Name: Nuclear Chain Reaction
Uploaded: 2019-07-19
Duration: 8 min 15 s
Description: Have you ever made a row of dominoes fall by pushing the first one? It is like a chain reaction. Learn about another type of chain reaction that happens on t...

Question

Derive an expression of mean life of a radio active substance.

Accepted Answer

Mean life is the average lifetime of all the nuclei of a particular unstable atomic species. This time interval may be thought of as the sum of the lifetimes of all the individual unstable nuclei in a sample, divided by the total number of unstable nuclei present.

Let a radioactive element contains $N_{0}$ atoms at a time $t_{0}$ and after time $t . N$ atoms are left. Let a small number $d N$ disintegrate in time $dt$ . Hence the lifetime of each of these atoms lies between $t and t + dt$ . If $d t$ is taken very small, then the age of each atom can be taken as $t$ ,then the total age of $dN$ atom= $dN . t$ .

Therefore, the total lifetime of all the atoms = $\int_{0}^{N_{o}} t d N$ .

By definition, the average lifetime of all atoms of the sample $τ = \frac{\int_{0}^{N_{0}} t d N}{N_{0}}$ .............(1)

But by radi0 active law, we have, $dN = - λNdt = - λ (N_{0} e^{- λt})$ , hence, we get,

$τ = \frac{\int_{0}^{N_{0}} - λN 0 e^{- λt} dt . t}{N_{0}} = λ \int_{0}^{\infty} {te}^{- λt} dt$ .........................(A)

Note that the limit of integration is changed in terms of time. Integrating by parts, we get,

$τ = - λ {[\frac{t . e^{- λt}}{- λ}]}_{0}^{\infty} - [\frac{. e^{- λt}}{- λ} dt] = λ [0 + \frac{1}{λ} \int_{0}^{\infty} e^{- λt} dt] = \int_{0}^{\infty} e^{- λt} dt$ .

Or $τ = {[\frac{c^{- λ t}}{- λ}]}_{0}^{\infty} = 0 - \frac{1}{λ} = \frac{1}{λ} o r τ = \frac{1}{λ}$ .

i.e the average lifetime of a radioactive substance is reciprocal of the decay constant

Further as $λ = \frac{0.6931}{T} or τ = \frac{T}{0.6931} = 1.44 T$ or average life = $1.44$ half life.

Embibe Experts Solutions for Chapter: Nuclei, Exercise 1: assam board-2018

Embibe Experts Physics Solutions for Exercise - Embibe Experts Solutions for Chapter: Nuclei, Exercise 1: assam board-2018

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