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

When a substance decays by `alpha` and `beta` emission simultaneously, the average rate of disintegration `lambda_(av)` is given by `lambda_(delv) = lambda_(alpha) + lambda_(beta)` when `lambda_(alpha) = ` disintegration constant for `alpha`-emission only `lambda_(beta)` = disintegration constant for `beta` -emissin only
Mean life a given by `T_(m) = (1)/(lambda) , lambda_(delv) = lambda_(alpha) + delta_(beta) rArr (1)/(T_(m)) = (1)/(T_(alpha)) + (1)/(T_(beta)) = (1)/(1620) + (1)/(405) = (1)/(324)`
`lambda_(delv)xxt = 2.303 "log" (N_(0))/(N_(t)) 1/324 t = 2.303 "log" (100)/(25) rArr t = 2.303 xx 324 "log" 4 = 449` years.