Question

A radioactive element \({}^{242}_{92}X\) emits two \(\alpha\)-particles, one electron, and two positrons. The product nucleus is represented by \({}^{234}_{P}Y.\) The value of \(P\) is _______.

Hint:

For radioactive decay:

• \(\alpha\)-particle emission decreases atomic number by 2 and mass number by 4.
• \(\beta^-\) emission increases atomic number by 1.
• \(\beta^+\) emission decreases atomic number by 1.

Answer 1

Correct Answer: 87

1. Effect of Emission:
   • Emission of an \(\alpha\)-particle reduces the mass number by 4 and the atomic number by 2.
   • Emission of an electron (\(\beta^-\)) increases the atomic number by 1.
   • Emission of a positron (\(\beta^+\)) decreases the atomic number by 1.
2. Calculate Atomic Number (\(P\)):
   • After two \(\alpha\)-particles:

\[ \text{Atomic number} = 92 - 2 \times 2 = 88, \quad \text{Mass number} = 242 - 2 \times 4 = 234. \]

• After one electron (\(\beta^-\)):

\[ \text{Atomic number} = 88 + 1 = 89. \]

• After two positrons (\(2\beta^+\)):

\[ \text{Atomic number} = 89 - 2 = 87. \]

Final Answer: 87