Radioactive decay series begin with a very long lived isotope. For example, the half-life of ²³⁸U is 4.5 × 10⁹ years. Each series is identified by the name of the long-lived parent isotope of highest mass.
(a) The uranium-238 radioactive decay series is sometimes referred to as the 4n + 2 series because the masses of all 13 members of this series can be expressed by the  equation M = 4n+ 2, where M is the mass number and n is an integer. Explain why the masses are correlated in this way.
(b) Two other radioactive decay series identified in minerals in the Earth’s crust are the thorium-232 series and the uranium-235 series. Do the masses of the isotopes in these series conform to a simple mathematical equation? If so, identify the equation.
(c) Identify the radioactive decay series to which each of the following isotopes belongs:

(d) Evaluation reveals that one series of elements, the 4n + 1 series, is not present in the Earth’s crust. Speculate why.

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