Lesson9

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Exponential Decay and Decay Constant Consider a large number of radioactive atoms N 0 at time t = 0. At any time t , number of atoms left is N . n = number of atoms disintegrated in time dt . n Ndt n = λNdt λ = decay constant Since all the kinds of decays are included here, the proportional con- stant λ is called the total decay constant. Definition of λ λ = ( n N ) dt The rate of probability that an individual atom will decay is called the total radioactive decay constant ( λ ). Assume, the change in N , in time dt is dn . dn = ( N - n ) - N = - n dn = - λNdt When, t = 0 -→ t = t , N 0 -→ N N N 0 dn N = - λ t 0 dt N = N 0 e - λt 2
Activity The number of disintegrations per second is called the Activity . A = - dn dt ( negative sign is because of N decreasing) Units: SI units of activity: Becquerel ( Bq ) Bq is defined as the number of disintegration per second. 1 Bq = 1 dps Since, dn dt = - λN The activity, A = λN The strength of a radioactive source is measured by its activity. The traditional unit of activity is curie ( Ci ). The Curie is defined as the number of disintegrations from 1 g of 226 88 Ra . 1 Ci = 3 . 7 × 10 10 Bq Since, N = N 0 e - λt λN = λN 0 e - λt A = A 0 e - λt 3

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Total Activity If the nucleus has more than one type of disintegrations, (say, type A , type B etc. ) then, λ = λ A + λ B + λ C + · · · Total activity, = A + B + C + · · · The activity of the i th mode, λ i N = λ i N 0 e - λt Mean Life time( τ ) The time needed for an initial N 0 nuclei to decay to 1 e of N 0 , is called the mean life time ( τ ). N = N 0 e - λt N 0 e = N 0 e - λt τ = 1 λ Units: s Half Life time ( T 1 2 ) The time required for one-half of the initial number of nuclei N 0 , to decay is called the half-life time. N = N 0 e - λt N 0 2 = N 0 e - λt 4
T 1 2 = ln 2 λ T 1 2 = 0 . 693 λ Units: s Relation between Mean Life time( τ ) and Half Life time ( T 1 2 ) τ = 1 λ and T 1 2 = 0 . 693 λ T 1 2 = 0 . 693 τ τ = 1 . 44 T 1 2 The following diagram illustrates the exponential decay and the con- cepts of mean life time and half life time. 5

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Specific Activity(SA) The specific activity (SA) of a radioactive sample,is defined as the ac- tivity per unit mass. SA = dA dm where dA = Activity and, dm = mass Units: Bq.g - 1 or Ci.g - 1 From the definition of a mole, the number of atoms per gram is, N A M where N A = Avogadro’s number = 6 . 023 × 10 23 mol - 1 and M = Atomic weight ( g.mol - 1 ) Since, A = λN A M = λ N A M N A = 6 . 023 × 10 23 mol - 1 Then, Specific activity, SA = A M = 6 . 023 × 10 23 λ M Also, λ = 0 . 693 T 1 2 SA = 4 . 17 × 10 23 MT 1 2 6
If T is in seconds, here units of SA is, Bq.g - 1 . Example: The half-life of 215 At is 100 μs . A sample of 215 At , initially contains 5 mg . What is its initial activity and activity after 150 μs . Solution: given T 1 2 = 100 × 10 - 6 s and mass = 5 × 10 - 3 g . Need to know the number of atoms in 5 × 10 - 3 g . Weight of 1 mole of 215 At is 215 g .

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