Activity of a radioactive sample
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Activity (A) of a radioactive material/sample is the number of disintegrations per second for the sample. we can also define activity as the the number of unstable atomic nuclei that decay per second in a given sample. Equivalently, it can be looked at as the magnitude of decay rate. Activity also represents the number of radiations emitted i.e the number of alpha,beta and gamma rays emitted that are in proprtion to the number of disintegrations.
The activity of a radioactive nuclide depend on two parameters; The quantity of radioactive material and its half-life. Indeed, the more the atomic nuclei of the radioisotope the higher the activity in terms if disintegrations per unit time, on the other hand, for a given number of radioactive nuclei, half-life is inversely proportional to its activity (equation 1); nuclides with longer half-lives have lower activities. As a radioactive nuclide decays to daughter nuclide, its quantity and activity gradually decrease with time.
Units of activity
Activity has the units disintegrations per second (dps). Activity can also be reported in other units such as Curie (Ci) and Becquerel (Bq). These two other units can be expressed in terms of disintegrations per second (dps) using the interconversions illustrated below.
Since activity is the number of disintegrations per unit time, we can express activity in terms of the total number of radioactive nuclei N and the decay constant λ,
or = 1
but we also know that the equation for the exponential decay is written as;
Using equations 1 and 2, we obtaing an equation of actvity as , where is the number of nuclei at =0.
Similarly, we can write exponential decay equation 2 interms of activity;
Equation 3 gives the number of nuclei that actually decayed is identical to the equation;
Where n is the number of half-lives. In these two equations (3 and 4), and are identical and both represents the fraction of the radioactive nuclei that remained, it is also called the decay factor.
Simplifying quation 4 yields the equation;
= or = 5
From equation 5, its clear that in n half-lives, the fraction of activity that decayed is given by;
By considering an initial atoms of Fluorine-18 (half-life= 110 minutes),
a) How many atoms of this radioisotope will remain undecayed after 220 minutes?
By shall utilize equation 5,
b) What is the fraction that remained?
An alternative method is calculating , = , we have = 110 minutes
c) Fraction that decayed
This is given by
Specific activity is defined as the activity per unit mass of atoms of a given radioactive nuclide. it is normally reported in the units Bq/g or Ci/g.
For a given radioactive nuclide, the number of atoms in 1 g, denoted (), is given by the relation
The number of atoms in 1 kg (1000g) of a radioactive nuclide is therefore given by
= , where is atomic weight in g/mole
From equation 7, We can draw a deduction that the mass of radioactive nuclide can be computed using the equation;
= , where is the number of atoms per gram, is molar mass in g/mol and is the avogadros constant ()
Specific activity interms of decay constant (λ)and half-life ()
Specific activity in (Bq/g) can expressed in terms of decay constant using the relation;
We also know that = , subsituting this in equation 8, we obtain an equation of specific activity with half-life component;
equation 9 can be simplified by using the numerical values of and , equation 9 thjerefore reduces to;
= , 10
The units of half-life in equation 10 should strictly be in seconds. But if the units of half-life is in years then equation 10 can further be sinplified to yield;
1. Given that the half-life of Cs-137 is 30.17 years, calculate its specific activity.
We solve this problem by using equation 11, because the units of half-life is in years, remember that mass number is of molar mass, we therefore use it as molar mass.
2. Calculte the half-life of a radioactive nuclide (Mass number 87) that has a specific activity of .
Equation 9 can be rearrange to give half-life in terms of other quantities;
= , 12
Equation 12 gives half-life in seconds and if need be, we convert to years
We first convert activity to to the units Bq/g by multiplying by 1000 to yield 3200 Bq/g
= 47 billion years
L'Annunziata, M. F. (Ed.). (2012). Handbook of radioactivity analysis. Academic press
Wahl, A. C., & Bonner, N. A. (1951). Radioactivity applied to chemistry (Vol. 72, No. 5, p. 405). LWW.
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