The half-life of a radioactive substance is approximately 68.7 days. A sample has an initial count rate of 106 counts per minute. When will the count rate drop to 53 counts per minute? Explain the sig

From the table, at Time = 0 days, the initial count rate ($N_0$) is $106 \text{ min}^{-1}$.

After one half-life, the count rate will be half of the initial count rate:

We need to find the time when the count rate is $53 \text{ min}^{-1}$.
Looking at the table:

• At 62 days, the count rate is $56 \text{ min}^{-1}$.
• At 82 days, the count rate is $47 \text{ min}^{-1}$. The count rate of $53 \text{ min}^{-1}$ falls between these two points. We can use linear interpolation to estimate the half-life ($T_{1/2}$):

The scientists recommended that the environment would be safe for use again if the half-life value is less than 60 days. Since the calculated half-life is approximately 68.67 days, which is greater than 60 days, the environment is not yet safe for use according to the scientific recommendation. The people should be advised to continue to avoid the area for a longer period than initially estimated, as the radioactive materials are decaying slower than the safety threshold.

The radioactive substance X has a mass number of 208 and an atomic number of 104.

The decay involves emitting 3 alpha ($\alpha$) particles, 4 beta ($\beta$) particles, and gamma ($\gamma$) rays to form nuclide Y.

• An alpha particle is $^{4}_{2}\text{He}$.
• A beta particle is $^{0}_{-1}e$.
• A gamma ray is $^{0}_{0}\gamma$ (it carries no mass or charge). The decay equation is:

The total mass number must be conserved on both sides of the equation.

The total atomic number must be conserved on both sides of the equation.

The number of neutrons is calculated as Mass Number (A) - Atomic Number (Z).