State the low of radioactive decay. Hence derive the relation N= `"Noe"^(-lambdat)`. Represent it graphically.

Explanation

Radioactive Decay Law. The number of nuclei undergoing the decay per unit time is directly proportional to t he number of undecayed nuclei present a that instant.
LetN, be the number of nuclei present at any instant 't' Let,'dN' be the number of nuclei that disintegrtated in short interval of time 'dt'
According to law of decay,
`-(dN)/(dt)propN`
(-ve sign indicates, decreases in number of nuclei)
`:. (dN)/(N)=-lambda N " "...(i)`
Where, `lambda` is decay constant or disintegration constant.
`(dN)/(N)=-lambdadt`
Now intergrating both sides,
`int (dN)/(N)int - lambda dt`
`"log"_(e)N=-lambdat+C" "...(ii)`
Where, C integration constant.
At `t=0,N=N_(0)`
`log_(e)N_(0)=0+C`
`log_(e)N-log_(e)N_(0)=-lambdat`
`log_(e)((N)/(N_(0)))=-lambdat`
`(N)/(N_(0))=e^(-lambdat)`
`N=N_(0)e^(-lambdat)`
The above expression shows that number of nuclei of the given radioactive substance decreases exponentially with time