The mean lives of a radioactive substance are 1620 years and 405 years of `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.

Explanation

When a substance decays by `alpha and beta` emission simultaneously, the average rate of disintegration `lambda_(av)` is given by
`lambda_(av)=lambda_(alpha)+lambda_(beta)`
where `lambda_(alpha)=` disintegration constant for `alpha-` emission only.
`lambda_(beta)=` disintegration constant for `beta-=` emission only.
Mean life is given by
`T_(m)=(1)/(lambda)`
`rArr" "lambda_(av)=lambda_(alpha)+lambda_(beta)rArr " "(1)/(T_(m))=(1)/(T_(alpha))+(1)/(T_(beta))`
`=(1)/(1620)+(1)/(405)=3.08xx10^(-3)`
`lambda_(av)t=2.303log""(100)/(25)`
`(3.08xx106(-3))t=2.303log""(100)/(25)`
`rArr" "t=2.303xx(1)/(3.08xx10^(-3))log4="450.17 years"`