JEE Main · 2017 · 02 Apr 2017 · Offline · Modern Physics

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) $t = T lo g (1.3)$
(B) $t = \frac{T}{l o g ( 1.3 )}$
(C) $t = \frac{T}{2} \frac{l o g 2}{l o g 1.3}$
(D) $t = T \frac{l o g 1.3}{l o g 2}$ ✓ Correct

Answer: (D)

Solution

The parent decays as

N_{A} = N_{0} e^{- λ t}

while the daughter builds up as

N_{B} = N_{0} (1 - e^{- λ t})

. Their ratio being

0.3

gives

\frac{1 - e ^{- λ t}}{e ^{- λ t}} = 0.3

, so

1 = 1.3 e^{- λ t}

and

e^{- λ t} = \frac{1}{1.3}

. Taking logarithms,

λ t = lo g (1.3)

, and since

λ = \frac{lo g 2}{T}

this rearranges to

t = T \frac{lo g 1.3}{lo g 2}

JEE Main · 2017 · 02 Apr 2017 · Offline · Single correct option

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