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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=Tlog1.3log2Since the radioactive nucleus has half life T, ⇒λ=ln2T A→λB At t=0, Number of A=N0 and Number of B=0 Therefore at time =t, Number of A=N0e−λt Number of B=N0−N0e−λt=N0(1−e−λt) ⇒Number of BNumber of A=N0(1−e−λt)N0e−λt=0.3 ⇒eλt−1=0.3 ⇒eλt=1.3⇒λt=ln(1.3) ⇒t=ln1.3λ=T ln1.3ln2 =T log1.3log2

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