Animated Solution for Physics - Atoms and Nuclei: In the options given below, let $E$ denote the rest mass energy of a nucleus and $n$ a neutron. The correct option is

Imagine a heavy, unstable nucleus, trembling with excess energy, just waiting for the right moment to split apart. This is the heart of nuclear fission, a process that powers stars and reactors alike. When a massive nucleus like Uranium-236 undergoes fission, it breaks into two lighter, more stable nuclei, releasing a few neutrons and a tremendous amount of energy in the process.

But where does this energy come from? To answer that, we must look to one of the most famous equations in all of physics.

Albert Einstein taught us that mass and energy are two sides of the same coin, beautifully connected by the equation $E=m{c}^2$. In any spontaneous nuclear reaction, the energy released is not created out of nowhere; it comes at the expense of mass.

This means that the total rest mass of the initial reactants must be strictly greater than the total rest mass of the final products. The "missing" mass, often called the mass defect, is converted directly into the kinetic energy of the fragments and the energy of the emitted photons. Therefore, for a fission reaction to occur spontaneously and release energy, the rest mass energy of the parent nucleus must be greater than the sum of the rest mass energies of the daughter nuclei and the emitted neutrons.

Let's apply this powerful principle to the options provided in the question. We are looking for a valid fission reaction where the energy of the parent nucleus is greater than the products.

Let's examine the reaction in option (a):

First, we must verify if this reaction is even possible by checking the conservation laws: 1. Conservation of Mass Number (Nucleons): The total mass number on the left is $236$. On the right, it is $137+97+2(1)=236$. The mass number is perfectly conserved. 2. Conservation of Atomic Number (Charge): The total atomic number on the left is $92$. On the right, it is $53+39=92$. The atomic number is also perfectly conserved.

Since this is a valid spontaneous fission reaction, it must release energy. Consequently, the rest mass energy of the Uranium-236 nucleus must be strictly greater than the combined rest mass energies of the Iodine-137 nucleus, the Yttrium-97 nucleus, and the two neutrons.

Mathematically, this is expressed as:

If we look at options (c) and (d), they also represent valid nuclear fission reactions in terms of nucleon and charge conservation. However, they use the "less than" ($<$) sign. This would imply that the rest mass of the reactants is less than the products, meaning energy would need to be absorbed for the reaction to happen. Since spontaneous fission releases energy, these options are physically incorrect.

Therefore, option (a) is the only correct choice. It presents a valid fission reaction and correctly captures the essence of mass-energy conservation.