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Q.

The nuclei of a certain radioactive element are produced at a constant rate of  α per second. This element has a decay constant λ . Initially N0  nuclei of the element are present.What is the number N of nuclei of this element at a time t?If in the above question α=2 λ N0  calculate the number of nuclei left in the element after one half life period of the element.

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answer is [OBJECT OBJECT], [OBJECT OBJECT].

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Detailed Solution

dNdt=α−λN∫N0NdNα−λN=∫0tdtlnα−λNα−λN0−λ=tλt=lnα−λN0α−λNeλtα−λN=α−λN0αeλt−λNeλt=α−λN0N=αeλt+λN0−αλeλt=1λα+λN0e−λt−αe−λtN=1λα1−e−λt+λN0e−λt λt=ln 2λN0−λN02λN0−λNλln 2λ=ln λN02λN0−λN2=λN02λN0−λN4λN0−2λN=λN0N=32N0
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