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).`

Explanation

For `alpha`-emission and `beta`-emission respetively, according to parallel disintegration `lamda=lamda_(alpha)+lamda_(beta)`
`" "=(1)/(T_(alpha))+(1)/(T_(beta))=(1)/(1620)+(1)/(405)=(5)/(1620)yr^(-1)`
`" "(because"Mean life"=(1)/("Decay constant"))`
If `3//4` th sample will decay i.e., remaining `1//4` th
After `n` half-life,
`" "N=N_(0)((1)/(2))^(n)rArr (N_(0))/(4)=N_(0)((1)/(2))^(n)rArrn=2`
Time during 3/4 th sample will decay, `t=nT_(1//2)=n(In2)/(lamda)`
`" "=2xx(0.693)/(5//1620)=449 yr`