Question

Accumulation of Stable Atoms

A graph with number of half-lives on the x-axis from 0 to 10 and number of radioactive atoms from 0 to 100. The orange graph is a curve starting at the origin and curves up to (10, 100).

The graph shows the accumulation of stable atoms during the decay of a radioactive substance. Why does the graph approach the horizontal line y = 100?

Answer 1

Because the total number of atoms is fixed (100), the number of stable atoms S(t) is just the initial total minus the remaining radioactive ones R(t). For t measured in half‑lives,
R(t) = 100·(1/2)^t, so
S(t) = 100 − 100·(1/2)^t = 100(1 − (1/2)^t).

As t → ∞, (1/2)^t → 0, so S(t) → 100. The curve therefore approaches the horizontal line y = 100 (a horizontal asymptote): eventually almost all atoms have decayed to the stable form, and the stable count cannot exceed the initial total of 100.