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

ln(2) = 0.6931

The

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 = 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)/

To calculate the

- Take the half-life and divide by
*ln*2 (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

- 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 10
^{23}) 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 10^{23}atoms/mole) = 2.66 x 10^{21}atoms

- 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 10
^{21}atoms x 1.37 x 10^{-11}per second = 3.65 x 10^{10}atoms per second

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.

- Decay of carbon-14 used in radioactive dating

- Decay rate of one gram or uranium
- Origin of atmospheric argon

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
*

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

- What is the crustal abundance of potassium-40 in parts per million?

- What was the crustal abundance of potassium-40 just after the Earth formed, 4.5 billion years ago?
- How much potassium-40 has decayed away in the last 4.5 billion years, in kg?

- How much argon-40 was produced by the decay of potassium-40 in the Earth's crust?

- 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

How Much Astatine Is Present in a Uranium Ore Rock?

Secular Equilibrium and Radioactive Decay

©2013 Gray Chang