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Consider a radioactive disintegration according to the equatio

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Solving time: 3 mins

Consider a radioactive disintegration according to the equation $A \to B \to C$ . Decay constant of $A$ and $B$ is same and equal to $λ$ . Number of nuclei of $A, B$ and $C$ are $N_{0}, 0, 0$ respectively at $t = 0$ . Find
(a) number of nuclei of $B$ as function of time $t$ .
(b) time $t$ at which the activity of $B$ is maximum and the value of maximum activity of $B$ .

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Optics And Modern Physics (DC Pandey)

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Consider a radioactive disintegration according to the equation

A \to B \to C

. Decay constant of

A

and

B

is same and equal to

λ

. Number of nuclei of

A, B

and

C

are

N_{0}, 0, 0

respectively at

t = 0

. Find
(a) number of nuclei of

B

as function of time

t

.
(b) time

t

at which the activity of

B

is maximum and the value of maximum activity of

B

12 mins ago

Discuss this question LIVE

12 mins ago

Text solutionVerified

(a)

At t = 0 At t Here, or A N_{0} N_{1} N_{1} = N_{0} e^{- λ t} \frac{d N _{2}}{d t} = λ N_{0} e^{- λ t} - λ N_{2} B 0 N_{2} \frac{d N _{2}}{d t} = λ (N_{1} - N_{2}) C 0 N_{3}

d N_{2} + λ N_{2} d t = λ N_{0} e^{- λ t}

∴ e^{λ t} d N_{2} + λ N_{2} e^{λ t} d t = λ N_{0} d t

d (N_{2} e^{λ t}) = λ N_{0} d t

∴ N_{2} e^{λ t} = λ N_{0} t + C

t = 0, N_{2} = 0

∴ C ∴ = 0 N_{2} = λ N_{0} (t e^{- λ t})

(b) Activity of

B

R_{2} = λ N_{2} = λ^{2} N_{0} (t e^{- λ t})

For maximum activity,

\frac{d R _{2}}{d t} = 0

∴ ∴ t = \frac{1}{λ} R_{m a x} = \frac{λ N _{0}}{e}

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Question Text	Consider a radioactive disintegration according to the equation $A \to B \to C$ . Decay constant of $A$ and $B$ is same and equal to $λ$ . Number of nuclei of $A, B$ and $C$ are $N_{0}, 0, 0$ respectively at $t = 0$ . Find (a) number of nuclei of $B$ as function of time $t$ . (b) time $t$ at which the activity of $B$ is maximum and the value of maximum activity of $B$ .
Topic	Nuclei
Subject	Physics
Class	Class 12
Answer Type	Text solution:1
Upvotes	76

Text solutionVerified

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