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Question:

At a given instant, say t = 0, two radioactive substances A and B have equal activities. The ratio RB/RA of their activities after time t itself decays with time t as e-λt. If the half-life of A is ln2, the half-life of B is:

2ln2

ln2/4

ln2/2

4ln2

Solution:

Half life of A = ln2
t1/2 = ln2/λ
λA = 1
at t = 0 RA = RB
NAe-λAt = NBe-λBt
NA = NB at t = 0
at t = t
RB/RA = N0e-λBt/N0e-λAt = e-λBt/e-λAt = e- (λB - λA)t = e-λt
λB - λA = λ
λ = 3
λB = 3 + λA = 4
t1/2 = ln2/λB = ln2/4