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X and Y are two radioactive nuclides - AQA - A-Level Physics - Question 28 - 2021 - Paper 2
Question 28
X and Y are two radioactive nuclides. X has a half-life of 3.0 minutes and Y has a half-life of 9.0 minutes. Two freshly prepared samples of X and Y start decaying ... show full transcript
Worked Solution & Example Answer:X and Y are two radioactive nuclides - AQA - A-Level Physics - Question 28 - 2021 - Paper 2
Step 1
Calculate the number of half-lives for X and Y in 18 minutes
Answer
For nuclide X with a half-life of 3.0 minutes:
Number of half-lives = ( \frac{18\text{ min}}{3.0\text{ min}} = 6 ) half-lives.
For nuclide Y with a half-life of 9.0 minutes:
Number of half-lives = ( \frac{18\text{ min}}{9.0\text{ min}} = 2 ) half-lives.
Step 2
Determine the remaining nuclei after decay for X and Y
Answer
The remaining nuclei after decay for nuclide X can be calculated using the formula:
( N_X = N_{X0} \left( \frac{1}{2} \right)^{n_X} )
Where ( N_{X0} ) is the initial number of nuclei of X, and ( n_X ) is the number of half-lives.
Thus, after 6 half-lives:
( N_X = N_{X0} \left( \frac{1}{2} \right)^{6} = N_{X0} \cdot \frac{1}{64} )
For nuclide Y, the remaining nuclei:
( N_Y = N \left( \frac{1}{2} \right)^{n_Y} = N \left( \frac{1}{2} \right)^{2} = N \cdot \frac{1}{4} )
Step 3
Set N_X equal to N_Y and solve for N_{X0}
Answer
Setting the remaining nuclei equal to each other:
( N_{X0} \cdot \frac{1}{64} = N \cdot \frac{1}{4} )
Multiplying both sides by 64 and simplifying:
( N_{X0} = N \cdot \frac{64}{4} = 16N )
Thus, the initial number of radioactive nuclei in the sample of X is ( 16N ).