JEE Main 2021 · 1 September · evening · Single correct · +4 / −1
The half life period of radioactive element x is same as the mean life time of another radioactive element y. Initially they have the same number of atoms. Then :
- (a) x-will decay faster than y.
- (b) y-will decay faster than x.
- (c) x and y have same decay rate initially and later on different decay rate.
- (d) x and y decay at the same rate always.
Answer and solution
Answer (b) y-will decay faster than x.
Solution
Physics sir · 4 steps
Write both lifetimes in terms of λ and compare: the larger decay constant decays faster.
- Two time scales. Half-life is T_(1/2) = (ln 2)/λ = 0.693/λ. Mean life is τ = 1/λ. For the same element, the half-life is always shorter than the mean life.
- Set them equal. The half-life of x equals the mean life of y.
0.693/λₓ = 1/λ_y ⇒ λ_y = λₓ/0.693 ≈ 1.44 λₓ
- Compare rates. With the same starting number N₀, the initial activity is λ N₀. So y starts at about 1.44 times the activity of x, and its fraction remaining, e^(-λ t), falls faster at every moment.
- So. y has the larger decay constant, so y decays faster than x. That is option (b).
The idea to keep. Turn half-life and mean life into λ before comparing: T_(1/2) = 0.693/λ and τ = 1/λ. Equal times mean unequal λ, and the bigger λ decays faster.
Answer from the NTA final answer key. Solution by PrepWiser.