[Solved] Q.229. A radionuclide A1 with decay constant λ1 transforms into a radionuclide A2 with decay constant λ2. Assuming that at the initial moment the preparation contained only the radionuclide A1, find: (a) the equation describing accumulation of the radionuclide A2 With time; (b) the time interval after which the activity of radionuclide A2 reaches the maximum value. - Problems in General Physics

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Problems in General Physics

PART SIX. ATOMIC AND NUCLEAR PHYSICS

Q.229. A radionuclide A1 with decay constant λ1 transforms int

Problems in General Physics

Class 11

IE Irodov

Chapter 1: PART ONE. PHYSICAL FUNDAMENTALS OF MECHANICS

277 questions

Chapter 2: PART TWO. THERMODYNAMICS AND MOLECULAR PHYSICS

9 questions

Chapter 3: PART THREE. ELECTRODYNAMICS

46 questions

Chapter 4: PART FOUR. OSCILLATIONS AND WAVES

155 questions

Chapter 5: PART FIVE. OPTICS

215 questions

Chapter 6: PART SIX. ATOMIC AND NUCLEAR PHYSICS

271 questions

Q.228. A

P^{32}

radionuclide with half-life T = 14.3 days is produced in a reactor at a constant rate

q = 2.7 \times 1 0^{9}

nuclei per second. How soon after the beginning of production of that radionuclide will its activity be equal to

A = 1.0 \times 1 0^{9}

dis/s?

Q.229. A radionuclide A₁ with decay constant λ₁ transforms into a radionuclide A₂ with decay constant λ₂. Assuming that at the initial moment the preparation contained only the radionuclide A₁, find:
(a) the equation describing accumulation of the radionuclide A₂ With time;
(b) the time interval after which the activity of radionuclide A₂ reaches the maximum value.

Q.231. A radionuclide A₁ goes through the transformation chain A₁→ A₂ → A₃ (stable) with respective decay constants λ₁ and λ₂. Assuming that at the initial moment the preparation contained only the radionuclide A₁ equal in quantity to N₁₀ nuclei, find the equation describing accumulation of the stable isotope A₃.

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Hard

Solving time: 5 mins

Q.229. A radionuclide A₁ with decay constant λ₁ transforms into a radionuclide A₂ with decay constant λ₂. Assuming that at the initial moment the preparation contained only the radionuclide A₁, find:
(a) the equation describing accumulation of the radionuclide A₂ With time;
(b) the time interval after which the activity of radionuclide A₂ reaches the maximum value.

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Text solutionVerified

(a) Suppose

N_{1}

and

N_{2}

are the number of two radionuclides

A_{1}, A_{2}

at time t. Then

\frac{d N _{1}}{d t} = - λ_{1} N_{1} \dots (1)

\frac{d N _{2}}{d t} = λ_{1} N_{1} - λ_{2} N_{2} \dots (2)

From (1)

N_{1} = N_{10} e^{- λ_{1} t}

where

N_{10}

is the initial number of nuclides

A_{l}

at time

t = 0

From (2)
\qquad\left(\frac{d N_{2}}{d t}+\lambda_{2} N_{2}\right) e^{\lambda_{2} t}=\lambda_{1} N_{10} e^{-\left(\lambda_{1}-\lambda_{2}\right) t
or

(N_{2} e^{λ_{2} t}) = const \frac{λ _{1} N _{10}}{λ _{1} - λ _{2}} e^{- (λ_{1} - λ_{2}) t}

since

N_{2} = 0

t = 0

Constant

N_{2} = \frac{λ _{1} N _{10}}{λ _{1} - λ _{2}}

Thus

= \frac{λ _{1} N _{10}}{λ _{1} - λ _{2}} (e^{- λ_{2} t} - e^{- λ_{1} t})

(b) The activity of nuclide

A_{2}

λ_{2} N_{2}

. This is maximum when

N_{2}

is maximum. That happens when

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

This requires

λ_{2} e^{- λ_{2} t_{m}} = λ_{1} e^{- λ_{1} t_{m}}

t_{m} = \frac{l n ( λ _{1} / λ _{2} )}{λ _{1} - λ _{2}}

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Question Text	Q.229. A radionuclide A₁ with decay constant λ₁ transforms into a radionuclide A₂ with decay constant λ₂. Assuming that at the initial moment the preparation contained only the radionuclide A₁, find: (a) the equation describing accumulation of the radionuclide A₂ With time; (b) the time interval after which the activity of radionuclide A₂ reaches the maximum value.
Topic	Nuclei
Subject	Physics
Class	Class 12
Answer Type	Text solution:1
Upvotes	100

Text solutionVerified

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