# Exponential decay of Radium 226

The text below is verbatim of what my question is.

The half life of $$\textrm{Ra-}226$$ is approximately $$1599$$ years. If the amount left after $$1000$$ years is $$1.7$$ grams, what is the amount left after $$2000$$ years? Give an approximation to three decimal places. If you use a formula you must derive it first.

So, I already know the decay formula which is $$y= Ce^{kt}$$

but where I get stuck is when trying to solve for one of the variables, because when I use the current amount of $$y=1.7$$ grams at $$t=1000$$ like so $$1.7=Ce^{1000k}$$ I still have unknown variables and I cannot solve for $$k$$ or $$C$$ without knowing the other.

You have not used the fact that the half life is $$1599$$ years. That allows you to derive the value of $$k$$, because at $$t=1599$$ the exponential factor is $$\frac 12$$

• So doing what you're saying would be $y=\frac{1}{2}e^{1599k}$ but I still have Y to solve for. Sep 25, 2019 at 4:08
• No, you need to derive $k$ from $1599$. You are told that if you start with $1$, after $1599$ years you have $\frac 12$. That is the definition of half-life Sep 25, 2019 at 4:14

The half-life $$t_{1/2}$$ represents the moment at which your initial quantity reaches half.

Thus, $$y(1599)=\frac{1}{2}C$$.

This means, $$1/2=e^{1599k} \Longrightarrow k=-\frac{\ln 2}{1599}$$

Now it is easy to solve for $$C$$.

Having $$C$$ and $$k$$ you can get your desired output.

If you want a more intuitive solution you can think of it like this. In every period of a single half-life the amount of radioactive material decreases to half its original value. Hence the amount left at time $$t$$ is given by: $$A(t) = A_0(\frac 12)^N$$, where $$N$$ is the number of half-lives elapsed.

Here, a thousand years have elapsed since the "initial" $$1.7$$ gram, so $$N=\frac{1000}{1599}$$, and you can immediately compute $$A(t) = 1.7(\frac 12)^{\frac{1000}{1599}}=1.102$$ gram.