Half-life, \(t_{1 / 2}\), is the time required for a quantity to shrink to half its initial value. Radioactive elements are often defined by their half-life radioactive decay given by \(N(t)=N_{0} e^{-\frac{t}{\tau}}\), where \(N_{\mathrm{O}}\) is the initial quantity and \(\tau\) is the element's mean lifetime. The half-life is related to the mean lifetime by \(t_{1 / 2}=\tau\) In 2. suppose a particular element has a half-life of 200 ps. Find the mean lifetime of the decay. If there are \(100 \mathrm{gm}\) of a particular element at \(t=0\), how many are there after \(1 \mathrm{~ns}\) ?