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Decay Rate
N vs t
Contents:
•Probability of decay and
Activity
•Whiteboard
•Half life and
exponential decay
•Whiteboard
•Radiometric dating
Remaining nuclei
100
90
80
70
60
50
40
30
20
10
0
0
20
40
60
Time
80
100
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)
A = -N/t = N
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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
Whiteboards:
Activity and decay probability
1|2
TOC
What is the activity if you have a  of 3.19 x
10-10 s-1, and you have 5.12 x 1023 undecayed 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
W
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
W
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 = N = Noe-t
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Half life
T1/2 - Half life - time for half nuclei to decay
1/ N = N e-T1/2
2 o
o
1/ = e-T1/2
2
2 = eT1/2
ln(2) = T1/2
T1/2 = ln(2)/
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Half life and exponential decay
A = -N/t
A = N = Noe-t
N = Noe-t
ln(2)
T1/2 =
/
Half lives occur over
and over
N vs t
Remaining nuclei
100
90
80
70
60
50
Half life here is 20 s
40
30
20
10
0
0
20
40
60
Time
80
100
TOC
Half life and exponential decay
A = N = Noe-t
N = Noe-t
A = -N/t
ln(2)
T1/2 =
/
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)
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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
Whiteboards:
Half Life and Decay
1|2|3|4|5|6
TOC
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
W
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
W
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
W
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
W
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
W
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
W
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 nearly pure crystals
Magma is product of fission within the earth
Lead - Earth’s age - Uranium/Lead - Clean room - lead in fuel.
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