Einstein's equation E = mc² is pivotal in understanding nuclear reactions as it establishes a direct relationship between mass and energy. This equation indicates that mass can be converted into energy and vice versa, which is fundamental in nuclear physics. In nuclear reactions, either fission or fusion, a small amount of mass is converted into a large amount of energy. For instance, in nuclear fission, when a nucleus splits, the mass of the products is slightly less than the mass of the original nucleus. This mass difference, though small, when multiplied by the square of the speed of light (a very large number), results in a significant amount of energy release. Similarly, in nuclear fusion, the mass of the resultant nucleus is less than the sum of the masses of the initial fusing nuclei. This mass loss is again converted into energy. Thus, E = mc² is not just a theoretical formula; it is the cornerstone that explains the tremendous energy produced in nuclear reactions, which is vastly more efficient than chemical reactions.

The stability of a nucleus is closely related to its neutron-to-proton (n/p) ratio. A stable nucleus usually has a balanced n/p ratio that allows for a strong nuclear force to overcome the repulsive electromagnetic force between protons. For lighter elements (up to iron), a 1:1 ratio of neutrons to protons is typically stable. As nuclei get heavier, more neutrons are needed to maintain stability, leading to ratios greater than 1:1. This is because the additional neutrons add to the nuclear strong force without increasing the repulsive electromagnetic force (as neutrons are electrically neutral). However, if there are too many neutrons, the nucleus becomes unstable, leading to beta decay, where a neutron is transformed into a proton. Conversely, a deficiency of neutrons in heavy nuclei can lead to instability and alpha decay. Therefore, the n/p ratio is a critical factor in determining the nuclear stability of different isotopes.

The concept of half-life is central in nuclear physics as it measures the stability and rate of decay of radioactive substances. The half-life of a radioactive isotope is the time required for half of its atoms to decay into other forms. This concept is crucial in understanding the behaviour of radioactive materials, whether in natural decay processes, nuclear medicine, or radioactive dating techniques. The half-life is determined experimentally by measuring the rate of decay of the isotope. This is typically done by detecting the radiation emitted as the radioactive nuclei decay and plotting these data over time. The time at which the activity, or radiation emission rate, falls to half its initial value is the half-life. Different isotopes have vastly different half-lives, ranging from fractions of a second to millions of years. The half-life is a constant property for a given isotope and is independent of the amount of substance or its external conditions.

In nuclear reactors, the energy released from fission is primarily used to generate electricity. The process begins with a nuclear fission reaction in the reactor core, where uranium or plutonium nuclei split into smaller nuclei, releasing a considerable amount of energy. This energy is initially in the form of kinetic energy of the fission products and neutrons. The core of the reactor contains a coolant, often water, which absorbs this kinetic energy, heating up in the process. This heated coolant is then used to produce steam in a secondary loop, which is isolated from the radioactive materials. The steam drives turbines connected to generators, thereby converting thermal energy into mechanical energy and finally into electrical energy through electromagnetic induction. The steam is then condensed back into water and recycled through the system. This process of harnessing the energy from nuclear fission to produce electricity is efficient and produces large amounts of energy from a relatively small amount of fuel.

The key differences between a nuclear reactor and a nuclear bomb lie in their design and the nuclear processes they utilize. A nuclear reactor is designed for a controlled, sustained nuclear chain reaction, whereas a nuclear bomb is designed for an uncontrolled, explosive chain reaction. In a reactor, the fission process is controlled by using control rods that absorb neutrons and by moderating the speed of neutrons to sustain a steady rate of reaction. This controlled environment allows for the gradual release of energy, which is used to generate electricity. In contrast, a nuclear bomb is designed to achieve a supercritical mass instantaneously, with no control mechanisms to slow down the reaction. This results in an extremely rapid, uncontrolled chain reaction, releasing a massive amount of energy in a very short time, leading to an explosion. The design of a reactor includes multiple safety systems to contain radiation and prevent accidents, whereas a bomb is designed for maximum destruction. Additionally, the materials used in a bomb are often different, requiring a much higher purity of fissionable material than a reactor.