Radioactive decay is a first-order chemical reaction which follows an exponential decay in the reactant, the radioactive element. Starting with an original number of radioactive atoms n_{0} at time zero, the number of radioactive atoms left, n, after a certain time t is given by

**n = n _{0} e^{-kt }**

Where k is the rate constant for this decay reaction. The decay of ^{238}U has a rate constant of k = **1.54 x 10 ^{-10} year^{-1} for the reaction **

^{238}U → α + ^{234}Th

Where α is an alpha particle, a fast moving helium ^{4}He^{2+} nucleus. Calculate the fraction of the uranium decayed after 4.51×10^{9} years, roughly the age of the Earth.

Using the “exp” nomenclature for clarity

Thus during the lifetime of the Earth about half of the ** ^{238}U** originally present has been converted to thorium-234