Question

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

t = T log (1.3)

$straight t space equals space fraction numerator straight T over denominator log space left parenthesis 1.3 right parenthesis end fraction$
$straight t space equals space fraction numerator straight T space log space 2 over denominator 2 space log space 1.3 end fraction$
$straight t space equals space straight T space fraction numerator log space 1.3 over denominator log space 2 end fraction$

Solution

straight t space equals space straight T space fraction numerator log space 1.3 over denominator log space 2 end fraction

At time t
$straight N subscript straight A space equals space fraction numerator straight N subscript 0 over denominator 1.3 end fraction$

also let initially there are total N₀ number of nuclei
N_A + N_B = N₀
$straight N subscript straight A space equals space fraction numerator straight N subscript 0 over denominator 1.3 end fraction Also space as space we space know straight N subscript straight A space space equals space straight N subscript 0 straight e to the power of negative λt end exponent fraction numerator straight N subscript 0 over denominator 1.3 end fraction space equals space straight N subscript 0 straight e to the power of negative λt end exponent fraction numerator 1 over denominator 1.3 end fraction space equals space straight e to the power of negative λt end exponent rightwards double arrow space calligraphic l space n left parenthesis 1.3 right parenthesis space equals lambda t t space equals space fraction numerator </div> <div style=$

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