Law of radioactive decay : At any instant, the rate of radioactive disintegration is directly proportional to the number of nuclei of the radioactive element present at that instant.We know by radioactive decay law\\\\(\\dfrac { dN }{ dt } \\propto N\\\\)or \\\\(\\dfrac { dN }{ dt } =-\\lambda N\\\\)where \\\\(\\lambda\\\\) is the constant of proportionality called radioactive decay constant. Let \\\\( { N }_{ 0 }\\\\) be the number of nuclei present at time \\\\(t=0\\\\), and \\\\(N\\\\) the number of nuclei present at time \\\\(t\\\\).\\\\(\\int _{ { N }_{ 0 } }^{ N }{ \\dfrac { dN }{ N } } =-\\int _{ 0 }^{ t }{ \\lambda dt } =-\\lambda \\int _{ 0 }^{ t }{ dt }\\\\)\\\\( \\log _{ e }{ N } -\\log _{ e }{ { N }_{ 0 } } =-\\lambda t\\\\)\\\\(\\log _{ e }{ \\left( \\dfrac { N }{ { N }_{ 0 } } \\right) } =-\\lambda t\\\\)\\\\(\\dfrac { N }{ { N }_{ 0 } } ={ e }^{ -\\lambda t }\\\\)\\\\(N={ N }_{ 0 }{ e }^{ -\\lambda t }\\\\)