  Question

State the law of radioactive decay. Hence derive the expression $$N={ N }_{ 0 }{ e }^{ -\lambda t }$$ where symbols have their usual meanings.

Solution

Law of radioactive decay : At any instant, the rate of radioactive disintegration is directly proportional to the number of nuclei of the radioactive element present at that instant.We know by radioactive decay law$$\dfrac { dN }{ dt } \propto N$$or $$\dfrac { dN }{ dt } =-\lambda N$$where $$\lambda$$ is the constant of proportionality called radioactive decay constant. Let $${ N }_{ 0 }$$ be the number of nuclei present at time $$t=0$$, and $$N$$ the number of nuclei present at time $$t$$.$$\int _{ { N }_{ 0 } }^{ N }{ \dfrac { dN }{ N } } =-\int _{ 0 }^{ t }{ \lambda dt } =-\lambda \int _{ 0 }^{ t }{ dt }$$$$\log _{ e }{ N } -\log _{ e }{ { N }_{ 0 } } =-\lambda t$$$$\log _{ e }{ \left( \dfrac { N }{ { N }_{ 0 } } \right) } =-\lambda t$$$$\dfrac { N }{ { N }_{ 0 } } ={ e }^{ -\lambda t }$$$$N={ N }_{ 0 }{ e }^{ -\lambda t }$$ Physics

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