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  • Define half-life.
  • Define dating.
  • Calculate age of old objects by radioactive dating.

Unstable nuclei decay. However, some nuclides decay faster than others. For example, radium and polonium, discovered by the Curies, decay faster than uranium. This means they have shorter lifetimes, producing a greater rate of decay. In this section we explore half-life and activity, the quantitative terms for lifetime and rate of decay.

Half-life

Why use a term like half-life rather than lifetime? The answer can be found by examining [link] , which shows how the number of radioactive nuclei in a sample decreases with time. The time in which half of the original number of nuclei decay is defined as the half-life    , t 1 / 2 size 12{t rSub { size 8{1/2} } } {} . Half of the remaining nuclei decay in the next half-life. Further, half of that amount decays in the following half-life. Therefore, the number of radioactive nuclei decreases from N size 12{N} {} to N / 2 size 12{N/2} {} in one half-life, then to N / 4 size 12{N/4} {} in the next, and to N / 8 size 12{N/8} {} in the next, and so on. If N size 12{N} {} is a large number, then many half-lives (not just two) pass before all of the nuclei decay. Nuclear decay is an example of a purely statistical process. A more precise definition of half-life is that each nucleus has a 50% chance of living for a time equal to one half-life t 1 / 2 size 12{t rSub { size 8{1/2} } } {} . Thus, if N size 12{N} {} is reasonably large, half of the original nuclei decay in a time of one half-life. If an individual nucleus makes it through that time, it still has a 50% chance of surviving through another half-life. Even if it happens to make it through hundreds of half-lives, it still has a 50% chance of surviving through one more. The probability of decay is the same no matter when you start counting. This is like random coin flipping. The chance of heads is 50%, no matter what has happened before.

The figure shows a radioactive decay graph of number of nuclides in thousands versus time in multiples of half-life. The number of radioactive nuclei decreases exponentially and finally approaches zero after about ten half-lives.
Radioactive decay reduces the number of radioactive nuclei over time. In one half-life t 1 / 2 size 12{t rSub { size 8{1/2} } } {} , the number decreases to half of its original value. Half of what remains decay in the next half-life, and half of those in the next, and so on. This is an exponential decay, as seen in the graph of the number of nuclei present as a function of time.

There is a tremendous range in the half-lives of various nuclides, from as short as 10 23 size 12{"10" rSup { size 8{ - "23"} } } {} s for the most unstable, to more than 10 16 size 12{"10" rSup { size 8{"16"} } } {} y for the least unstable, or about 46 orders of magnitude. Nuclides with the shortest half-lives are those for which the nuclear forces are least attractive, an indication of the extent to which the nuclear force can depend on the particular combination of neutrons and protons. The concept of half-life is applicable to other subatomic particles, as will be discussed in Particle Physics . It is also applicable to the decay of excited states in atoms and nuclei. The following equation gives the quantitative relationship between the original number of nuclei present at time zero ( N 0 size 12{N rSub { size 8{0} } } {} ) and the number ( N size 12{N} {} ) at a later time t size 12{t} {} :

N = N 0 e λt , size 12{N=N rSub { size 8{0} } e rSup { size 8{ - λt} } } {}

where e = 2.71828 ... size 12{e=2 "." "71828" "." "." "." } {} is the base of the natural logarithm, and λ size 12{λ} {} is the decay constant    for the nuclide. The shorter the half-life, the larger is the value of λ size 12{λ} {} , and the faster the exponential e λt size 12{e rSup { size 8{ - λt} } } {} decreases with time. The relationship between the decay constant λ size 12{λ} {} and the half-life t 1 / 2 size 12{t rSub { size 8{1/2} } } {} is

Practice Key Terms 8

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Source:  OpenStax, Basic physics for medical imaging. OpenStax CNX. Feb 17, 2014 Download for free at http://legacy.cnx.org/content/col11630/1.1
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