## Half-Life

Read this text. Pay attention to the figure with the red and blue circles, which illustrates how samples shrink by half during each successive half-life.

Note that Equation 1 shows the equation for the radioactive decay rate constant.
If you know the rate constant for the decay of a given material, you can determine the half-life of the material using this equation. Also pay attention to Equation 2, which shows how to determine the initial concentration of the material if you know the half-life and the rate constant for the material. Carefully read the step-by-step example of this type of calculation in the text.

It is interesting to note that there are no stable atoms with an atomic number greater than 83 (**bismuth**). The more massive atoms only exist because certain isotopes are very long lived and therefore on a normal time scale of years appear
stable. Thus, **uranium**-238, uranium-232 and **thorium**-232 are very long-lived isotopes, while **technetium** does not have any stable isotopes.

Since unstable isotopes decay (via either α or β decay) we need a way to express the relative stability of isotopes or the time for radioactive decay. This is expressed by an isotopes half life (t_{1/2}). Empirical observation shows that the number
of atoms of a radioactive element that disintegrates per unit time is a constant fraction of the total number of atoms.

Based upon this the time required for
^{1}/_{2} of the atoms of a radioactive element (X) to decay to a daughter element (Y) is defined as the half life. For example, **polonium**-216 decays to
**lead**-206
with a half life of 0.16 seconds. Thus, as shown in the figure below, a sample of eight polonium-216 decays in 0.16 s to mixture of four lead-206 atoms and a residual of four polonium-216. After a further 0.16 s two of the remaining polonium-216 decay
to lead-206, then in another 0.16 s one of the two remaining polonium-216 atoms decay to a lead-206 atom.

The following table shows selected isotopes and their half life:

Isotope |
Half life |

Cerium–142 |
5 x 10^{15} years |

Radium–226 |
1590 years |

Radon–222 |
3.82 days |

Polonium–216 |
0.16 s |

The rate of radioactive decay can be expressed as a rate constant (k):

The relation of the initial concentration (C_{0}) and the concentration at time t (C_{t}) can be expressed by the following:

It should be noted that

Using these equations the rate constant (k) can be calculated for a particular isotope. For example, colbolt-60 which is used in cancer therapy, decays to nickel-60 with loss of a β particle with a half life of 5.2 years:

Using Equation 1,

As an alternative knowing the rate constant we can calculate the fraction or percentage of **cobalt**-60 isotope that will remain in 15 years.

From Equation 2,

The fraction remaining after 15 years is therefore determined as follows:

Thus, 14% of a sample of cobalt-60 isotope remains after 15 years.

Source: Andrew R. Barron, http://www.vias.org/genchem/nuclear_chem_31328_04_05.html

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