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Question

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=t0,dNdt is minimum. Then the number of nuclei of element A at time t=t0 is:

A
2t0λt20λ2
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B
t0λt20λ2
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C
2t0λt20λ
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D
t0λt20λ
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Solution

The correct option is A 2t0λt20λ2
If the nuclei of radioactive element are produced at rate t2 and if λ is the decay constant and N is the number of nuclei of element then we have:
dNdt=t2λN
Given that at time t=to, dNdt is minimum.
d2Ndt2=0
d2Ndt2=2tλdNdt=2tλ(t2λN)=0
2tλt2λ2N=0
N=2tλt2λ2
This is the case when t=to.
Hence we have,
N=2toλto2λ2

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