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A radioactive nucleus can decay by two d...

A radioactive nucleus can decay by two different processes. The half-life for the first process is `t_1` and that for the second process is `t_2`. Show that the effective half-life `t` of the nucleus is given by
`1/t = 1/t_1 + 1/t_2`.

Text Solution

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Let N be the total number of atoms of 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 `(dN_(1))/(dt) = lamda _(1) N and (dN_(2))/(dt) = lamda _(2)N`
When `lamda _(1) and lamda _(2)` are the decay constnats for the first and second processes respectively.
The initial rate of disintegrations of the radioactive sample by both the processes.
` = (dN _(1))/(dt)+ (dN_(2))/(dt) = lamda _(1) N + lamda _(2) N = (lamda _(1) + lamda _(2)) N.`
If `lamda` is the effective decay constnat of the radioactive sample, its initial rate of disintegration.
`(dN)/(dt) = lamdaN`
But ` (dN)/(dt)= (dN_(1))/( dt) + (dN_(2))/(dt)`
` lamda N = (lamda _(1) + lamda _(2)) N`
`lamda = lamda _(1) + lamda _(2)`
`(0.693)/(T_(1)) + (0.693)/(T _(2)) = (0.693)/(T)`
`1/T = (1)/( T _(1)) + (1)/( T _(2)) , T = (T _(1) T _(2))/( T _(1) + T _(2))`
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