Many transuranium elements, such as plutonium-232, have very short half-lives. (For \({ }^{232} \mathrm{Pu}\), the half-life is 36 minutes.) However, some, like protactinium-231 (half-life \(=3.34 \times\) \(10^{4}\) years), have relatively long half-lives. Use the masses given in the following table to calculate the change in energy when 1 mole of \({ }^{232}\) Pu nuclei and 1 mole of \({ }^{231}\) Pa nuclei are each formed from their respective number of protons and neutrons.

We'll calculate mass difference for the two nuclei separately. To do this, we'll first calculate the total mass of protons and neutrons and then subtract the mass of the nucleus. For \({ }^{232}\mathrm{Pu}\): Number of protons = 94 Number of neutrons = 232 - 94 = 138 Mass of 1 proton = 1.00728 amu Mass of 1 neutron = 1.00867 amu Total mass of protons and neutrons = 94 x 1.00728 amu + 138 x 1.00867 amu = 94.28544 amu + 139.19646 amu = 233.4819 amu Mass of 1 mole of \({ }^{232}\mathrm{Pu}\) nucleus = 232.037 amu (given) Mass difference = 233.4819 amu - 232.037 amu = 1.4449 amu For \({ }^{231}\mathrm{Pa}\): Number of protons = 91 Number of neutrons = 231 - 91 = 140 Total mass of protons and neutrons = 91 x 1.00728 amu + 140 x 1.00867 amu = 91.66348 amu + 141.2148 amu = 232.87828 amu Mass of 1 mole of \({ }^{231}\mathrm{Pa}\) nucleus = 231.036 amu (given) Mass difference = 232.87828 amu - 231.036 amu = 1.84228 amu

We will use the Einstein's mass-energy equivalence formula, assuming \(c^2 = 9 \times 10^{16} m^2/s^2\) to convert the mass difference (in amu) to energy difference (in Joules). 1 amu = 1.66 x 10^{-27} kg For \({ }^{232}\mathrm{Pu}\): Mass difference = 1.4449 amu = 1.4449 x 1.66 x 10^{-27} kg = 2.40054 x 10^{-27} kg Energy difference = \(E = mc^2\) = 2.40054 x 10^{-27} kg x 9 x 10^{16} m^2/s^2 = 2.16049 x 10^{-10} J For \({ }^{231}\mathrm{Pa}\): Mass difference = 1.84228 amu = 1.84228 x 1.66 x 10^{-27} kg = 3.05818 x 10^{-27} kg Energy difference = \(E = mc^2\) = 3.05818 x 10^{-27} kg x 9 x 10^{16} m^2/s^2 = 2.75236 x 10^{-10} J These are the energy changes for 1 mole of \({ }^{232}\mathrm{Pu}\) and \({ }^{231}\mathrm{Pa}\) nuclei, respectively.