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At a given instant, say t = 0, two radioactive substances A and B have equal activities. The ratio R_B/R_A of their activities after time t itself decays with time as e⁻³t. If the half-life of A is ln 2, the half-life of B is -

1. ln 2 / 4
2. 4 ln 2
3. 2 ln 2
4. ln 2 / 2

Correct answer: ln 2 / 4

Solution

The ratio of activities decaying as e⁻³t indicates that the decay constant of substance B is three times that of substance A. Since the half-life is inversely related to the decay constant, if A has a half-life of ln 2, B's half-life must be ln 2 divided by 3, which simplifies to ln 2 / 4.

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