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

Explanation

Law of radioactive decay : The number of nuclei undergoing the decay per unit time is proportional to the number of unchanged nuclei present at any instant t, dN be the number of nuclei that disintegrated in short interval of time dt. Then according to decay law.
` ( dN ) /( dt ) prop - N `
or ` ( dN ) /( dt ) - lamda N " " ` ... ( i)
where ` lamda ` is known as decay constant or disintegration constant.
From equation ( i) , ` ( dN )/( N ) = - lamda dt `
Integrating both sides, we get
` int ( dN ) /( N ) = int - lamda dt `
` log _ e N = - lamda t + C " " ` ... ( ii)
where C is constant of integration whose values depends on initial conditions.
At ` t = 0 , " " N = N _ 0 `
` therefore log _ e N _ 0 = 0 + C `
` rArr C = log _ e N _ 0 `
Substituting the value in equation (ii), we get
` log _ e N = - lamda t + log _ e N _ 0 `
` log _ e N - log _ e N _ 0 = - lamda t `
` log _ e (( N) /(N _ 0 )) = -lamda t `
` ( N ) /( N _ 0 ) = e ^ ( - lamda t ) `
` N = N _ 0 e ^( - lamda t ) `
This expression shows that number of nuclei of given radioactive substance decreases exponentially with time.