Atoms and Nuclei

The amount of radioactive substance remaining after time $t$ is given by:

$$N = N_0 \left(\frac{1}{2}\right)^n$$

where $n$ is the number of half-lives elapsed:

$$n = \frac{t}{t_{1/2}} = \frac{12\,\text{years}}{4\,\text{years}} = 3$$

Calculating the remaining mass:

$$N = N_0 \left(\frac{1}{2}\right)^3 = \frac{N_0}{8}$$

Matching components of a nuclear reactor with their functions:

• Uranium ($^{235}\text{U}$): Used as nuclear fission fuel (Matches III).
• Moderator (heavy water, graphite): Slows down fast fission neutrons to thermal energies to sustain controlled fission (Matches II).
• Control rods (cadmium, boron): Absorbs excess neutrons to regulate or halt the nuclear reaction rate (Matches I).
• Coolant (water, liquid sodium): Extracts and transfers thermal energy from reactor core to steam turbines to generate electricity (Matches IV).

Correct matching is (A)-(III), (B)-(II), (C)-(I), (D)-(IV). Hence, option 3 is the correct answer.