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.