A radioactive nucleus has a half-life of 20 years, starting with 100,000 particles, how many particles will be left exactly at the end of 40 years

Analyze the question: We have a radioactive sample with a half-life of 20 years and an initial count of 100,000 particles. We need the number of undecayed particles after 40 years.

Concept: The half-life is the time it takes for half of the radioactive nuclei in a sample to decay. The formula is:

N = N₀ × (1/2)^(t/T)

where

• N = amount remaining
• N₀ = initial amount
• t = elapsed time
• T = half-life

Solution:

1. Find the number of half-lives: t/T = 40 ÷ 20 = 2 half-lives

2. Apply the formula: N = 100,000 × (1/2)² N = 100,000 × 1/4 N = 25,000

Step-by-step check:

• After 20 years (1st half-life): 100,000 → 50,000
• After 40 years (2nd half-life): 50,000 → 25,000

Answer: 25,000 particles remain, which is option C.

Tip: For half-life questions, first work out how many half-lives have passed (n = time ÷ half-life). Then simply halve the amount n times. Remember that 2 half-lives leave 1/4, 3 half-lives leave 1/8, and so on. Option D (50,000) is the trap answer for students who stop after only one half-life.