Modern Physics 3 Question 39

38. 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 N0 nuclei of the element.

(1998,8 M)

(a) Calculate the number N of nuclei of A at time t.

(b) If α=2N0λ, calculate the number of nuclei of A after one half-life of A and also the limiting value of N as t.

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Answer:

Correct Answer: 38. (a) 1λ[α(αλN0)eλt] (b) (i) 32N0 (ii) 2N0 39. (a) 14.43s (b) 40s40.5.95L

Solution:

  1. (a) Let at time t, number of radioactive nuclei are N. Net rate of formation of nuclei of A

dNdt=αλN or dNαλN=dt or N0NdNαλN=0tdt

Solving this equation, we get

N=1λ[α(αλN0)eλt]

(b) (i) Substituting α=2λN0 and t=t1/2=ln(2)λ in Eq. (i) we get, N=32N0

(ii) Substituting α=2λN0 and t in Eq. (i), we get

N=αλ=2N0 or N=2N0



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