Rates (Kinetics) of Radioactive Decay
The rate is not affected by temperature, pressure or chemical environment.
For radioactive nuclides it is impossible to say when a particular nucleus will decay. What we can measure is the time it
takes for a certain percentage of nuclei in a sample to decay. The number of nuclei that will decay over a given time
period is determined by Poisson statistics.
The rate of nuclear decay is measured by the activity, A, of the sample, often expressed as the number of
disintegrations observed per unit time. This is analogous to the Rate in chemical kinetics.
Rate of decay = Activity = A = –dN/dt = kN. (units are disintegrations/time)
Where k is the decay constant, and N is the number of radioactive nuclei in the sample.
Question: What is the order for radioactive decay kinetics?
The becquerel (Bq), the SI unit for activity, is defined as one nuclear disintegration per second (dps).
This is an extremely low activity and not often used.
1 Bq = 1 dps
The Curie (Ci), an older but still common activity unit is defined as 3.7 x 1010 Bq = 3.7 x 1010 dps (Originally defined as
the activity of 1.0 g of Radium-226.)
Specific activity: Decay rate per gram (Ci/g)
A “hot” sample has a high activity (high k) (danger!), while a “cold” sample has a low activity (low k).
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Nuclear Chemistry
Kinetics of Radioactive Decay
−
dN
= A = kN
dt
The decay process of radioactive nuclei always follows first order kinetics. For radioactive decay, concentration is
replaced in the integrated rate law equation by the number of radioactive nuclei, N. We can also use the mass of
radioactive nuclei, m, or the activity of the nuclei, A.
We can derive the following formulas from first order kinetics:
N t = N 0 e− kt
mt = m0 e− kt
At = A0 e− kt
Nt
= e− kt
N0
mt
= e− kt
m0
At
= e− kt
A0
⎛N ⎞
ln ⎜ t ⎟ = −kt
⎝ N0 ⎠
⎛m ⎞
ln ⎜ t ⎟ = −kt
⎝ m0 ⎠
⎛A ⎞
ln ⎜ t ⎟ = −kt
⎝ A0 ⎠
ln ( N t ) = −kt + ln(N 0 )
ln ( mt ) = −kt + ln(m0 )
ln ( At ) = −kt + ln(A0 )
m = MASS of radioactive nuclei
A = ACTIVITY of radioactive nuclei
N = # of radioactive nuclei
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Half-Lives of Radioactive Decay
The rates of decay of nuclei are
commonly expressed in terms of their
half-lives. Each isotope has its own
characteristic half-life.
Questions:
• What is the half-life of strontium-90?
• Calculate the decay constant for strontium-90.
Strontium-90 occurs in the fall-out after a nuclear bomb test or an accidental release of
radioactive materials in the air from a nuclear power plant (Japan, 2011). It is chemically similar
to calcium, and can be easily incorporated into bones making exposure very dangerous.
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Kinetics of Radioactive Decay Example Problems
Starting with 1.0 g of strontium-90:
1.
What is the initial activity in Bq? (Molar Mass = 89.91 g/mole)
2.
What mass of strontium-90 remains after 115 years?
3.
What will be the activity after 4 half-lives?
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Kinetics of Radioactive Decay -Example Problem
Cobalt-60 is used as a radiation source for treatment of cancerous tumors. It has a half-life of 5.26 yr. The
cobalt-60 in a radiotherapy unit must be replaced when its radioactivity falls to 75% of the original sample. If
the original sample was purchased in August 2014, when will it be necessary to replace the cobalt-60?
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Radiometric Dating with C-14
Under the right conditions the age of a sample can be determined by measuring the activity of a radioactive isotope.
The best known example is C-14 dating of organic based material. C-14 is a beta emitter with a half-life of 5715 years.
It is formed in the upper atmosphere and is found in about 1 in every 1012 carbon atoms.
14
1
14
1
7
0
6
1
Production:
N + n→ C+ p
Living organism
activity:
15.3 d/min*g C
Dead organism
activity:
<15.3 d/min*g C
Decay: 146 C → 147 N + −10 e
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How has radiocarbon dating changed the way
scientists are able to interpret and understand history?
■
Before the 1950’s there was no way of knowing the
precise age of an artifact or site; we had to depend on
historical records and context.
■
Revolutionized the approach to dating organic
archeological objects almost overnight.
■
C-14 dating was one of the most critical discoveries of
20th century science: The 1960 Nobel Prize in chemistry
was given to Willard Libby for his work with radiocarbon
dating.
•
This dating technique has limitations:
! Assumes constant C-14 amounts in the atmosphere
over thousands of years. This is not true, the C-14 has
varied with variation in cosmic ray activity. For the
best accuracy, data needs to be corrected for
fluctuations in atmospheric C-14. Carbon-14 dating of
tree rings can be used to determine the variation of
C-14 in the atmosphere over a period of time.
Ötzi, the Iceman
Killed in mountain region of Austria/Italy
5,300 years ago.
Click Link for more Information:
Iceman
! Precision: About ±100 yr
(“Recent” human activities have impacted dating for
more recent objects.)
! TIme Limit ≈ 50,000 yr (about 10 half-lives)
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Nuclear Chemistry
Famous things that have been radiocarbon dated...
Shroud of Turin
The proposed burial cloth
of Jesus
■
■
A small sample (50 mg) was sent to 3 labs -Tucson (USA), Oxford (England) and Zurich (Switzerland).
Results all very consistent- between AD 1260 and AD 1390.
■
■
Fits closely with first appearance in history (mid 14th century).
Strongly suggests that the artifact is from the Middle Ages, rather than a genuine 2000 year old burial cloth.
The Dead Sea Scrolls
Nuclear Chemistry
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Radiocarbon date 100BC - 100AD.
■
Close to dates written on them.
■
Close to dates estimated based on writing
style.
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Radiocarbon Dating
The charred bones of a sloth in a cave in Chile represent the earliest evidence of the human presence in the
southern tip of South America. A sample of bone from the sloth has a 14C activity of 5.22 disintegrations per
minute per gram of carbon (d/min*g C) If a living organism has specific of 15.3 d/min*g C, how old are the bones?
The half-life of 14C is 5730 years.
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