Half-Life: Definition, Equation

Nuclides are characterized by their atomic number (number of protons) and atomic mass number (total number of protons and neutrons). The number of protons dictates what element it is, and the total number of protons and neutrons determines the isotope.

Radioisotopes (radioactive isotopes) are atoms that have an unstable nucleus and are prone to nuclear decay. They are in a high-energy state and want to jump to a lower-energy state by releasing that energy, either in the form of light or other particles. A radioisotope's half-life, or the amount of time it takes one half of a radioisotope's atoms to decay, is a very useful measure to know.

Radioactive elements tend to be on the last row of the periodic table, and the last row of the rare earth elements.

Radioactive isotopes have unstable nuclei, where the binding energy keeping the protons and neutrons tightly locked together is not quite strong enough to hold permanently. Imagine a ball sitting at the peak of a hill; a light touch will send it rolling down, as if to a state of lower energy. Unstable nuclei can become more stable by releasing some of their energy, either in the form of light or other particles such as protons, neutrons and electrons. This energy release is called radioactive decay.

The decay process can take many forms, but the basic types of radioactive decay are: ​alpha​ decay (emission of an alpha particle/helium nucleus), ​beta​ decay (emission of a beta particle or electron capture) and ​gamma​ decay (emission of gamma rays or gamma radiation). Alpha and beta decay transmute the radioisotope into another nuclide, often called a daughter nuclide. All three decay processes create ionizing radiation, a type of high-energy radiation that can be damaging to living tissue.

In alpha decay, also called alpha emission, the radioisotope emits two protons and two neutrons as a helium-4 nucleus (also known as an alpha particle). This causes the radioisotope's mass number to go down by four and its atomic number to go down by two.

Beta decay, also called beta emission, is the emission of an electron from a radioisotope as one of its neutrons turns into a proton. This does not change the nuclide's mass number, but does increase its atomic number by one. There is also a kind of beta decay that is almost an inverse of the first: the nuclide emits a positron (the positively charged antimatter partner of an electron), and one of its protons turns into a neutron. This lowers the nuclide's atomic number by one. Both the positron and electron would be considered beta particles.

A special kind of beta decay is called electron-capture beta decay: One of the nuclide's innermost electrons is captured by a proton in the nucleus, turning the proton into a neutron and emitting an ultra-tiny, super-fast particle called an electron neutrino.

Radioactivity is usually measured in one of two units: the becquerel (bq) and the curie. Becquerels are the standard (SI) units of radioactivity, and represent a rate of one decay per second. Curies are based on the number of decays per second of one gram of radium-226, and are named after the celebrated radioactivity scientist Marie Curie. Her discovery of radium's radioactivity led to the first use of medical x-rays.

The half-life of a radioactive isotope is the average amount of time it takes about one-half of the atoms in a sample of radioisotope to decay. Different radioisotopes decay at different rates and can have wildly differing half-lives; these half-lives can be as short as a few microseconds, such as in the case of polonium-214, and as long as a few billion years, such as uranium-238.

The important concept is that a given radioisotope will ​always​ decay at the same rate. Its half-life is an inherent characteristic.

It may seem strange to characterize an element by how long it takes half of it to decay; it makes little sense to talk about the half-life of a single atom, for instance. But this measure is useful because it is not possible to determine exactly which nucleus will decay and when – the process can only be understood statistically, on average, over time.

In the case of one atomic nucleus, the common definition of half-life can be inverted: the probability of that nucleus decaying in less time than its half-life is about 50%.

There are three equivalent equations that give the number of nuclei remaining at time ​t​. The first is given by:

\(N(t) = N_0(1/2)^{t/t_{1/2}}\)

Where ​_t_1/2​ is the half-life of the isotope. The second involves a variable ​τ_​, which is called the mean lifetime, or the characteristic time:

\(N(t) = N_0e^{-t/τ}\)

The third uses a variable ​λ​, known as the decay constant:

\(N(t) = N_0e^{-λt}\)

The variables ​_t_1/2​, ​τ​ and ​λ_​ are all related by the following equation:

\(t_{1/2} = ln(2)/λ = τ × ln(2)\)

Regardless of which variable or version of the equation you use, the function is a negative exponential, meaning it will never reach zero. For each half-life that passes, the number of nuclei is halved, becoming smaller and smaller but never quite vanishing – at least, this is what happens mathematically. In practice, of course, a sample is made up of a finite number of radioactive atoms; once the sample is down to a single atom, that atom will eventually decay, leaving no atoms of the original isotope behind.

Scientists can use radioactive decay rates to determine the ages of old objects or artifacts.

For example, carbon-14 is constantly replenished in living organisms. All living things have the same ratio of carbon-12 to carbon-14. That ratio changes once the organism dies because the carbon-14 decays while the carbon-12 remains stable. By knowing the decay rate of carbon-14 (it has a half-life of 5,730 years), and measuring how much of the carbon-14 in the sample has transmuted into other elements relative to the amount of carbon-12, then it is possible to determine the ages of fossils and similar objects.

Radioisotopes with longer half-lives can be used to date older objects, although there must be some way to tell how much of that radioisotope was in the sample originally. Carbon dating can only date objects fewer than 50,000 years old because after nine half-lives, there is usually too little of the carbon-14 remaining to take an accurate measure.

If the half-life of seaborgium-266 is 30 seconds, and we start with 6.02 × 10²³ atoms, we can find how much is left after five minutes by using the radioactive decay equation.

To use the radioactive decay equation, we plug in 6.02 × 10²³ atoms for ​_N₀​, 300 seconds for ​t​ and 30 seconds for ​t_1/2_​.

\((6.02 × 10^{23})(1/2)^{(300/30)} = 5.88 × 10^{20}\)

What if we only had the beginning number of atoms, the final number of atoms, and the half-life? (This is what scientists have when they use radioactive decay to date ancient fossils and artifacts.) If a sample of plutonium-238 started with 6.02 × 10²³ atoms, and now has 2.11 × 10¹⁵ atoms, how much time has passed given that the half-life of plutonium-238 is 87.7 years?

The equation we have to solve is

\(2.11\times 10^{15}=(6.02\times 10^{23})(1/2)^{\frac{t}{87.7}}\)

and we must solve it for ​t​.

Dividing both sides by 6.02 × 10²³, we get:

\(3.50\times 10^{-9}=(1/2)^{\frac{t}{87.7}}\)

We can then take the log of both sides and use the rule of exponents in log functions to get:

\(-19.47 = (t/87.7)log(1/2)\)

We can solve this algebraically to obtain t = 2463.43 years.