Two species of radioactive atoms are mixed in equal number. The disintegration constant of the first species is \lambda and of the second is \lambda/3. After a long time the mixture will behave as a species with mean life of approximately

KVPY SX SB Exam 2013 Question Paper

Question : 31 of 120

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Two species of radioactive atoms are mixed in equal number. The disintegration constant of the first species is

λ

and of the second is

λ/3

. After a long time the mixture will behave as a species with mean life of approximately

Solution:

avg life

={∫\tdN_1+\tdN_2}/{2\N_0}

where

\dN_1 = \λ\N_0\e^{-\λt}\dt

\dN_2=\λ/3\N_0\e^{-\λ/3\t}\dt

avg. life

={∫↖{∞}↙{0}(\λ\N_0\r^{\λ\t}+\λ/3\N_0\e^{-\λ/3\t})dt}/{2\N_0}

Intergrating we got
avg life

=2/\λ

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