You might wonder how it is possible to determine the half-life of long-lived radioactive isotopes such as ²³⁸U. With a half-life of more than 10⁹ years, the radioactivity of a sample of uranium will not measurably change in your lifetime. In fact, you can calculate the half-life using the mathematics governing first-order reactions. It can be shown that a 1.0-mg sample of ²³⁸U decays at the rate of 12 α emissions per second. Set up a mathematical equation for the rate of decay, ΔN/Δt = −kN, where N is the number of nuclei in the 1.0-mg sample and ΔN/Δt is 12 dps. Solve this equation for the rate constant for this process, and then relate the rate constant to the half-life of the reaction. Carry out this calculation, and compare your result with the literature value, 4.5 × 10⁹ years.