When one of the naturally occurring radioactive heavy elements like ²³⁸U decays, a decay chain follows, leading to the creation of various unstable,radioactive descendant nuclides. Consider a sample known to be pure ²³⁸U at time zero. Suppose that none of its decay products migrate away from the original sample. Use the fact that the lifetime of the original nuclide, in this case ²³⁸U, is much longer than the lifetimes of any of its descendants to show that a quasi equilibrium develops in which the amounts of ²³⁸U and all of its descendant are steadily decreasing at a rate determined by the lifetime of ²³⁸U. Show that the ratio of the abundance of each descendant nuclide to the abundance of ²³⁸U is very nearly constant in time, and is given by the ratio of the descendant’s lifetime to the lifetime of ²³⁸U. Show that in this equilibrium every radioactive species contributes equally to the radioactivity(in decays per second) of the sample.