.5% of the original amount.

Thus, we see that $\displaystyle N=N_0e^{-kt}\implies .955N_0=N_0e^{-k}$, where we replaced $\displaystyle N$ with $\displaystyle .955N_0$.

This now simplifies to $\displaystyle .955=e^{-kt}\implies \ln(.955)=-k\implies \color{red}\boxed{k\approx 0.046}$

Now we can find the half life:

$\displaystyle \lambda=\frac{\ln 2}{0.046}\implies\color{red}\boxed{\lambda\approx 15.05~\text{years}}$

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Now, can you find $\displaystyle t$, such that $\displaystyle .25=e^{-0.046t}$?? [this was what the second part of the question was asking for]

Does this make sense?

--Chris

3. thank you so much!!