Given two radioactive elements, a parent having a half life T 1 and a daughter product having a half life T 2 . If T 1 >> T 2 and if initially there is only the parent element present, how long will it take it terms of T 2 until these elements are within 1 percent of their equilibrium value.
Text Solution
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Sol. A 
B 
C
Initially, N 0 zero zero
At instant t, N 1 N 2
Net rate of formation of B is
= λ 1 N 1 – λ 2 N 2
Solution of this differential equation is
N 2 = 
[e – λ 1t – e – λ 2t ]
If λ 2 >> λ 1 , then N 2 = 
[1 – e – λ 2t ]
λ 1 = 
and λ 2 = 
where T 1 and T 2 are the half lives of the parent and daughter nuclei.
At equilibrium N 2 λ 2 = N 1 λ 1
The equilibrium value of the daughter nuclei is
N 2 = 
Thus, 
= N 2 [1 – e – λ 2t ]
According to the given condition,
= 0.99 or e – λ 2t = 1 – 0.99 = 0.01
or – λ 2 t = ln |0.01| = – 4.6t = 
= 
T 2 = 6.64 T 2 t = 6.64 T 2
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