# 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.

Q: A radioactive sample cam decay by two different process. The half-life for the first process is T1 and that for the second process is T2 .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 dN1/dt and dN2/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 λ1 and λ2 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.

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

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

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

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

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

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

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