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Nuclei of a radioactive element A are being produced at a cons

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Solving time: 5 mins

Nuclei of a radioactive element $A$ are being produced at a constant rate $α$ . The element has a decay constant $λ$ . At time $t = 0$ , there are $N_{0}$ nuclei of the element. (a) Calculate the number $N$ of nuclei of $A$ at time $t$ . (b) If $α = 2 N_{0} λ$ , calculate the number of nuclei of $A$ after one half-life of $A$ , and also the limiting value of $N$ as $t \to \infty$ .

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Optics And Modern Physics (DC Pandey)

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Nuclei of a radioactive element

A

are being produced at a constant rate

α

. The element has a decay constant

λ

. At time

t = 0

, there are

N_{0}

nuclei of the element. (a) Calculate the number

N

of nuclei of

A

at time

t

. (b) If

α = 2 N_{0} λ

, calculate the number of nuclei of

A

after one half-life of

A

, and also the limiting value of

N

t \to \infty

10 mins ago

Discuss this question LIVE

10 mins ago

Text solutionVerified

(a) Let at time

t

, number of radioactive nuclei are

N

.
Net rate of formation of nuclei of

A

\frac{d N}{d t} = α - λ N or \frac{d N}{α - λ N} = d t

\int_{N_{0}}^{N} \frac{d N}{α - N} = \int_{0}^{t} d t

Solving this equation, we get

N = \frac{1}{λ} [α - (α - λ N_{0}) e^{- λ t}]

(b) (i) Substituting

α = 2 λ N_{0}

and

t = t_{1/2} = \frac{l n 2}{λ}

in Eq. (i), we get

N = \frac{3}{2} N_{0}

(ii) Substituting

α = 2 λ N_{0}

and

t \to \infty

in Eq. (i), we get

N = \frac{α}{λ} = 2 N_{0} N = 2 N_{0}

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Question Text	Nuclei of a radioactive element $A$ are being produced at a constant rate $α$ . The element has a decay constant $λ$ . At time $t = 0$ , there are $N_{0}$ nuclei of the element. (a) Calculate the number $N$ of nuclei of $A$ at time $t$ . (b) If $α = 2 N_{0} λ$ , calculate the number of nuclei of $A$ after one half-life of $A$ , and also the limiting value of $N$ as $t \to \infty$ .
Topic	Nuclei
Subject	Physics
Class	Class 12
Answer Type	Text solution:1
Upvotes	126

Text solutionVerified

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