Q: A radioactive sample cam decay by two different process. The half-life for the first process is T_{1} and that for the second process is T_{2} .find the effective half-life T of the radioactive sample.

Sol: Let N be the total number of atoms of the radioactive sample initially . Let dN_{1}/dt and dN_{2}/dt be the initial rates of disintegrations of the radioactive sample by the two processes respectively.

Then , $\large \frac{dN_1}{dt} = \lambda_1 N \; and , \frac{dN_2}{dt} = \lambda_2 N $

Where λ_{1} and λ_{2} are the decay constants for the first and second processes respectively.

The initial rate of disintegrations of the radioactive sample by both the processes.

$\large \frac{dN_1}{dt} + \frac{dN_2}{dt} = \lambda_1 N + \lambda_2 N $

If λ is the effective decay constant of the radioactive sample, its initial rate of disintegration.

$\large \frac{dN}{dt} = \lambda N $

$\large \frac{dN}{dt} = \frac{dN_1}{dt} + \frac{dN_2}{dt} $

$\large \lambda N = (\lambda_1 + \lambda_2 )N $

$\large \lambda = (\lambda_1 + \lambda_2 ) $

$\large \frac{0.693}{T} = \frac{0.693}{T_1} + \frac{0.693}{T_2} $

$\large \frac{1}{T} = \frac{1}{T_1} + \frac{1}{T_2} $

$\large T = \frac{T_1 T_2}{T_1 + T_2} $