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Obtain the relation between the decay constant and half life of a radioactive sample.

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#### 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

N_{0}=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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