A certain radio nuclide starts getting produced at a constant rate = 1 0 4 s − 1 by some mechanism at a certain moment t = 0 . At this moment, there are already 1 0 5 number of radio-nuclides present

Question

A certain radio nuclide starts getting produced at a constant rate = 1 0 4 s − 1 by some mechanism at a certain moment t = 0 . At this moment, there are already 1 0 5 number of radio-nuclides present in the sample. The mean life of the radionuclide is 20 seconds. At t = 40 sec , what will be dN/dt, rounded to the nearest whole number? [Given that e 2 = 7.4 ]

TestHub · Accepted Answer

dt dN ​ = r − λ N ⇒ ∫ N 0 ​ N ​ r − λ N dN ​ = ∫ 0 t ​ dt ⇒ ln r − λ N 0 ​ r − λ N ​ = − λ t ​ ( r − λ N ) = ( r − λ N 0 ​ ) e − λ t ⇒ d t d N ​ = r − λ N = ( r − λ N 0 ​ ) e − λ t = ( 1 0 4 − 20 1 0 5 ​ ) e − t /20 ​ = 5000 e − 2 = 5000/7.4 = + 676 per second Ans. ] ​

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