The amount #N# of a radioactive element left after time #t# while initial amount is #N_0# is given as
#{dN}/dt\propN#
#{dN}/N=-kdt\quad (\text{since N decreases w.r.t. time}\ t)#
#\int_{N_0}^N {dN}/N=-k\int_0^t dt#
#N=N_0e^{-kt}#
where, #k# is a constant
As per given data, the #N_0=250\ mg# of a radioactive element decays to #N=220\ mg# in time #t=12\ \text{hrs}# then we have
#220=250e^{-k(12)}#
#k=1/12\ln(250/220)#
If #t_{\text{1/2}}# is half life of radioactive element then its amount becomes #N_0/2# in one half line hence we have
#N_0/2=N_0e^{-k(t_{\text{1/2}})}#
#kt_{\text{1/2}}=\ln 2#
setting the value of #k# we have
#1/12\ln(250/220)t_{\text{1/2}}=\ln2#
#t_{\text{1/2}}=\frac{12\ln 2}{\ln(250/220)}#
#=65.0672\ \text{hrs}#
