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Question

# At a given instant, say t=0, two radioactive substances A and B have equal activities. The ratio RBRA of their activities after time t itself decays with time t as e−3t. If the half-life of A is ln2, the half-life of B is:

A
ln22
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B
4ln2
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C
2ln2
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D
ln24
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Solution

## The correct option is D ln24Given, Half life of A, (t1/2)A=ln2 As we know that, (t1/2)A=ln2λA ∴λA=1 at t=0, RA=RB NAe−λAT=NBe−λBT NA=NB at t=0 At t=t, RBRA=NBe−λB tNAe−λA t ⇒e−3t=e−(λB−λA)t (∵NA=NB) ⇒λB−λA=3 ⇒λB=3+λA=3+1=4 ∴(t1/2)B=ln2λB=ln24 Hence, option (C) is correct.

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