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The mean lives of a radioactive substanc...

The mean lives of a radioactive substance are 1620 years and 405 years for `alpha` emission and `beta` emission respectively. Find out the time during which three fourth of a sample will decay if it is decaying both by `alpha`-emission and `beta`-emission simultaneously. `(log_e4=1.386).`

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Here, `tau_(alpha)=1620` years, `tau_(beta)=405"years"`.
`t=? N=N_0-3/4N_0=(N_0)/4`
If `lambda_(alpha)` and `lambda_(beta)` are decay constant of alpha and beta emission respectively, then,
`lambda_(alpha)=1/(tau_(alpha))=1/1620yr^-1`
`lambda_(beta)=1/(tau_(beta))=1/405yr^-1`
Total decay constant `lambda=lambda_(alpha)+lambda_(beta)`
`=1/1620+1/405=(5)/1620=1/324yr^-1`
form `N=N_0e^(-lambda t)`,
`1/4N_0=N_0e^(-lambdat)`
`t=(log_e4)/lambda=1.386/(1//324)yr=449.1 years`
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