TOPIC 2: Half-life Equation

Now let’s talk about half-life.

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The concept of half-life is a new term that tends to confuse people.  The radiation that is given off by a radioisotope results from the transformation or decay of an atom into another element.  If we were to select one particular atom, we would not know when it would decay.  It could happen now, next week, or year in the future.  It is impossible to tell.  However, if we had a large population of atoms, We can measure how long it takes for half of them to decay.  Does this mean that the other half will decay in the next half-life?  No.  It means that of the atoms that are left, half of them will decay in the next half-life. Each radionuclide has its own half-life.  Some radionuclides have half-lives measured in a fraction of a second and some for billions of years.

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For example, after one half-life, fifty percent of the original activity remains.  Then, after another half-life (half of a half), only twenty-five percent of the original activity is left.  In the mathematical decay model you can keep dividing the activity in half indefinitely, but in the physical world that is not the case. In a finite amount of radioactive material with a finite number of unstable atoms, eventually a point will be reached where one unstable atom will remain and it will decay leaving nothing.  In practical terms, after many half-lives there will be no detectable activity left.  The amount of radioactive material will enter a concept of “nothingness” where the amount of radioactivity cannot be measured, is no longer a hazard, is not of any environmental consequence and is not regulated.

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Let’s take a look at how we can use this concept of decay in the real world.  Carbon is a very common element and essential for any living thing. C-14 is created in the atmosphere and has a very constant concentration throughout the planet in the form of CO2.  Plants take in CO2.  Animals and humans eat the plants.  Humans also eat the animals that eat the plants.  So, all living things have a known concentration of C-14 in their tissues.

At death, no more C-14 is taken into the tissue.  C-14 dating uses the half-life in determining how long it has been since persons, plants, or animals died.  So, how can the age of an Egyptian mummy be determined?  When the Egyptian died, they no longer consumed food.  Thus, the person stopped replenishing the amount of radioactive carbon in their body.   The existing radioactive C-14 began to decay.  Since C-14 has a half-life of 5,730 years, a sample of tissue can be analyzed to determine the C-14 concentration.  If the amount of radioactivity is one-half that of a living human being, then 5,730 years has passed since our Egyptian died. There is a decay calculation formula that can provide the number of years that has passed.

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The Math Primer goes into more detail about the half-life equation.  There are 4 variables in this equation:

At is the activity after some period of time, t.

Ao is the original activity.

Since At is that activity some time AFTER Ao, At will always be smaller than Ao.

t1/2 is the Half-Life of the radioactive material.  The Half-Life is unique to each radionuclide.

The time, t, and the half-life, t1/2, must be in the same units.  If the time lapsed is in days, then the half-life has to be defined in days too.

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The Math Primer is an intrigal part of this training.  Download and use it as a working tool during your training.