The radioactivity of a sample is `A_1` at time `t_1` and `A_2` at time `t_2` If the mean life of the specimen is `T`, the number of atoms that have disintegrated in the time interval of `(t_2 - t_1)` is :

Options

A. `R_1t_1 - R_2t_2`
B. `(R_1-R_2)^(-1)`
C. `(R_1-R_1)/(T)`
D. `(R_1-R_2)T`

Answer

D

Explanation

Activity R `( " or " (dN)/(dt)) = lamdaN` numerically
For the same element , `lamda` does not change
`therefore R_1 = lamda N_1 " and " R_2 lamdaN_2`
`N_1 " and" N_2` are the undecayed nuclei at time `t_1 " and " t_2` and
`therefore t_2 gt t_1 therefore N_1 gt N_2`
`therefore` The number of atoms distintegrated in time `t_2 - t_1 ` is
`N_1 - N_2 = (R_1)/(lamda) - (R_2)/(lamda) = (R_1-R_2)/(lamda)`
But `T = (0.693)/(lamda) therefore 1/lamda = (T)/(0.693)`
`therefore N_1 - N_2 = ((R_1 - R_2)T)/(0.693)`
`therefore N_1 - N_2 prop(R_1 -R_2) T`