Determine the original mass of a radioactive substance given its half-life and remaining mass, list two medical uses of radioactivity, and solve nuclear equations.

Given $M_t = 15 \text{ kg}$ and $n = 5$:

The original mass of the radioactive substance is $\boxed{480 \text{ kg}}$.

11. (b)
Two uses of radioactivity in medicine are:

• Sterilization of medical equipment (e.g., using gamma radiation).
• Diagnosis of diseases (e.g., using radioisotopes as tracers for imaging organs).

12.
The nuclear equation is:

The values are $\boxed{X = 90}$ and $\boxed{Y = 231}$.

13. (a) (i)
To draw a graph of disintegration rate against time, plot the following points with Time (hours) on the x-axis and Rate of disintegration (counts s⁻¹) on the y-axis:

• (0, 730)
• (5, 570)
• (10, 455)
• (15, 365)
• (20, 292)
• (25, 232) Draw a smooth curve connecting these points.

13. (a) (ii)

Initial rate at time = 0 hours is 730 counts s⁻¹.

Half of 730 counts s⁻¹ is $730 / 2 = 365$ counts s⁻¹.

From the table, at 15 hours, the rate of disintegration is 365 counts s⁻¹.
The half-life of bismuth is $\boxed{15 \text{ hours}}$.

13. (a) (iii)
If half the amount of sample of bismuth were used, the initial disintegration rate would be halved. The curve would start at 365 counts s⁻¹ (instead of 730 counts s⁻¹) and follow the same decay pattern, but all disintegration rate values would be half of the original values. The half-life would remain the same.

13. (b)
One application of radioactivity in:

• (i) Medicine: Radiotherapy for cancer treatment.
• (ii) Agriculture: Pest control through sterilization of male insects.
• (iii) Tracers: Detecting leaks in underground pipes.
• (iv) Nuclear power station: Generating electricity through nuclear fission.

13. (c)
Two dangers associated with radioactivity are:

• Genetic mutations and birth defects in offspring due to damage to reproductive cells.
• Radiation sickness from high doses, leading to nausea, vomiting, hair loss, and potentially death.

14. (a)
The two charged particles emitted by a radioactive isotope are:

• Alpha particle: $\boxed{^{4}_{2}\alpha}$ or $\boxed{^{4}_{2}He}$ (positive charge)
• Beta particle: $\boxed{^{0}_{-1}\beta}$ or $\boxed{^{0}_{-1}e}$ (negative charge)

14. (b)
The nuclear equation is:

The mass number of the resulting nuclide is $\boxed{210}$ and the atomic number is $\boxed{84}$.

15.
The diagram in Figure 6 is not provided in the image, so this question cannot be answered.