Earth's Atmosphere

The nuclei of some atoms can decay. An example is

• Each second, there is a certain small chance that any given K^40 nucleus will decay.
• 88% of the time, it decays into an calcium nucleus.
• 12% of the time, it decays into an argon nucleus.

Argon is interesting because it has very different chemistry from potassium.

• The potassium nucleus had charge + 19.
• So it was surrounded by 19 electrons.
• An argon nucleus has charge + 18.
• So it is surrounded by only 18 electrons.
• This makes argon chemically different from potassium.
• In fact, potassium is highly reactive, while argon is inert!

How often does the decay happen?

• Decay of any individual K nucleus is a matter of chance.
• In any given second, a K nucleus has a 1 in 4.1 x 10^16 chance of decaying.
• In 1.3 x 10^9 years, a K nucleus has a 1 in 2 chance of decaying.
• Potassium has a half life of 1.3 x 10^9 years.

So

• After 1 half life, 1/2 of the K atoms are still K atoms.
• After 2 half lives, 1/4 of the K atoms are still K atoms.
• After 3 half lives, 1/8 of the K atoms are still K atoms.
• After 4 half lives, 1/16 of the K atoms are still K atoms.

Here is a picture, in which represents either of the decay products, calcium or argon.

Here is a puzzle. K^40 atoms are red. Ar^40 and Ca^40 atoms are blue. For the atoms shown, assuming that you started with just K atoms and no atoms have escaped, how many half lives have elapsed?

Answer: There are 20 total K and Ar and Ca atoms, and 5 K atoms are left. 5 is 1/4 of 20. To get to 1/4 takes 2 half lives.

---