The half-life of a radioactive substance is 5000 years. In how many years, its activity will decay to 0.2 times `alpha` its initial value ? Given `log_(10)5=0.6990`.

Explanation

T = 5000 years , t = ?
Activity, `A = N lambda = 0.2` times initial value
Initial activity `A_(0)=N_(0)lambda`
In radioactivity,
`N=N_(0)e^(-lambda t)rArr (N)/(N_(0))=e^(-lambda t)=0.2`
Put `0.2 = (1)/(5), (1)/(5)= e^(-lambda t)` (or) `- lambda t = - log_(e)^(5)`
(or) `t= "log"_(e)^(5)(5)/(lambda)`
Radioactive decay constant `A = (log_(e )2)/(T)`
`= (2.303 log_(10)^(2))/(5000)=(0.693)/(5000)`
Time taken to decay
`t=(log_(e)^(5))/(lambda)=(2.303 log_(10)^(5))/(0.693)xx5000`
`therefore t=(2.303xx0.6990xx5000)/(0.693)=(8049)/(0.693)`
`=1.161xx10^(4)` years