and solved by the methods of calculus to yield N = Noe−λt (3)
which is the number N of undecayed nuclei at any time t when the initial number at t = 0 is No. The symbol “e” is the natural exponential number 2.7138.
Since the activity of a sample is the time rate of change of N, the activity A of a sample is defined by
A= ΔN = dN = -λN =λN (4) Δt dt
and is proportional to the number of radioactive nuclei. Since the activity is proportional to the number of radioactive nuclei, the activity will have the same time behavior and will change at same rate as the number of nuclei. Therefore, the activity is given by
and
A=λNoe−λt (5) A = A e−λt (6)
where Ao is the initial activity of the sample at time t = 0, and is equal to λNo. The activity of a sample can be measured as a function of time and the rate constant can be determined experimentally.
The SI unit of activity is the becquerel and is defined as the activity of a radionuclide that has a decay rate of one spontaneous nuclear transition per second. Historically, the standard unit of activity is the Curie (Ci) and is still the most often used unit for expressing activity. One Curie is equal to 3.7 x 1010 becquerels. (1 Ci = 3.7 x 1010 Bq.) Typical exempt sources that can be purchased without a license have activities of a few microcuries, i.e. 10-6 Ci.
A useful concept in nuclear methods is the time that it takes the activity of a given radioactive sample to decrease by half of its original activity. This time is known as the nuclear half-life and can be used to help identify an unknown radioisotope. The nuclear half-life τ depends on the decay rate constant λ so that the larger the decay rate, the smaller the half-life. The nuclear half-life τ is defined such that if the initial activity is Ao attimet=0,thentheactivityattimet=τwillbeA=1⁄2Ao and
1A =Ae−λτ . (7) 2oo
2
o
Radioactive Half-life of Barium-137m
This equation may be solved for the half-life by simplifying and taking the natural logarithm of both sides of the equation. By dividing both sides of Equation (7) by Ao and by taking the inverse of each side, Equation (7) becomes
and and
12 = e − λ τ
2−1 =e−λτ
( 8 )
(9) (10)
(11) (12)
2=eλτ . Taking the natural logarithm of each side of this equation yields
and
ln2=lneλτ , ln 2 = λτ ,
τ = ln 2 . λ
(13) Since ln 2 = 0.693, the nuclear half-life τ can be computed from a measurement of the
decay rate constant λ and the simple relationship τ = 0.693 . (14)
λ
The decay rate constant for a particular isotope may be found by Louis J. Sheehan, Esquire.
simple 330.sim.0002 Louis J. Sheehan, Esquire
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