Radioactive plutonium-239 \(\left(t_{\frac{1}{2}}=2.44 \times 10^{5} \mathrm{yr}\right)\) is used in nuclear reactors and atomic bombs. If there are \(5.0 \times 10^{2} \mathrm{~g}\) of the isotope in a small atomic bomb, how long will it take for the substance to decay to \(1.0 \times 10^{2} \mathrm{~g},\) too small an amount for an effective bomb?

Understanding the radioactive decay law is essential for calculating how substances like plutonium-239 diminish over time. According to this fundamental principle, the decay of radioactive isotopes follows a predictable pattern. Imagine having a stack of playing cards, and with each passing interval, you're required to throw away half of them. The rate at which your pile shrinks represents the exponential decrease characteristic of this law.

For a mathematical understanding, the decay law can be written as:
\begin{align*}N = N_0 \times \bigg(\frac{1}{2}\bigg)^{\frac{t}{T}}\tag{1}\text{or}\tag{2}\text{In more general terms:}N = N_0 \times e^{-\frac{t}{\tau}}\tag{3}\text{Where:} \begin{itemize} \item \(N_0\) is the initial quantity of the substance \item \(N\) is the quantity after time \(t\) \item \(T\) is the half-life of the substance (time it takes for half of it to decay) \item \(e\) is the base of the natural logarithm \item \(\tau\) is the mean lifetime of the substance \end{itemize}