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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

8 mins ago

Discuss this question LIVE

8 mins ago

Text solutionVerified

(a) Let

R_{A_{0}}

and

R_{B_{0}}

be the initial activities of

A

and B. Then,

R_{A_{0}} + R_{B_{0}} = 1 0^{10} dps

Activity of

A

after time

t = 20

days (two half-lives of

A

) is

R_{A} = (\frac{1}{2})^{2} R_{A_{0}} = 0.25 R_{A_{0}}

Similarly, activity of

B

after

t = 20

days (four half-lives of

B

) is

R_{B} = (\frac{1}{2})^{4} R_{B_{0}} = 0.0625 R_{B_{0}}

Now, it is given that

R_{A} + R_{B} = 20%

1 0^{10}

0.25 R_{A_{0}} + 0.0625 R_{B_{0}} = 0.2 \times 1 0^{10} dps

Solving Eqs. (i) and (ii), we get

R_{A_{0}} = 0.73 \times 1 0^{10} dps

and

R_{B_{0}} = 0.27 \times 1 0^{10} dps

(b)

\frac{R _{A_{0}}}{R _{B_{0}}} = \frac{λ _{A} N _{A_{0}}}{λ _{B} N _{B_{0}}} = \frac{( t _{1/2} ) _{B}}{( t _{1/2} ) _{A}} \cdot \frac{N _{A_{0}}}{N _{B_{0}}}

∴ \frac{N _{A_{0}}}{N _{B_{0}}} = (\frac{R _{A_{0}}}{R _{B_{0}}}) \frac{( t _{1/2} ) _{A}}{( t _{1/2} ) _{B}} = (\frac{0.73}{0.27}) (\frac{10}{5}) = 5.4

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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	54

Text solutionVerified

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