The law of radioactive disintegration states that the rate of radioactive disintegration of a radio-element at any instant is proportional to the number of undecayed radioelement N present at that instant in the sample considered.
Mathematically
-(dN)/(dt)propN
"-ve sign represents decrease with time t"
So -(dN)/(dt)=lambdaN
=>-(dN)/N=lambdadt
Where lambda is the radioactive constant
Integrating we get
=>-int(dN)/N=lambdaintdt
=>-lnN=lambdat+k
Where k = integration constant.
If at t=0 the number of undecayed atoms be N_o then
=>-lnN_o=lambdad xx0+k
=>-lnN_o=k
So
=>-lnN=lambdat-lnN_o
=>-lambdat=lnN-lnN_o=ln(N/N_o)
=>N/N_o=e^(-lambdat)
=>N=N_oe^(-lambdat)
By radioactive disintegration law it is obvious that we cannot predict which specfic atom of the radioelement present in the sample is going to be disintegrated first. An atom may disintegrate at t=0 or t = oo So life of a radioelement is expressed as mean or average life which is defined as follows.
"Aerage life"(tau)
="Total life of all atoms in the sample"/"Total number of atoms"
Now
N_o->"Initial number of atoms of radioelement"
N->"No.of atoms of radioelement after t sec "
dN->"No.of atoms disintegrated in the"
" time interval t and t+dt"
Since dt represents infinitesimally small time then total life of dN atoms will be tdN and average life
tau=1/N_o int_0^(N_o)tdN
(As t varies from 0 to oo, N varies from N_o ->0 )
Now to change the variable we substitute dN obtained by differentiating N=N_oe^(-lambdat)
So dN=-lambdaN_oe^(-lambdat)
=1/N_o int_(oo)^0t(-lambdaN_oe^(-lambdat)dt)"
=(lambdaN_o)/N_o int_0^(oo)te^(-lambdat)dt
=lambdaint_0^(oo)te^(-lambdat)dt
=lambda[tinte^(-lambdat)dt-intd/(dt)(t)inte^(-lambdat)dt]_0^(oo)
=lambda[tinte^(-lambdat)dt+1/lambdainte^(-lambdat)dt]_0^(oo)
=lambda[-t/lambdae^(-lambdat)-1/lambda^2e^(-lambdat)]_0^(oo)
=1/lambda
