(a) Calculate the energy released when an U-238 isotope decays to Th-234. The atomic masses are \(\begin{array}{llll}\text { given by } & \text { U-238: } & 238.0508 & \text { amu; } & \text { Th-234: }\end{array}\) 234.0436 amu; He-4: 4.0026 amu. (b) The energy released in (a) is transformed into the kinetic energy of the recoiling Th- 234 nucleus and the \(\alpha\) particle. Which of the two will move away faster? Explain.

In nuclear physics, the term 'mass defect' refers to the difference between the mass of a nucleus and the sum of the masses of its individual nucleons (protons and neutrons). This discrepancy arises because some mass is converted into binding energy, which holds the nucleus together.

To understand the concept of mass defect, we need to realize that a nucleus is actually lighter than its constituent protons and neutrons when they are separate. The energy equivalent of this 'missing' mass is what accounts for the stability of the nucleus. When a nuclear reaction such as decay occurs, the initial mass (parent nucleus) and the final mass (decay products) will differ by the mass defect.

Let's apply this to an exercise: when an Uranium-238 nucleus decays to Thorium-234 and Helium-4 (an alpha particle), we calculate the mass defect by subtracting the mass of the decay products from the mass of the original Uranium-238 nucleus. This mass defect then allows us to calculate the energy released during the decay.

One of the most famous equations in physics is Einstein’s mass-energy equivalence formula, stated as \(E=mc^2\). This principle implies that mass (m) and energy (E) are interchangeable; a small amount of mass can be converted into a large amount of energy, with the 'c' representing the speed of light squared, which is a very large number.

The conversion factor from mass to energy in atomic mass units (amu) to mega-electron-volts (MeV) is 931.5 MeV/amu. This factor is crucial in nuclear physics calculations, particularly in quantifying the energy released or absorbed during nuclear reactions. In the case of the decay of Uranium-238, the energy released is calculated by converting the mass defect to energy, using this equivalence. The operation provides us with the total energy that would be theoretically available if the mass were fully converted into energy.

The principle of conservation of momentum states that for a closed system, the total momentum remains constant over time, provided no external forces are acting on the system. In the context of nuclear reactions, this principle can explain the behavior of particles after decay.

According to conservation of momentum, if the parent nucleus is initially at rest, the momentum of the system is zero. After decay, the sum of the momenta of the products must also be zero. This means the momentum (mass times velocity) of each product must be equal in magnitude but opposite in direction.

In the exercise, even though the alpha particle is lighter than the Thorium-234 nucleus, it moves away faster because it has less mass. The heavier Thorium-234 nucleus moves more slowly to ensure that the momentum of the system is conserved and remains at zero post-decay.

A nuclear reaction involves the change in the composition of a nucleus and can lead to transmutation of elements or isotopes. Nuclear reactions can release or absorb energy and often involve the emission of particles such as alpha particles, beta particles, or neutrons.

In a decay process like the one involving Uranium-238, the nucleus emits an alpha particle and transforms into a different element, Thorium-234. This particular reaction type is known as alpha decay, and is one common way unstable atoms (radioisotopes) achieve greater stability.

Understanding nuclear reactions provides insight into the behavior of radioactive elements, the energy associated with different types of decay, and the fundamental concepts governing the transformations of atomic nuclei. The energy released or absorbed during such reactions has both practical and theoretical implications, from nuclear power generation to the study of elemental formation across the universe.