Question

A radioactive nucleus emits \( \alpha \) particles and \( \beta \) particles in succession. The ratio of number of neutrons to that of protons is \( A = \) mass number, \( Z = \) atomic number

• A. \( \frac{A - Z - 13}{Z - 2} \)
• B. \( \frac{A - Z - 13}{Z - 1} \)
• C. \( \frac{A - Z - 15}{Z - 1} \)
• D. \( \frac{A - Z - 11}{Z - 2} \)

Hint:

For radioactive decay processes, track the changes in mass and atomic numbers to find the resulting number of neutrons and protons.

Answer 1

The Correct Option is C

Step 1: Understanding the process.
When a radioactive nucleus emits an \( \alpha \) particle, its mass number decreases by 4 and its atomic number decreases by 2. When it emits a \( \beta \) particle, the mass number remains the same, but the atomic number increases by 1.
Step 2: Calculating neutrons and protons.
Before the emissions, the number of neutrons is \( N = A - Z \). After emitting an \( \alpha \) particle, the new mass number is \( A - 4 \) and the new atomic number is \( Z - 2 \). After emitting a \( \beta \) particle, the atomic number becomes \( Z - 1 \), and the number of neutrons becomes \( N - 2 \). The ratio of neutrons to protons is: \[ \frac{N - 2}{Z - 1} = \frac{A - Z - 15}{Z - 1} \]
Step 3: Conclusion.
The ratio is \( \frac{A - Z - 15}{Z - 1} \), so the correct answer is (C).