Question

If a thorium nucleus of mass number 232 and atomic number 90 emits six $\alpha$-particles and four $\beta$-particles, then the ratio of number of protons to neutrons in the final nucleus is:

• A. 39:65
• B. 41:63
• C. 41:104
• D. 63:104

Hint:

Alpha decay reduces both A and Z; beta increases Z only.

Answer 1

The Correct Option is B

Step 1: Initial nucleus: \[ A = 232,\quad Z = 90 \]
Step 2: Effect of 1 alpha decay: \[ A \to A - 4,\quad Z \to Z - 2 \]
Step 3: For 6 alpha decays: \[ A = 232 - 24 = 208,\quad Z = 90 - 12 = 78 \]
Step 4: Effect of beta-minus decay: \[ Z \to Z + 1,\quad A \text{ unchanged} \]
Step 5: For 4 beta decays: \[ Z = 78 + 4 = 82,\quad A = 208 \]
Step 6: Neutrons: \[ N = A - Z = 208 - 82 = 126 \]
Step 7: Ratio: \[ Z:N = 82:126 = 41:63 \]