Uranium-235 Lead-207 7.13 × 10⁸ 7 Alpha + 4 beta Uranium-238 Lead-206 4.51 × 10⁸ 8 Alpha + 6 beta In a large number of radioactive nuclei of a given kind, a certain fraction will decay in a specific length of time. Let’s take this fraction as one-half and measure the time it takes for half the nuclei to decay. This time it is called the HALF-LIFE of that particular nucleus and there are various accurate physical ways of measuring it. During the interval of one half-life, one-half of the nuclei will decay, during the next half-life half of what’s left will decay, and so on. We may tabulate it like this: Elapsed time Amount left of (Number of what was half-lives) originally present 1 ½ 2 ¼ 3 ⅛ 4 ¹/₁₆ 5 ¹/₃₂ 6 ¹/₆₄ 7 ¹/₁₂₈ ... ... In other words, after seven half-lives, less than 1% of the original amount of material will still be radioactive and the remaining 99%+ of its atoms will have been converted to atoms of another nuclide. This kind of process can be made the basis of a clock. It works, in effect, like the upper chamber of an hourglass. Mathematically it is written: N = N₀e^{-λt} where N = the number of radioactive atoms present in the system now, N₀ = the number that was present when t = 0, (in other words, at the time the clock started), e = the base of natural (or Napierian[8]) logarithms (the numerical value of e = 2.718 ...), λ (lambda) = the decay rate of the radioactive material, expressed in atoms decaying per atom per unit of time, t = the time that has elapsed since the origin of system, expressed in the same units. Obviously, in ordinary computations that would not be enough information to calculate the time, because there still are two unknowns, _N₀_ and _t_. In a closed system, however, the atoms that have decayed do not disappear into thin air. They merely change into other atoms, called daughter atoms, and remain in the system. And at any point in time, there will be both PARENT and DAUGHTER atoms mixed together in the material. The older the material, the more daughters and the fewer parents. Some daughters are also radioactive, but this does not change the basic situation. Thus it follows that N₀ = N + D where D = the number of daughter (decayed) atoms. We may then substitute into the first equation N = (N + D) e^{-λt} and solve t = 1/λ · ln(1 + D/N) where ln = the natural logarithm, the logarithm to base e.