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Question
Obtain the relation between the decay constant and half life of a radioactive sample.
Solution
The number of atoms at any instant in a radioactive sample is given by
N=N0e−λt
where
N=total number of atoms at any instant
N0=number of atoms in radioactive substance at t=0
λ=decay constant
t=time
When t=T (Where T is the half life of the sample)
`N=N_0/2`
`=>N_0/2=N_0e^(-lambdat)`
`=1/2=e^(-lambdaT)`
`=>e^(lambdaT)=2`
Taking log on both the sides, we get
`lambdaT=log_e2=2.303 lod_10 2`
`=>T=(2.303 lod_10 2)/lambda`
`=>T=(2.303 xx 0.3010)/lambda`
`=>T=0.6931/lambda`
Thus, half life of a radioactive substance is inversely propotional to decay constant.
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