Radioactive Decay Activity: the number of atoms that decay per unit time: (disintegrations per second). Units: Becquerel (Bq) = 1 dps Curie (Ci) [old unit] = 3.7 x 10 10 Bq exactly (originally defined as the activity of 1.0 g of radium Exponential Decay: Activity (A) of a radioactive nuclide decreases exponentially with time. Let N = # atoms present dN = - λN dt The constant of proportionality, λ, has units of sec -1 . A = = ? dt dN λN Each radioactive nuclide has a unique decay constant λ. = N dN λdt ∫∫ ?= dt N dN λ ln N = -λt + c When t = 0, N 0 atoms are present - implies that ln N 0 = c ln N = - λt + ln N 0 ln = 0 N N -λt = 0 N N e -λt or N = N 0 e -λt or A = A 0 e -λt 1 Half-Life (t 1/2 or T) When N = o N 2 1 2/1 0 0 2 1 t e N N λ? = 2/1 2 1 t e λ? = - ln 2 = - λt? 0.693 = λt? t? 693.0 =λ λ 693.0 2/1 =t Image removed. Fig. 4.1 in Turner J. E. Atoms, Radiation, and Radiation Protection, 2 nd ed. New York: Wiley-Interscience, 1995. 2 Specific Activity Specific Activity (SA) defined as activity per unit mass. Units: g Bq or g Ci A = λN N = # of atoms g atoms mole grams M 1002.6 23 == mole atoms x g N SA = = g A g λN λ M x SA 23 1002.6 = Example: Specific activity of radium M = mole g 226 t? = 1600 y λ = t? 693. SA = mole g 226 mole atoms 1002.6 23 ? ? ? ? ? ? x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? sec60 1 min60 1 24 1 365 1 1600 693. mh h d d y y SA = sec 1066.3 10 ?g atomsx = g Bqx 10 1066.3 1 Ci = 3.66 x 10 10 dps 1 Ci orig. defined as activity associated with 1 g of Radium. Ci is now defined as 3.7 x 10 10 dps exactly. 3 Count rates - vs half-life Example: Compound A: t? = 45 min Compound B: t? = 45 years Given 10 10 atoms of each - find the activity (A) A = λN λ = t? 693.0 [λ = 2.5 x 10 -4 sec -1 ] A A = () ? ? ? ? ? ? min sec60 min45 693.0 10 10 atoms A A = 2.56 x 10 6 Bq A B = ( )( )()()()60602436545 693.0 y 10 10 [λ = 4.8 x 10 -10 sec -1 ] A B = 4.8 Bq 239 Pu t? = 24,065 y 235 U t? = 7.038 x 10 8 y 4 Serial Radioactive Decay N 1 → N 2 N 10 = # parent atoms present at t = 0. N 20 = # daughter atoms present at t = 0. General Case ttt eAeeAANN dt dN 221 20 12 2 1022211 2 )( λλλ λλ λ λλ ??? +? ? =???= Secular equilibrium (T 1 >> T 2 ) Simplifying assumptions: A 20 = 0 T 1 is large, ∴ λ 1 is small; λ 2 - λ 1 = λ 2 1e 1 - ? tλ General Case simplifies to )1( 2 102 t eAA λ? ?= after ~ seven half-lives (of N 2 daughter), A 2 = A 10 0 2 ≈ ? t e λ Image removed. Fig 4.4 in [Turner]. 5 Radon Decay Radium 1620y Radon 3.82d Polonium 3.05m (Radium A) Lead 26.8m (Radium B) Astatine 2s Bismuth 19.7m (Radium C) Polonium 0.000164s (Radium C) Radon 0.035s Thallium 3.1m Lead 22y Mercury 8m Thallium 4m Bismuth 5d Lead Polonium 138d 0.000002% 0.00013% 226 222 218 214 210 206 Mass Number α β α α α α αα α α α 0.021% 0.019% β β β β β β ββ Short-lived radon daughters ? Radon itself, due to its fairly short half-life ( 222 Rn) is not a major concern. ? Radon is also an inert gas and is typically exhaled after breathing it in (although some will dissolve in the blood). ? The concern is over the daughter products of radon that are particulate (attached to aerosol particles), α-emitting, and decay within hours to 210 Pb (T 1/2 = 22 years). 6 Transient equilibrium (T 1 ≥ T 2 ) General Case ttt eAeeAANN dt dN 221 20 12 2 1022211 2 )( λλλ λλ λ λλ ??? +? ? =???= Simplifying assumptions: A 20 = 0 after ~ 10t?s << t e 2 λ? t e 1 λ? A 2 = A 10 t 1 - 12 2 e λλ λ λ ? by definition: 110 1 AeA t = ?λ A 2 = A 1 22 2 λλ λ ? or 12 2 1 2 λλ λ ? = A A - at equilibrium A 1 and A 2 present in a constant ratio Image removed. Fig. 4.5 in [Turner]. 7 No equilibrium (T 1 < T 2 ) [no simplifying assumptions possible] ttt eAeeAANN dt dN 221 20 12 2 1022211 2 )( λλλ λλ λ λλ ??? +? ? =???= Image removed. Fig. 4.6 in [Turner]. 8 The 99m Tc Generator: Transient equilibrium in action ? 99 Mo is adsorbed on an alumina column as ammonium molybdate (NH 4 MoO 4 ) ? 99 Mo (T = 67 hrs) decays (by β -decay) to 99m Tc (T = 6 hrs) ? 99 MoO 4 ion becomes the 99m TcO 4 (pertechnetate) ion (chemically different) ? 99m TcO 4 has a much lower binding affinity for the alumina and can be selectively eluted by passing physiological saline through the column. Image removed. 9 Image removed. N N O - OO - O O - O O - O EDTA ethylenediaminetetraacetate DTPA Image removed. Chelator Kits Image removed. 10