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Q.

Nuclei of radioactive element  A are produced at rate ‘t2’ at any time  t. The element  A has decay constant λ. Let  N be the number of nuclei of element   A at any time  t. At time t=to,dNdt is minimum. Then the number of nuclei of element  A at time  t=t0 is:

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a

2to−λto2λ2

b

to−λto2λ2

c

2to−λto2λ

d

to−λto2λ

answer is A.

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Detailed Solution

The rate of formation of radioactive nuclei is t2  Rate of decay of radioactive nuclei is  λN∴dNdt=t2−λN For dNdt to be minimum; d2Ndt2=0⇒d2Ndt2=2t−λdNdt=2t−λt2−λN=0⇒N=2t−λt2λ2⇒N=2t0−λt02λ2 at t=t0
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