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Chapter 1.2: Transformation Kinetics

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Chapter 1.2: Transformation Kinetics Chapter 1: Radioactivity Chapter 1.2: Transformation Kinetics 200 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity http://www.world‐nuclear.org/info/inf30.html 201 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Serial Transformation In many situations, the parent nuclides produce one or more radioactive offsprings in a chain. In such cases, it is important to consider the radioactivity from both the parent and the daughter nuclides as a function of time. • Due to their short half lives, 90Kr and 90Rb will be completely transformed, results in a rapid building up of 90Sr. • 90Y has a much shorter half‐life compared to 90Sr. After a certain period of time, the instantaneous amount of 90Sr transformed per unit time will be equal to that of 90Y. •Inthiscase,90Y is said to be in a secular equilibrium. 202 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Exponential DecayTransformation Kinetics • Different isotopes are characterized by their different rate of transformation (decay). • The activity of a pure radionuclide decreases exponentially with time. For a given sample, the number of decays within a unit time window around a given time t is a Poisson random variable, whose expectation is given by t Q Q0e •Thedecayconstant is the probability of a nucleus of the isotope undergoing a decay within a unit period of time. 203 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Why Exponential Decay? The decay rate, A, is given by Separate variables in above equation, we have The decay constant is the probability of a nucleus of the isotope undergoing a decay within a unit period of time. 204 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Why Exponential Decay? (Continued) where c is a constant. It can be determined by the boundary condition. For example, we assume that when t=0, N=N0 ,thenwehave Therefore Similarly, we can write in terms of radioactivity A as The decay constant is the probability of a nucleus of the isotope undergoing a decay within a unit period of time. 205 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Characteristics of Exponential Decay –Half‐life Half‐life The time required for any given radioisotope to decrease to one‐half of its original quantity is defined as the half‐life, T. 206 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Characteristics of Exponential Decay –half‐life The relationship between half‐life T and decay constant can be derived by writing 207 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Characteristics of Exponential Decay – Average or Mean Life It is sometimes useful to characterize a radioactive source in terms of the average or mean life of the given isotope,. It can be understood as 1 208 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Characteristics of Exponential Decay – Average or Mean Life (Continued) The average or mean life, is given by Since ax ax e xe dx 2 (ax 1) a 209 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Characteristics of Exponential Decay – Average or Mean Life (Continued) Remember that ax ax e xe dx 2 (ax 1) a Then et 1 2 (t 1) 0 210 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Units for Radioactivity The Becquerel (Bq) – SI standard unit for radioactivity The Becquerel is the quantity of radioactive material in which one atom is transform per second. 1Bq 1tps 1Curie(Ci) 3.71010 Bq Note that a Becquerel is not the number of particles emitted by the radioactive isotope in 1 s. 211 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Specific Activity (SA) Specific activity of a sample is defined as its activity per unit mass, given in units of Bq/g or Ci/g. Specific activity for pure radioisotopes is defined as the number of Becquerels per unit mass. 6.031023 atoms / mole Specific Activity = Bq / g Ag / mole Activity per The probably of an atom unit mass Number of atoms per unit mass decaying within a unit time span SA can be related to the half‐life (T) of the radionuclide by 4.181023 SA Bq / g AT 212 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Specific Activity (Continued) An example: 213 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Specific Activity (Continued) Solution: activity from Hg per mL SA Hg weight of Hg per mL 4.181023 4.181023 SA Bq / g Bq / g 5.21014 Bq / g AT 20346.5d 24h / d 3600s / h 214 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Specific Activity (Continued) Solution: 4.181023 4.181023 SA Bq / g Bq / g 5.21014 Bq / g AT 20346.5d 24h / d 3600s / h 215 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Specific Activity (Continued) Solution: 216 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Serial Transformation In many situations, the parent nuclides produce one or more radioactive offsprings in a chain. In such cases, it is important to consider the radioactivity from both the parent and the daughter nuclides as a function of time. • Due to their short half lives, 90Kr and 90Rb will be completely transformed, results in a rapid building up of 90Sr. • 90Y has a much shorter half‐life compared to 90Sr. After a certain period of time, the instantaneous amount of 90Sr transformed per unit time will be equal to that of 90Y. •Inthiscase,90Y is said to be in a secular equilibrium. 217 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Indoor Radon 218 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Naturally Occurring Radioactivity – Health Concerns of Radon Gas 219 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity http://www.world‐nuclear.org/info/inf30.html 220 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity General Case Consider a more general case, in which (a) the half‐life of the parent can be of any conceivable value and (b) no restrictions are applied on the relative half‐lives of both the parent and the daughter. The number of atoms of the parent A and the daughter B at any given time t are therefore related by 221 NPRE 441, Principles of Radiation Protection Proof of the Previous Serial Decay Equation From Cember, p123‐124 222 Proof of The Serial Decay Equation (Continued) e 223 Proof of The Serial Decay Equation (Continued) e e 224 Chapter 1: Radioactivity General Case Consider a more general case, in which (a) the half‐life of the parent can be of any conceivable value and (b) no restrictions are applied on the relative half‐lives of both the parent and the daughter. The number of atoms of the parent A and the daughter B at any given time t are therefore related by 225 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity General Case Consider a more general case, in which (a) the half‐life of the parent can be of any conceivable value and (b) no restrictions are applied on the relative half‐lives of both the parent and the daughter. The number of atoms of the parent A and the daughter B at any given time t are therefore related by 226 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Activity Peaking Times Under General Case Q e Qtotal QA e e QB tmt tmd t 227 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Activity Peaking Time Under General Case The peak‐reaching‐time for the activity from the daughter can be derived as the following: Start from the equation for the general case Differentiate respect to t and set to zero and therefore 228 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Activity Peaking Time Under General Case Similarly, the peak reaching times for the total activity is … The total activity is Since and e then Differentiate respect to t and set to zero, we have Solving for t, it can be shown that 229 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Activity Peaking Times Under General Case Q e Qtotal e e QA QB tmt tmd 230 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Further Discussions on Serial Transformations 231 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Further Discussions on Serial Transformations 232 Chapter 1: Radioactivity Activity Peaking Times Under General Case 233 NPRE 441, Principles of Radiation Protection Proof of The Serial Decay Equation (Continued) 234 Chapter 1: Radioactivity Further Discussions on Serial Transformations 235 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Several Special Cases 236 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Secular Equilibrium: TA >> TB (λA << λB)and t > 7TB For the following serial transformation: where A << B and TA >> TB, B is said to be in secular equilibrium.Forexample 237 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Secular Equilibrium: TA >> TB (λA << λB) TA >> TB 7TB General Case Secular Equilibrium 238 Chapter 1: Radioactivity Secular Equilibrium: TA >> TB (λA << λB) From this relationship, one can see that TA >> TB 1.Asthetimegoesby,e‐t decreases and QB approaches QA. At equilibrium, we have 7TB and QA QB 2. Since A has a relatively long half life, QA may be considered as a constant. So the total activity converges to a constant. 239 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Activity Peaking Time Under General Case Qtotal Q QA0 A QB tmt tmd 240 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity Transient Equilibrium: TA TB (λA ≤ λB)and t > tmd A and B are in Transient Equilibrium Qtotal QA0 QA QB t t mt md Transient Equilibrium General case 241 NPRE 441, Principles of Radiation Protection Chapter 1: Radioactivity No Equilibrium: When TA < TB and λA > λB • The half‐life of the daughter exceeds that of the parent, no equilibrium is possible.
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