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Question

# Half lives of α and β emission of a radioactive material are 30 years and 60 years respectively. When material decays giving α and β emission simultaneously, then time in which 78th of the material decays is

A
20 yeras
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B
60 yeras
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C
50 yeras
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D
90 yeras
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Solution

## The correct option is B 60 yeras Given. Half life (T1/2) for α -decay =30 year Half life (T1/2) for β−decay=60 year and also, decay constant =0.693T1/2 λα=0.69330 and λβ=0.69360 Now, for simultaneous decay λ net =λ1+λ2+…so ,λnet=λα+λβ Given, 78 th decay in time t So, left will be (1−78)=18 Now, At=A0e−λt18=1e−λnett18=e−λnettln8=λnett λnet =λα+λB 0.69330+0.69360∴0.693=ln2 λnet =ln2(3)60λnet=ln220 Now, t=ln8ln2×20 year =3ln2ln2×20t=60 year.

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