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Half Life Calculation of Radioactive DecayAtomic Physics
Stability of Isotopes
Decay of Uranium-238
Activity (A) ... is the number of nuclei in a
given sample that will decay in a given time.
Usually measured in decays/s, becquerels (Bq).
Half LifeHalf-life of a radioactive isotope is the time taken
for half of the atoms of an element to decayEg) each radioactive isotope has its own half life
31
238 9 292
227 292
Hydrogen 12.3
5730 years
Uranium 4.50 10
1.10 min
) 131 1/2 life is 8.07 days
original mass
H years
Carbon C
U years
U
eg iodine
12.0 g t = 0 (1)
1 6.0 g 8.07 days ( ) 1
21
3.0 g 16.14 days ( ) 2 decays4
decay
1 1.5 g 24.21 days ( ) 3 decays
8
Half Life... is the time required for half of
the radioactive nuclei in a sample to decay.
Example:Half-life for iodine-131 is 192 h.Initial mass of sample: 20 gAfter 192 h, 10 g of I-131 remains
(the rest is decay products)After another 192 h, 5.0 g of I-
131 remains.
http://videos.howstuffworks.com/hsw/17819-physics-the-nature-of-radioactive-decay-video.htm
http://www.youtube.com/watch?v=6X-zjmEZO4o
http://www.youtube.com/watch?v=xhOtKurHayo
Equation
Equation to determine the mass remaining after some time period
1( )2
n is the number of half-lives or decays
n =
N is the undecayed mass remaining
N is the original mass
no
o
N N
time
halflife time
Example
Argon-39 undergoes beta decay, with a half-life of 269 years. If a sample contains 64.0 g of Ar-39, how many years will it take until only 8.00 g of Ar-39 remain?
Ignoring any other decays that may occur, what element is the rest of the sample transmuted into?
Solution: Number of ½ lives:64.0 g x ½ = 32.0 g One ½
Life2.0 g x ½ = 16.0 g Two ½
Lives16.0 g x ½ = 8.00 g Three
½ Livest = 3 x t1/2 = 3 x 269 y= 807 y
Solution:beta decay:
39 39 018 19 1Ar K v
Product: Potassium-39
Eg) For Iodine-131 which has a halflife of 8.02 days, determine the mass remaining after 72.2 days having started with a mass of 12.0g.
The amount remaining is 2.34 x 10-2 g.
9
2
72.21( ) n = 9.002 8.07
112.0( )
20.0234375
2.34 10
no
daysN N
days
N
N g
N g
Radioactive Decay of Iodine-131
Graph For HalfLife Calculation
Be able to interpret these graphs for half life time.
Radioactive DatingBy measuring the relative
amounts of different isotopes in a material, the age of the material can be determined.
Carbon dating, using carbon-14, is the most well known example.
Carbon-14 has a half-life of 5730 years.
ExampleA sample of bone contains one
quarter of the C-14 normally found in bone. What is the bone’s approximate age?
Solution:The age of a sample with half the
normal amount of C-14 would be approximately the same as the half life of C-14 (half the C-14 will have decayed).
¼ = ½ • ½ so ¼ is two half-lives.time = 2 • 5730 y = 11460 y
Why carbon dating worksCarbon dating works for bone,
and wood, etc.The proportion of C-14 to C-12 in
the atmosphere is well known.A living tree will have the same
proportion of C-14 to C-12 as it constantly absorbs carbon from the air.
Why carbon dating works
When the tree dies (ie use the wood to make a tool) it no longer absorbs carbon.
Decay of C-14 starts to reduce the amount of C-14 in the wood.
Amount of stable C-12 remains constant.
Why carbon dating worksWhen there is half the usual
amount of C-14 remaining, the wood is about 5730 years old (one half life).
Accurate measurements need to account for variations in proportion of C-14 to C-12 over the centuries.
Carbon dating has been verified by comparing to known dates.
Fractional half-livesA bone fragment has 40% of the
original C-14 remaining. What is its age?
The age will be:1.32 • one half life = 1.32 • 5730
years≈ 7500 years
1.32
1N = No
2
140% = 100%
2
n
Why carbon dating worksCarbon dating does not provide
accurate results for materials older than about 50 000 years, or fairly recent materials.
This is because there is either not enough C-14 left to accurately measure or not enough has decayed yet.