(a) Define the terms (i) half-life `(T_(1//2) )` and (ii) average life `(tau)`. Find out their relationships with the decay constant `( lambda)`.
(b) A radioactive nucleus has a decay constant, `lambda = 0.3465` (day)`""^(-1)`. How long would it take the nucleus to decay to 75% of its initial amount?

Explanation

(a) Definition :
(i) Half life : Time taken by a radioactive nuclei to reduce to half of the initial number of radionuclei.
(ii) Average life : Ratio of total life time of all radioactive nuclei to the total number of nuclei in the sample.
Relation between half life and decay constant :
`T_(1//2) = (0.693)/( lambda)`
Relation between average life and decay constant
`tau = (1)/(lambda)`.
(b) `N= N_(0) e^(- lambda t)`
`(3)/(4) N_(0) = N_(0) e^(-(0.3465) t) " "(because N= 75% "of" N_(0) )`
`N= (3)/(4) N_(0)`
`e^((0.3465)t) = (4)/(3)`
`0.3465 xx t= log_(e) (4//3)`
`=2.303 [ log4-log3]`.
`=2.303 [0.6020-0.4771]`
`=2.303 xx 0.1249`
`t=(2.303 xx 0.1249) /( 0.3465)`
`therefore t=0.83` days or 19.92 hours