Step 1: Understand the decay process.
Each -particle emission reduces:
Mass number by 4
Atomic number by 2
Each emission increases:
Atomic number by 1
Mass number remains unchanged
Step 2: Apply mass number conservation.
Initial mass number = 238.
After 7 emissions:
Final nucleus has mass number 210, which matches .
Thus mass balance is satisfied.
Step 3: Apply atomic number conservation.
Initial atomic number = 92.
After 7 emissions:
Let number of emissions be .
Each increases atomic number by 1.
Final atomic number becomes:
Given final atomic number of Pb = 82.
Final Answer: