Radioactive decay

An important physical quantity  at radioactive decay is the half-life.

The half-life T1/2  of a radioactive decay is the time in which one-half of the radioactive nuclei has disintegrated.

The number radioactive nuclei remaining after time t :

  1. N(t) = N(0). (1/2)t/T1/2

               N(0)  the number radioactive nuclei at t = 0 s

                 N(t)   the number radioactive nuclei at t = t s

               t         tijd

               T1/2    half-life

  1. N(t) = N(o) eλt      

             λ = ln 2/T1/2

λ          decay constant  

Remark :   t en T1/2  have the same unit of time

Example

The half-life of I-131 is 8.0 days.

i      Determine the percentage of the radioactive nuclei  that is disintegrated after 6 days.

ii     After what time is 80 % disintegrated ?

Solution I

         i     N(t) = N(0) (1/2)6/8 = N(0) 0.59        

               After 6 days, 59 %  of the number of radioactive nuclei are stil there.

              Therefore  41 % is disintegrated

ii      N(t) = N(0) (1/2)t/T1/2

         N(t)/N(0) = (1/2) t/T1/2    

        80 % disintegrate… 20 % remain

0.20 =  (1/2)t/T1/2

0.20 =   (1/2)t/T1/2

log 0.20 = t/T1/2 log (1/2)

-0.699 = (t/T1/2)(  -0.301)

t/T1/2 = (-0,699/-0.301) = 2,32

t = 2.32 T1/2 = 2.32 8 = 18,56 days

After 19 days 80 % is disintegrated

Solution II

i    N(t) = N(0) e – λt         

     λ = ln 2/T1/2

        λ = 0.693/8 = 0.0866           ( T1/2 in days)

     N(t) = N(0) e – (0.0866) 6

      N(t) =N(0) e – 0,5196  

      N(t) = N(0) 0,59

                    After 6 days,  59 % of the number of radioactive nuclei are still there.

                    Therefore  41 % is disintegrated.

ii     N(t) = N(o) e– λt

                      N(t)/N(0) = eλt

                            0.20 = eλt

                               ln (0.20) = – λt

 -1.609  = – 0.0866 t

 t = 18.58 days

 After  19 days, 80 % is disintegrated      

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