Rates (Kinetics) of Radioactive Decay

Rates (Kinetics) of Radioactive Decay The rate is not affected by temperature, pressure or chemical environment. For radioactive nuclides it is imposs...
Author: Heather Glenn
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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). 1

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

Nuclear Chemistry

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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. Nuclear Chemistry

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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?

Nuclear Chemistry

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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?

Nuclear Chemistry

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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: