(a) Which of the following are required characteristics of an isotope to be used as a fuel in a nuclear power reactor? (i) It must emit gamma radiation. (ii) On decay, it must release two or more neutrons. (iii) It must have a half-life less than one hour. (iv) It must undergo fission upon the absorption of a neutron. (b) What is the most common fissionable isotope in a commercial nuclear power reactor?

To determine which characteristics are needed for an isotope to be used as fuel in a nuclear power reactor, we will analyze each listed characteristic one by one. (i) It must emit gamma radiation: Gamma radiation is not a mandatory requirement for fuel isotopes in a nuclear reactor. Gamma radiation is emitted during the decay of some radioactive isotopes, but it is not a factor in determining whether an isotope can be used as fuel in a reactor. (ii) On decay, it must release two or more neutrons: This is an important characteristic for a nuclear fuel isotope. When an isotope undergoes fission, the release of two or more neutrons ensures that a chain reaction can be sustained. One of these neutrons will cause the fission of another nucleus, while the rest will be absorbed by other materials or escape the reactor. (iii) It must have a half-life less than one hour: This characteristic is not necessary for a nuclear fuel isotope. Having a short half-life means that the isotope will decay rapidly, which may not be ideal for maintaining a sustained chain reaction in a nuclear reactor. In fact, most nuclear fuels have half-lives on the order of years or even millennia, allowing them to provide energy for long periods. (iv) It must undergo fission upon the absorption of a neutron: This is a crucial characteristic for nuclear fuel isotopes. To sustain a chain reaction in a nuclear reactor, the fuel isotope must undergo fission upon the absorption of a neutron. This process releases energy which is then used to heat water, produce steam, and ultimately generate electricity.

Now let's move on to the second part of the problem, which requires us to identify the most common fissionable isotope used in commercial nuclear power reactors. The most common fissionable isotope used in commercial nuclear power reactors is Uranium-235 (\(^{235}\textrm{U}\)). This isotope has an abundant supply on Earth, and its fission releases a considerable amount of energy, making it an ideal fuel for nuclear reactors. Other isotopes, such as Plutonium-239 (\(^{239}\textrm{Pu}\)) and Thorium-232 (\(^{232}\textrm{Th}\)), are also used in some reactors but are less common than Uranium-235. So, to summarize: Characteristics (ii) and (iv) are required for an isotope to be used as a fuel in a nuclear power reactor. The most common fissionable isotope in a commercial nuclear power reactor is Uranium-235 (\(^{235}\textrm{U}\)).