Calculations Involving HalfLife and DecayRates
The halflife of a radionuclide is the time taken for half the radionuclide's atoms to decay.
Learning Objective

Explain what is a halflife of a radionuclide.
Key Points
 The halflife is related to the decay constant as follows:
$t_{1/2} = ln2/\lambda$ .  The relationship between the halflife and the decay constant shows that highly radioactive substances are quickly spent while those that radiate weakly endure longer.
 Halflives of known radionuclides vary widely, from more than 10^{19} years, such as for the very nearly stable nuclide 209 Bi, to 10^{23} seconds for highly unstable ones.
Terms

halflife
the time required for half of the nuclei in a sample of a specific isotope to undergo radioactive decay

radionuclide
A radionuclide is an atom with an unstable nucleus, characterized by excess energy available to be imparted either to a newly created radiation particle within the nucleus or via internal conversion.
Example
 A sample of _{14}C, whose halflife is 5730 years, has a decay rate of 14 disintegrations per minute (dpm) per gram of natural carbon. An artefact is found to have radioactivity of 4 dpm per gram of its present C. How old is the artefact? We have:
$N = N_o e^{t/\tau} \text{ where } N/N_o=4/14≈0.286 $ ,$\tau = t_{1/2}/ln2 \approx 8267 \text{ years, } t=−\tau lnN/N_o≈10360 \text{ years.}$
Full Text
The halflife of a radionuclide is the time taken for half of the radionuclide's atoms to decay. Taking
The halflife is related to the decay constant by substituting the condition
A halflife must not be thought of as the time required for exactly half of the atoms to decay.
The following figure shows a simulation of many identical atoms undergoing radioactive decay. Note that after one halflife there are not exactly onehalf of the atoms remaining; there are only approximately onehalf left because of the random variation in the process. However, with more atoms (the boxes on the right), the overall decay is smoother and less randomlooking than with fewer atoms (the boxes on the left), in accordance with the law of large numbers.
Radioactive decay simulation
A simulation of many identical atoms undergoing radioactive decay, starting with four atoms (left) and 400 atoms (right). The number at the top indicates how many halflives have elapsed
The relationship between the halflife and the decay constant shows that highly radioactive substances are quickly spent while those that radiate weakly endure longer. Halflives of known radionuclides vary widely, from more than 10^{19} years, such as for the very nearly stable nuclide 209 Bi, to 10^{−23} seconds for highly unstable ones.
The factor of ln(2) in the above equations results from the fact that the concept of "halflife" is merely a way of selecting a different base other than the natural base e for the lifetime expression. The time constant τ is the e^{1}life, the time until only 1/e remains  about 36.8 percent, rather than the 50 percent in the halflife of a radionuclide. Therefore, τ is longer than t_{1/2}. The following equation can be shown to be valid:
Since radioactive decay is exponential with a constant probability, each process could just as easily be described with a different constant time period that (for example) gave its 1/3life (how long until only 1/3 is left), or its 1/10life (how long until only 1/10 is left), and so on. Therefore, the choice of τ and t_{1/2} for markertimes is only for convenience and for the sake of uploading convention. These markertimes reflect a fundamental principle only in that they show that the same proportion of a given radioactive substance will decay over any time period you choose.
Mathematically, the nth life for the above situation would be found by the same process shown above  by setting
Key Term Reference
 Law
 Appears in these related concepts: TwoComponent Forces, Physics and Other Fields, and Models, Theories, and Laws
 atom
 Appears in these related concepts: Overview of Atomic Structure, Description of the Hydrogen Atom, and Stable Isotopes
 base
 Appears in these related concepts: Temple Architecture in the Greek Orientalizing Period, Rules for Exponent Arithmetic, and Rational Exponents
 decay
 Appears in these related concepts: Radioactive Decay Series: Introduction, Models Using Differential Equations, and Discovery of Radioactivity
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 nuclide
 Appears in these related concepts: Dating Using Radioactive Decay, Nuclear Stability, and Rate of Radioactive Decay
 period
 Appears in these related concepts: Frequency of Sound Waves, Sine and Cosine as Functions, and Tangent as a Function
 radioactive decay
 Appears in these related concepts: Alpha Decay, Nuclear Stability, and Tracers
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