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 \(^{232} \mathrm{Pu}\) nuclei and 1 mole of \(^{231} \mathrm{Pa}\) nuclei are each formed from their respective number of protons and neutrons. (Since the masses of \(^{232} \mathrm{Pu}\) and \(^{231} \mathrm{Pa}\) are atomic masses, they each include the mass of the electrons present. The mass of the nucleus will be the atomic mass minus the mass of the electrons.)

For this step, use the atomic masses given in the table for plutonium-232 (232.0381 u) and protactinium-231 (231.0359 u). Remember that the table gives the mass of the entire atom, including the electrons. Since we want to find the mass of the nucleus, we will need to subtract the mass of the electrons present.

To find the mass of the electrons, we will use the following information: - Atomic number of plutonium (Pu) = 94 - Atomic number of protactinium (Pa) = 91 - Mass of an electron = 9.10938356 × 10⁻³¹ kg (which is approximately 5.4858 x 10^-4 u) Multiply the atomic number of each element by the mass of an electron to find the mass of the electrons in the element. Mass of electrons in Pu = (94)*(5.4858 x 10^-4 u) Mass of electrons in Pa = (91)*(5.4858 x 10^-4 u)

Now, subtract the mass of the electrons from the atomic masses of the elements to obtain the mass of each nucleus: Mass of Pu nucleus = Atomic mass of Pu - Mass of electrons in Pu Mass of Pa nucleus = Atomic mass of Pa - Mass of electrons in Pa

To find the number of protons and neutrons in each nucleus, use the atomic numbers and mass numbers of the elements: - Number of protons in Pu = atomic number of Pu = 94 - Number of neutrons in Pu = mass number of Pu - atomic number of Pu = 232 - 94 = 138 - Number of protons in Pa = atomic number of Pa = 91 - Number of neutrons in Pa = mass number of Pa - atomic number of Pa = 231 - 91 = 140

Use the following information to calculate the mass of the protons and neutrons combined in each nucleus: - Mass of a proton = 1.007276466812 u - Mass of a neutron = 1.008664915 u Mass of protons and neutrons in Pu: 94 * (Mass of a proton) + 138 * (Mass of a neutron) Mass of protons and neutrons in Pa: 91 * (Mass of a proton) + 140 * (Mass of a neutron)

To find the change in energy, we can use Einstein's famous equation: \[E=mc^2\] Where E is the energy, m is the mass difference (mass of the combined protons and neutrons minus the mass of the nucleus), and c is the speed of light (2.998 x 10^8 m/s). First, calculate the difference in mass for Pu and Pa formations: Delta_mass(Pu) = Mass of protons and neutrons in Pu - Mass of Pu nucleus Delta_mass(Pa) = Mass of protons and neutrons in Pa - Mass of Pa nucleus Now, convert the mass difference from atomic mass units (u) to kilograms (kg) by multiplying by 1.6605 x 10^-27: Delta_mass(Pu) in kg = Delta_mass(Pu) in u * 1.6605 x 10^-27 Delta_mass(Pa) in kg = Delta_mass(Pa) in u * 1.6605 x 10^-27 Finally, calculate the change in energy for each nucleus formation: Change in energy for Pu formation = Delta_mass(Pu) in kg * c² Change in energy for Pa formation = Delta_mass(Pa) in kg * c²