Radioactive Decay: Half-Life, Average Life, and Decay Rate

half-life = (mean lifetime) * ln(2)

(2) = 0.6931

Why are half-life and average life different?
Problem Examples


To get the mean (or average) lifetime of a radioactive element, you take a sample of the radioactive atoms and wait for all of them to decay away, and keep track of how long each atom lasts. The sum of all the lifetimes of the atoms, divided by the original number of atoms, is the mean lifetime. In other words, the mean lifetime is simply the arithmetic average of the lifetimes of the individual atoms. The lifetime units are time units such as seconds or years.

The half-life of a radioactive element is the amount of time it takes for half of a sample of the element to decay away. It is smaller than the mean lifetime by a factor of ln(2), the natural logarithm of 2.

half-life = (mean lifetime)*ln(2) mean lifetime = (half-life) / ln(2)       ln(2) = 0.6931      

The half-life units are time units such as seconds or years.

The decay rate or decay constant is the fraction of the total mass that decays in one unit of time. It is equal to the inverse of the mean lifetime:

decay rate = 1 / (mean lifetime) mean lifetime =  1 / (decay rate)

For example, radium-226 has a half-life of 1,602 years, an mean lifetime of (1,602)/ln2 = 2,311 years, and a decay rate of 1/(2,311) = 0.000433 per year. In other words, 0.0433 percent of the radium decays away each year, or 433 parts per million per year. The decay rate units are the inverse of time units, for example, years-1 ("per year") or sec-1 ("per second"). 

To calculate the decay rate in becquerels (atoms per second) for a given mass of a radioactive element sample, do the following:
  1. Take the half-life and divide by ln2 (0.6931) to get the mean lifetime; convert the time units to seconds; and take the inverse to get the decay rate per second. For radium-226, the decay rate is 0.000433 per year / [ (365.25 days/year) * (24 hours/day) * (60 minutes/hour) * (60 seconds/minute) ] = 1.37 x 10-11 per second

  2. Take the mass of the element sample in grams, divide by the atomic mass to get moles, then multiply by Avagadro's number (6.02 x 1023) to get the number of atoms. For example, for 1.00 gram of radium-226, the number of atoms is [1.00 gram / (226 grams/mole)] x (6.02 x 1023 atoms/mole) =  2.66 x 1021 atoms

  3. Multiply the number of atoms by the decay rate per second to get the number of atoms decaying per second. This is the decay rate in becquerels. For 1.00 gram of radium-226, the decay rate is 2.66 x 1021 atoms x 1.37 x 10-11 per second = 3.65 x 1010 atoms per second
To convert the decay rate in becquerels to curies, divide by 3.7 x 1010. For example, for 1.00 gram of radium-226, the activity is 3.65 x 1010 / 3.7 x 1010 =  0.99 curie. The fact that this is close to 1.00 is not a coincidence. The original definition of the curie unit was based on the activity of radium emanation (radon) in equilibrium with 1 gram of radium. The difference is a rounding error.

Why are the half-life and average life different?

Imagine that you put 100 radioactive atoms in a box and observe them decay, one by one, until they are all gone. If you add up the lifetimes of the 100 individual atoms and divide the sum by 100, you get the mean lifetime, also known as the average lifetime, often designated τ (Greek letter tau). On the other hand, the half-life is the median lifetime -- half of the atoms live longer and half live shorter than this time. This is the time at which 50 of the atoms have decayed and 50 remain in the box.

The decay rate, often designated λ (Greek letter lambda), is the fraction of the total mass that decays in one unit of time. It is equal to the inverse of the mean lifetime.

Why are the average and median lifetimes different? There are always a few Methuselah atoms that live much longer than usual, purely by chance. These atoms pull up the average, but they have no effect on the median. Meanwhile, the shortest time that an atom can live is zero. There are no atoms with negative lifetimes that could pull the average down. So the average lifetime is somewhat longer than the median.

If you multiply the average lifetime by the natural logarithm of 2 (the number 0.6931), you get the half-life. You can read more about half-life, mean lifetime, and exponential decay in any algebra II book or the Wikipedia article on half-life.

Radioactive Decay Calculation Problem Examples

The following problems and solutions demonstrate the calculation of decay rates:

Decay of carbon-14 used in radioactive dating

A sample of wood taken from a freshly felled tree contains 10 grams of carbon. How many atoms of carbon-14 are decaying each second in the sample? The abundance of carbon-14 in atmospheric carbon and in living matter is 1.2 part in a trillion (1.2 x 10-12). The half-life of carbon-14 is 5,700 years. Solution

Decay rate of one gram of uranium

How many atoms decay each second in a 1.000 gram sample of natural uranium? Solution

Origin of atmospheric argon

The Earth's crust is about 2.1% potassium by weight. The crust has a total mass of about 2.4 x 1022 kg. All natural potassium is 0.012% potassium-40, a radioactive isotope with a half-life of 1.2 billion years; the rest consists of stable isotopes of potassium. Potassium-40 has two ways to decay, producing calcium-40 89% of the time and argon-40 11% of the time.
  1. What is the crustal abundance of potassium-40 in parts per million?
  2. What was the crustal abundance of potassium-40 just after the Earth formed, 4.5 billion years ago? 
  3. How much potassium-40 has decayed away in the last 4.5 billion years, in kg?
  4. How much argon-40 was produced by the decay of potassium-40 in the Earth's crust?
  5. How much argon now exists in the atmosphere?

More information about radioactivity:

How Much Astatine Is Present in a Uranium Ore Rock?

Secular Equilibrium and Radioactive Decay

2013 Gray Chang