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The mean lives of an unstable nucleus in...

The mean lives of an unstable nucleus in two different decay processes are 1620 yr and 405 yr, respectively. Find out the time during which three-fourth of a sample will decay.

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The correct Answer is:
D
Let at some instant of time t, number of nuclei are N. Then,
`((-dN)/(dt))_(net)=((-dN)/(dt))_1+((-dN)/(dt))_2`
If the effective decay constant is `lambda`, then
`lambdaN=Lambda_1N+lambda_2N`
or `lambda=lamba_1+lambda_2=1/1620+1/405=1/324year^-1`
Now, `N_0/4=N_0e^(-lambdat)`
:. `-lambdat=1n(1/4)=-1.386`
or `(1/324)t=1.386`
:. `t=449 yr`
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