X and Y are two radioactive nuclides - AQA - A-Level Physics - Question 28 - 2021 - Paper 2

To find the initial number of radioactive nuclei in sample X, we start by determining how many half-lives have passed for both isotopes after 18 minutes.

1. Calculate the half-lives passed for isotope X:

   The half-life of X is 3.0 minutes. In 18 minutes, the number of half-lives is:

   $ext{Half-lives for X} = \frac{18 ext{ minutes}}{3.0 ext{ minutes}} = 6$

   After six half-lives, the remaining quantity of X, starting with an initial quantity of, say, $X_0$, can be determined using the formula:

   $N_X = X_0 \left( \frac{1}{2} \right)^6$

2. Calculate the half-lives passed for isotope Y:

   The half-life of Y is 9.0 minutes. In 18 minutes, the number of half-lives is:

   $\text{Half-lives for Y} = \frac{18 ext{ minutes}}{9.0 ext{ minutes}} = 2$

   After two half-lives, the remaining quantity of Y, starting from N, is:

   $N_Y = N \left( \frac{1}{2} \right)^2 = N \times \frac{1}{4}$

3. Set the equations equal to find X's initial quantity:

   Since the number of nuclei in both samples is the same after 18 minutes, we can set the remaining quantities equal to each other:

   $X_0 \left( \frac{1}{2} \right)^6 = N \times \frac{1}{4}$

   Rearranging gives:

   $X_0 = 4N \left( \frac{1}{2} \right)^6 = 4N \times \frac{1}{64} = \frac{N}{16}$

   Thus, simplifying gives:

   $X_0 = 16N$

Therefore, the initial number of radioactive nuclei in the sample of X is 16N.