Saved Bookmarks
| 1. |
What is meant by average life of a radioactive element ? Derive an expression for it. What is the relation between average life and half life ? |
|
Answer» `"Average life"=("Sum of lives of ALLTHE atoms")/("Total number of atoms")` Since `N=N_(0)e^(-LAMBDA t)` `:. dN= - lambda N_(0)e^(-lambda t) dt`. It means that dN atoms disintegrate between time t and t + dt i.e. they have survived for time t. `:.` Total life of dNatoms `=t` dN Hence total life time of all the atoms `tau_("total")=int_(0)^(N_(0)) t dN = - lambda N_(0) int_(oo)^(0) t e^(-lambda t) dt` `=lambda N_(0) int_(0)^(oo) t e^(-lambda t) dt` `tau_("total")=lambdaN_(0)[t{(e^(-lambda t))/(-lambda)}_(0)^(oo) - int_(0)^(oo) e^(-lambda t) dt]` `=N_(0)[t{(e^(-lambda t))/(-lambda)}_(0)^(oo)-{(e^(-lambda t))/(-lambda)}_(0)^(oo)]` `=N_(0)[0-(e^(-oo))/(lambda)+(e^(0))/(lambda)]` or `tau_("total")=N_(0)[0+1/lambda ]=(N_(0))/(lambda)` `:.` average life or MEAN life, `tau_(a)=(tau_("total"))/(N_(0))=(N_(0)//lambda)/(N_(0))=1/lambda` Hence average life `tau_(a)` of the atom is the reciprocal of the radioactive disintegration constant `lambda`. Relation between average life and half life Since average life of a radioactive element is given by `tau_(a)=1/lambda`...(i) and half life of a radioactive element is given by `T=(0.693)/(lambda)` Using Eq. (i), we get `T=0.693 tau_(a)` |
|