Modern Physics 3 Question 3

3. In given time t=0, Activity of two radioactive substances A and B are equal. After time t, the ratio of their activities RBRA decreases according to e3t. If the half life of A is In2, the half-life of B will be

(Main 2019, 9 Jan II)

(a) 4ln2

(b) ln24

(c) ln22

(d) 2ln2

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Solution:

  1. Activity of radioactive material is given as

R=λN

where, λ is the decay constant N is the number of nuclei in the radioactive material.

For substance A,

RA=λANA=λAN0A( initially NA=N0A)

For substance B,

RB=λBNB=λBN0B (initially NB=N0B)

At t=0, activity is equal, therefore

λAN0A=λBN0B

The half-life is given by

T1/2=0.693λ=ln2λ

So, for substance A,

(T1/2)A=ln2λAln2=ln2λAλA=1

According to the given question,

at time t,

RBRA=e3t

Using Eqs. (i), (ii) and (iii)

RBRA=e3t=λBN0BeλBtλAN0AeλAte3t=e(λAλB)t3=λAλBλB=λA+3λB=1+3=4

The half-life of substance B is

(T1/2)B=ln2λB=ln24



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