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Mastering Decay Rates and Half-Life Calculations

Learn how to determine decay probability, activity, and half-life in radiometric dating using formulas and examples. Explore exponential decay, nuclear decay, and practical applications.

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Mastering Decay Rates and Half-Life Calculations

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  1. Decay Rate • Contents: • Probability of decay and Activity • Whiteboard • Half life and exponential decay • Whiteboard • Radiometric dating

  2. Probability and activity N - Number of un-decayed nuclei (number)  - Per second probability of a nuclei decaying (s-1) A - Activity - decays/sec (Becquerels (Bq) = s-1)

  3. Example - Radon 222 has an atomic mass of 222.02. How many grams of it do you have if your activity is 8.249 x 1016 decays/sec, and your decay probability is 2.098 x 10-6 s-1? (4) NA = 6.02 x 1023 atoms/mol n = N/NA n = (grams you have)/(molar mass) A = N 8.249 x 1016 s-1 = (2.098 x 10-6 s-1)N N = 3.93184 x 1022 nuclei n = (3.93184 x 1022)/(6.02 x 1023 mol-1) = 0.065312954 grams = n(molar mass) = (0.065312954)(222.02) = 14.5 g 14.5 g

  4. Whiteboards: Activity and decay probability 1 | 2

  5. What is the activity if you have a  of 3.19 x 10-10 s-1, and you have 5.12 x 1023 un-decayed nuclei? A = N = (3.19 x 10-10 s-1)(5.12 x 1023) = 1.63 x 1014 decays/s (or s-1) 1.63 x 1014 decays/sec

  6. What is the  if 1.27 x 1020 un-decayed nuclei generate 1420 decays per second? A = N 1420 s-1 = (1.27 x 1020)  = 1.12 x 10-17 s-1 1.12 x 10-17 s-1

  7. Half life and exponential decay •  - Per second probability of a nuclei decaying (s-1) • N - Number of un-decayed nuclei (number) • A = Activity - decays/sec (s-1) • A = -N/t • N = Noe-t - Exponential decay • No - Original value • N - Value at time t • t - Elapsed time • Diff EQ… • A= Noe-t

  8. Half life • T1/2 - Half life - time for half nuclei to decay • 1/2No= Noe-T1/2 • 1/2= e-T1/2 • 2 = eT1/2 • ln(2) = T1/2 • T1/2= ln(2)/

  9. Half life and exponential decay Half lives occur over and over Half life here is 20 s

  10. Example: Bi 211 has a half life of 128.4 s. What is the per-second probability of a nuclei decaying? If you start out with 32 grams of Bi 211, how much is left after 385.2 s? After what time is there 23 grams left? What is the activity when there is 23 grams left? (m = 210.987 u) Use formulas Cheat (385.2 s = 3 half lives)

  11. Example: Bi 211 has a half life of 128.4 s. What is the per-second probability of a nuclei decaying? If you start out with 32 grams of Bi 211, how much is left after 385.2 s? After what time is there 23 grams left? What is the activity when there is 23 grams left? (m = 210.987 u) so the  = ln2/128.4 = 0.005398343 s-1. and the N (just use grams) at 385.2 seconds would be: N = Noe-t = (32 g)e-(0.005398343 s-1)(385.2 s) = 4.0 g (How come it is exact??? – 385.2 = 3*128.4 so it is exactly 3 half lives. look for that on tests and stuff.) Use the same formula for finding the time it will be 23 g (less than one half life 128.4 s – right?) N = Noe-t (23 g) = (32 g)e-(0.005398343 s-1)t so t = 61 s Activity is just given by A = N, so we use chemistry to find N: N = (6.02x1023atoms/mol)(23 g)/210.987 g/mol) = 6.56249E+22 atoms so A = N = (0.005398343 s-1)(6.56249E+22 atoms) = 3.54266E+20 counts per second

  12. Whiteboards: Half Life and Decay 1 | 2 | 3 | 4 | 5 | 6

  13. Oregonium has a decay probability of 8.91 x 10-8 s-1. What is its half life in days? T1/2= ln(2)/= ln(2)/(8.91 x 10-8 s-1) = 7779429.636 s = 90.0 days 90.0 days

  14. What is the nuclear decay probability of a substance that has a half life of 96.23 minutes? T1/2= ln(2)/- (in data packet) T1/2= (96.23 min)(60 sec/min) = 5773.8 s = ln(2)/  = ln(2)/(5773.8 s) = 0.0001201 s-1 0.0001201 s-1

  15. Oregonium has a decay probability of 8.91 x 10-8 s-1. If you have 1250 grams of Oregonium initially, how many grams do you have after 30.00 days? (x24x3600) N = Noe-t - (in data packet) N = (1250) e-(8.91x 10-8)(30*24*3600) N = 992 g 992 g

  16. Tualatonium has a half life of 12 seconds. If you start with 64 grams of it, how much remains after a minute? (Cheat) 60 seconds is exactly 5 half lives, so divide in half five times 64/25 = 2.0 grams 2.0 grams

  17. Tigardium has a half life of 8.34 seconds. The initial activity is 1350 counts/second, after what time is the activity 125 counts/sec? = ln(2)/T1/2 = 0.083111173 s-1 N = Noe-t - (in data packet) N/No = e-t ln(N/No) = -t ln(No/N) = t ln(No/N)/ = t = ln(1350/125)/(0.083111173 s-1) t = 28.6 s 28.6 s

  18. A certain substance has an activity of 1245 counts/sec initially, and an activity of 938 counts/second after exactly 3.00 minutes. What is the half life of the substance? N = Noe-t - (in data packet) N/No = e-t ln(N/No) = -t ln(No/N) = t ln(No/N)/t =  = ln(1245/938)/(180. s) = .001573 s-1 T1/2= ln(2)/(.001573 s-1)= 441 s 441 s

  19. Radiometric dating • Know original proportion of unstable nuclei • Measure current proportion • Use N = Noe-t to calculate elapsed time • Carbon 14 dating: • C14 created in atmosphere (T1/2 = 5730 Y) • Living things absorb C14 in known amounts • They die, and quit absorbing C14 • Rocks can be dated • Hardening forms purecrystals • Magma is product of fission within the earth (Rock cycle) • Lead - Earth’s age - Uranium/Lead - Clean room - lead in fuel.

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