At time t = 0, a material is composed of two radioactive atoms A and, B, where N_A(0) = 2N_B(0). The decay constant of both kind of radioactive atoms is $λ .$ However, A disintegrates to B and B disintegrates to C. Which of the following figures represents the evolution of N_B(t)/N_B(0) with respect to time t?

$[\begin{matrix} N_{A} (0) = N o . o f A a t o m s a t t = 0 & N_{B} (0) = N o . o f B a t o m s a t t = 0 \end{matrix}]$

Option 1 -

(a)

Option 2 -

(b)

Option 3 -

(c)

Option 4 -

(d)

3 Views | Posted 3 months ago

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1 Answer

Answered by

alok kumar singh | Contributor-Level 10

3 months ago

Correct Option - 1

Detailed Solution:

$\frac{d N_{B}}{d t} = λ N_{A} - λ N_{B} = 2 λ N_{B} (0) . e^{- λ t} - λ N_{B}$

$\Rightarrow \frac{d N_{B}}{d t} + λ N_{B} = 2 λ N_{B} (0) e^{- λ t}$

Solving it, $N_{B} (t) = N_{B} (0) (1 + 2 λ t) e^{- λ t}$

N_B (t) is maximum when,

$\frac{d N_{B}}{d t} = 0 \Rightarrow t = \frac{1}{2 λ}$

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