Nuclei practice · Mixed difficulty

Option C ($_{92}{U^{235}}$) is correct because Uranium-235 is a fissionable isotope, meaning it can sustain a nuclear chain reaction.

Option D ($4 \times 10^{17} kg/m^3$) is correct because nuclear density is approximately constant and independent of the mass number, and it can be calculated using the mass of a nucleon and the volume of the nucleus.

Option B (${F_1} = {F_2} = {F_3}$) is correct because the nuclear force is charge-independent and primarily depends on the separation between the nucleons.

Option C ($5\ln 2$) is correct because it is the time at which the activity reduces to half its initial value, derived from the radioactive decay law.

Option B ($2.0 \times 10^{-24} kg - m/s$) is correct because linear momentum is the product of mass and velocity.

Option A (4) is correct because it calculates the ratio of the remaining masses of substances A and B after 20 seconds, considering their different atomic weights and half-lives.

Option B (138 neutrons and 88 protons) is correct because the number of protons is the atomic number (88), and the number of neutrons is the mass number (226) minus the atomic number (88), which equals 138.

Option A (2 protons and 2 neutrons only) is correct because an alpha particle is the nucleus of a helium atom, consisting of 2 protons and 2 neutrons.

Option D ($\mathrm{40}$ microgram) is correct because it calculates the mass decay of ${U^{235}}$ per hour based on the power generated in the reactor.

Option C ($k = 1$) is correct because a critical condition in a nuclear reactor requires a self-sustaining chain reaction, which occurs when the neutron multiplication factor is equal to 1.

Option D (12 neutrons) is correct because the number of neutrons is the difference between the mass number and the atomic number.

Option D ($5.6 \times 10^{26} \mathrm{MeV}$) is correct because it represents the energy equivalent of 1 gram of substance, calculated using $E=mc^2$ and converting to MeV.

Option C (Production of more neutrons in fission) is correct because the continuous chain reaction in nuclear fission is sustained by the release of multiple neutrons during each fission event, which then trigger further fission reactions.

Option C ($1 \mathrm{~sec}$) is correct because the number of beta particles emitted is proportional to the number of radioactive nuclei remaining, and the ratio of emissions in consecutive intervals gives the decay factor.

Option A (C - 14) is correct because Carbon-14 dating is used for dating organic materials up to around 50,000 years, making it suitable for dating beyond 56,000 years, while other options are used for much longer timescales.

Option B is correct because an atom bomb releases energy through nuclear fission, where heavy nuclei split into lighter ones, releasing a large amount of energy.

Option B (10.2 MeV) is correct because the Q-value is calculated by finding the difference in mass between the reactants and products and then converting it to energy using $E=mc^2$.

Option A (7) is correct because the kinetic energy of the combined nucleus C is equal to the total kinetic energy of the reactants minus the excitation energy of the product.

Option A ($0.012\,\mathop \mathrm{A}\limits^ \circ \,$) is correct because the wavelength of the gamma rays produced is calculated using the energy equivalent of the electron and positron mass.

Option D ($\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }$ and $\mathrm{\gamma },\mathrm{\beta },\mathrm{\alpha }$ respectively) is correct because alpha particles have the highest ionization power and lowest penetration range, while gamma rays have the lowest ionization power and highest penetration range.

Option D is correct because radioactive decay is a random process, and the decay of one nucleus does not influence the decay of another. Therefore, the next nucleus can decay at any time.

Option A (a current will flow from $A$ to $B$) is correct because alpha particles are positively charged and beta particles are negatively charged. Irradiating end A with alpha particles deposits positive charge, while irradiating end B with beta particles deposits negative charge, creating a potential difference and causing current to flow from A to B.

Option C (Tritium) is correct because Tritium, an isotope of hydrogen, is radioactive due to its unstable nucleus.

Option C ($1/2\lambda$) is correct because it satisfies the given ratio of active nuclei after the specified time interval.

Option C is correct because Statement-1 is true (electrons are emitted from the nucleus during beta decay, but do not reside there), and Statement-2 is false (electrons do not experience the strong nuclear force).

Option C (Cadmium) is correct because cadmium is a strong neutron absorber and is commonly used in control rods to regulate the chain reaction in nuclear reactors.

Option B ($\mathrm{\gamma },\,\mathrm{\beta },\,\mathrm{\alpha }$) is correct because gamma rays have the highest penetrating power, followed by beta particles, and then alpha particles.

Option C ($\mathrm{20}\,\mathrm{ min}$) is correct because after two half-lives, the count rate reduces to 1/4 of its initial value.

Option B (y will decay faster than x) is correct because the mean lifetime is greater than the half-life, implying that element y has a higher decay constant and thus decays faster initially.

Option A (Short ranged attractive and charge independent) is correct because nuclear forces are strong, attractive forces that act over short distances and are independent of the charge of the nucleons.