Radioactive Radium has a half-life of approximately 1600 years. What percentage of the present amount remains after 100 years?
$\endgroup$ 11 Answer
$\begingroup$The formula for exponential decay is: $$\frac{dN}{dt}=-N\lambda$$
Solving the differential equation, we get the following:
$$N(t)=N(0)\cdot \rm{e}^{-\lambda t}$$
To get the percentage, we will start off with $N(0)=100$, solving for $t=100$ and $\lambda=\frac{\ln{2}}{1600}$ (which we find from the definition of half-life: $t_{\frac{1}{2}}=\frac{\ln{2}}{\lambda}$), we get:
$$N(100)=100\cdot\rm{e}^{-\frac{100\ln{2}}{1600}}=95.76$$
So $95.76\%$ remains after $100$ years.
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