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Question

A radioactive nucleus A with a half life T, decays into a nucleus B. At t=0, there is no nucleus B. At sometime t, the ratio of the number of B to that of A is 0.3. Then, t is given by

A
t=Tlog(1.3)
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B
t=T2log2log1.3
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C
t=Tlog1.3log2
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D
t=Tlog(1.3)
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Solution

The correct option is C t=Tlog1.3log2
Since the radioactive nucleus has half life T,
λ=ln2T
AλB

At t=0, Number of A=N0 and Number of B=0
Therefore at time =t,

Number of A=N0eλt

Number of B=N0N0eλt=N0(1eλt)

Number of BNumber of A=N0(1eλt)N0eλt=0.3

eλt1=0.3

eλt=1.3λt=ln(1.3)

t=ln1.3λ=T ln1.3ln2
=T log1.3log2

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