(a) Calculate the energy released in the neutron-induced fission \(n+^{238} \mathrm{U} \rightarrow^{96} \mathrm{Sr}+^{140} \mathrm{Xe}+3 n,\) given \(m\left(^{96} \mathrm{Sr}\right)=95.921750 \mathrm{u}\) and \(m\left(^{140} \mathrm{Xe}\right)=139.92164\). (b) This result is about \(6 \mathrm{MeV}\) greater than the result for spontaneous fission. Why? (c) Confirm that the total number of nucleons and total charge are conserved in this reaction.

Neutron-induced fission occurs when a nucleus captures a neutron and subsequently splits into smaller fragments, releasing additional neutrons and energy. This process is central to many nuclear reactors where controlled fission chains drive the production of energy. In the exercise, uranium-238 captures a neutron and fissions into strontium-96, xenon-140, and releases three more neutrons. The additional neutrons can continue the fission process by being absorbed by other uranium nuclei, sustaining a chain reaction.

Understanding neutron-induced fission also includes recognizing that the incoming neutron adds energy to the nucleus. This energy helps to overcome the repulsive forces among protons in the nucleus, enabling it to split into fission fragments.

Mass-energy equivalence, encapsulated by Einstein's famous equation, E=mc^2, posits that mass can be converted into energy and vice versa. In the realms of nuclear reactions, this principle is crucial because it explains how, during fission, a slight loss in mass (called the mass defect) is transformed into a significant amount of energy. The calculated mass difference between the initial and final states in the fission process is converted into energy using the conversion factor of 931.5 MeV/u, reflective of the high energy density of nuclear processes.

In the provided example, the mass difference calculated is converted into a sizable amount of energy (177.1 MeV), released as the kinetic energy of the fission products and as radiation. The mass-energy equivalence explains why even minute differences in mass result in the release of a large amount of energy, underpinning why nuclear power is so potent.

The conservation of nucleons is a fundamental principle in nuclear physics that states the total number of protons and neutrons (nucleons) remains constant in any nuclear reaction. This must be true for any process to comply with the conserved properties of nuclear reactions. The exercise illustrates this by calculating the number of nucleons before and after the fission event, showing that the total remains unchanged at 239.

It is essential to differentiate the conservation of nucleons from the conservation of mass. While the number of nucleons is conserved, their mass can slightly differ due to the binding energy resulting from the strong nuclear force between them. Thus, mass is not conserved in nuclear reactions (leading to energy release), while the number of nucleons is.

The conservation of charge is another cornerstone of nuclear reactions, necessitating that the sum of electric charges before and after a reaction remains equal. Charges are carried by protons within the nucleus, and during fission, the total charge must be preserved. In the exercise example, the nuclear charge is maintained at 92 from uranium-238 through to the products strontium-96 and xenon-140, combined with the uncharged neutrons.

This conservation underpins the predictability of nuclear reactions and is crucial for balancing nuclear equations. It assures that the electrically neutral neutron does not affect the total charge of the reacting system, and therefore, the charge balance can be checked using only the protons present before and after the reaction. Understanding this helps in verifying that a proposed nuclear reaction is physically plausible.