Half-life

Note: A separate article treats the Half-Life computer game.

The half-life of a radioactive isotope is the time it takes for half of the atoms in a pure sample of the isotope to decay into another element. It is a measure of the stability of an isotope; the shorter the half life, the less stable the atom. The decay of an atom is said to be spontaneous as one can only determine the probability of decay and not predict when an individual atom will decay. (Refer to the last section on the generalization of the concept of half-life to other scientific subjects)

All the atoms of a particular radioactive species have the same probability of disintegrating in a given time, so that an appreciable sample of radioactive material, containing many millions of atoms, always changes or disintegrates at the same rate. This rate at which the material changes is expressed in terms of the half-life, the time required for one half the atoms initially present to disintegrate, which is constant for any particular isotope.

The half-life is shorter than the average lifetime. The half-life is ln 2 ≈ 0.693 times the average life. If this seems strange, note that the life of half of the particles is only somewhere between 0 and the half-life, while the life of the other half can be anywhere between the half-life and infinite.

Half-lives of radioactive materials range from fractions of a second for the most unstable to billions of years for those which are only slightly unstable. Decay is said to occur in the parent nucleus and produce a daughter nucleus. Decay from a parent to a daughter nucleus may produce alpha, beta particles, and neutrinos. Gamma radiation may be produced as the nucleus is de-excited but this is only after the alpha or beta decay has taken place. Radioactive decay results in a mass loss, which is converted to energy (the disintegration energy) according to the formula E = mc². Often, the daughter nucleus is also radioactive, and so on down the line for several successive generations of nuclei until a stable one is finally reached. The three such naturally occurring series are shown in the following table:

Note: there are naturally occurring radioactive isotopes (such as C-14) but they are not part of a series.

Other applications of the half-life concept

The concept of half-life is not restricted to the decay of radioactive nuclei. The law is also useful in many processes where the rate of change of some property of a system depends itself on this property. In some chemical reactions, the rate of reaction depends on the concentration of a particular reactant. During the course of the reaction this concentration decreases, causing the rate of reaction also to go down. It is found that the time taken for the rate of reaction to halve is constant, if the reactant is said to be first order with respect to the rate. Enzyme-catalyzed reactions fall into this category. Accordingly, half-life is a common term in pharmacology, used to refer to the time it takes the body to metabolize or remove half the amount of an administered substance.

Half-life is also important in calculating populations, although it is only applicable where the resources available to the population remain surplus to the needs. In these situations the population and its demands increase rapidly, so in reality the resources are always a limiting factor.

Mathematical basis for half-life

The decay of radioactive isotopes can be modelled as an exponential decay, i.e.

x(t) = x(0)e^at

where x(t) represents the amount of the isotope, x(0) represents the value of x at the time t = 0, and a is a constant. In the case of exponential decay, this constant will be negative.

As half-life gives the length of time it takes for x to decay to half of its initial value, it can be modelled as:

where t_1/2 represents the time constant of the half-life, and is the value we wish to solve the equation for.

Half-life is independent of the initial value. That is, x(0) cancels in the above equation, leaving the solution to the equation as:

ln0.5 = at_{1 / 2}

  or : 

As a is a constant it can absorb the minus sign, and we can rewrite the half-life as follows:

References

• Physics for Scientists and Engineers, 4th edition, Raymond Serway, chp 45

See also

• radioactive decay
• nuclear fission
• radiometric dating
• lifetime