The radionuclide ^232 U alpha-decaps into ^228 Th. (a)...

The radionuclide ${ }^{232} \mathrm{U}$ alpha-decaps into ${ }^{228}$ Th. $(a)$ Find the energy released in the decay, (b) is it possible for ${ }^{232} \mathrm{U}$ to decay into ${ }^{231} \mathrm{U}$ by emituing a neutron? $(c)$ is it possible for ${ }^{232} \mathrm{U}$ to decay into ${ }^{231} \mathrm{~Pa}$ by emiting a proton? The atomic masses of ${ }^{231} \mathrm{U}$ and ${ }^{231} \mathrm{~Pa}$ are respectively $231.036270 \mathrm{u}$ and $231.035880 \mathrm{u} .$

Key Concepts

Alpha Decay

Alpha decay is a type of radioactive decay in which an unstable nucleus emits an alpha particle consisting of two protons and two neutrons, resulting in a new nucleus with its atomic number reduced by two and mass number reduced by four. This decay process is governed by energy and nuclear structure considerations and is common among heavy elements seeking greater stability.

Q-value Calculation (Energy Released in Nuclear Decay)

The Q-value of a nuclear reaction represents the net energy released or absorbed during the process. It is determined by the mass difference between the reactants and products, converted to energy using Einstein’s mass-energy equivalence principle (E=mc²). A positive Q-value indicates that energy is released, which is a driving factor for spontaneous decay.

Mass-Energy Equivalence and Mass Defect

Mass-energy equivalence is the principle that mass can be converted to energy and vice versa, as expressed by Einstein’s famous equation E=mc². In nuclear reactions, the measured mass of a nucleus is often less than the sum of the masses of its individual nucleons; this 'missing' mass, known as the mass defect, is released as energy during decay processes.

Conservation Laws in Nuclear Reactions

Nuclear reactions must obey several conservation laws including the conservation of energy, momentum, charge, and nucleon number. These laws determine the allowed decay pathways; for instance, a decay involving the emission of a neutron or proton must conserve mass and charge, and the final products must reflect specific constraints based on the initial nuclear composition.

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