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Two radio active nuclei X and Y decay in...

Two radio active nuclei X and Y decay into stable nucleus Z
`X to Z + 2alpha+ beta^-`
`Y to Z + alpha + 2 beta^+`
if `Z_1` and `Z_2` are atomic numbers of X and Y then

A

`Z_1=Z_2`

B

`Z_2-Z_1=1`

C

`Z_1-Z_2=1`

D

`2Z_1-Z_2=2`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the decay processes of the radioactive nuclei X and Y and determine the relationship between their atomic numbers \( Z_1 \) and \( Z_2 \). ### Step-by-Step Solution: 1. **Understanding the Decay of Nucleus X:** - The decay of nucleus X is given by the equation: \[ X \rightarrow Z + 2\alpha + \beta^- \] - Each alpha particle (\( \alpha \)) emission decreases the atomic number by 2 and the mass number by 4. - The beta minus (\( \beta^- \)) emission increases the atomic number by 1 but does not change the mass number. - Therefore, for nucleus X: - Decrease in atomic number due to 2 alpha particles: \( 2 \times 2 = 4 \) - Increase in atomic number due to 1 beta particle: \( +1 \) - Overall change in atomic number: \( -4 + 1 = -3 \) - Thus, the atomic number of nucleus Z from X is: \[ Z = Z_1 - 3 \] 2. **Understanding the Decay of Nucleus Y:** - The decay of nucleus Y is given by the equation: \[ Y \rightarrow Z + \alpha + 2\beta^+ \] - Each alpha particle (\( \alpha \)) emission decreases the atomic number by 2. - Each beta plus (\( \beta^+ \)) emission decreases the atomic number by 1. - Therefore, for nucleus Y: - Decrease in atomic number due to 1 alpha particle: \( 2 \) - Decrease in atomic number due to 2 beta particles: \( 2 \times 1 = 2 \) - Overall change in atomic number: \( -2 - 2 = -4 \) - Thus, the atomic number of nucleus Z from Y is: \[ Z = Z_2 - 4 \] 3. **Setting the Equations Equal:** - Since both reactions lead to the same stable nucleus Z, we can equate the two expressions for Z: \[ Z_1 - 3 = Z_2 - 4 \] 4. **Solving for the Relationship Between \( Z_1 \) and \( Z_2 \):** - Rearranging the equation gives: \[ Z_2 - Z_1 = 1 \] ### Final Answer: The relationship between the atomic numbers \( Z_1 \) and \( Z_2 \) is: \[ Z_2 - Z_1 = 1 \]
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