Calculation of Potassium-40 Decay Into Argon-40 in the Earth's Crust
The following problem shows how the radioactive decay of potassium-40
explains the presence of argon in the Earth's crust and atmosphere. The
follow-up explanation shows how to calculate the age of a rock using
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.
- 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?
Ignore the possible gain or loss of material over time due to mixing between the crust and the mantle.
Here are the solutions with detailed calculations. Your answers might be slightly different due to rounding.
1. Crustal abundance of potassium-40
The crustal abundance of potassium is 2.1% or 0.021. Of this,
only 0.012% (.00012) is radioactive potassium-40. Thus, the crustal
abundance of potassium-40 is:
0.012 x 0.00012 = 1.4 x 10-6
= 1.4 parts per million at the present
2. Crustal abundance of potassium-40 when the Earth formed
The amount of a radioactive material existing at time N(t)
depends on the original amount N0
, the half-life
of the material, and elapsed time t
according to the following equation
N(t) = N0(1/2)t/(half-life)
If we take the formation of the Earth as time 0 and the present time as 4.5 billion years, then we have:
1.4 = N0(1/2)4.5/1.2
where 1.4 is the present abundance of potassium-40 in parts per million, N0
is the abundance at time 0 (when the Earth formed), and the time values are in billions of years. As a result,
= 1.4 / [(0.5)3.75
] = 19 parts per million potassium-40 when the Earth formed
3. Quantity of potassium-40 that has decayed
Shortly after the Earth formed, the abundance of potassium-40 was 19
parts per million. Now it is 1.4 parts per million. That means that 17.6
parts per million of the crust, in the form of potassium-40, has
decayed away over the past 4.5 billion years. The total decayed mass is equal to
the total mass of the crust multiplied by 17.6 parts per million:
(2.4 x 1022
kg) x (17.6 x 10-6
) = 4.2 x 1017 kg of potassium-40 decayed
4. Quantity of argon-40 produced by the decay of potassium-40
Potassium-40 decays to calcium-40 89% of the time and argon-40 11% of
the time. Therefore, the amount of argon-40 produced by the decay of
potassium-40 is 11% of the total mass of potassium that has decayed:
4.2 x 1017
kg x 0.11 = 4.6 x 1016 kg of argon-40 produced in the Earth's crust
5. Quantity of argon in the Earth's atmosphere
Argon is a common gas, making up 0.9% by volume or 1.3% by mass of the
atmosphere. Only nitrogen and oxygen (and sometimes water vapor,
depending on the humidity) are more plentiful in the atmosphere.
The total mass of the argon in the Earth's atmosphere is
(total mass of atmosphere) x 1.3% =
(5.1 x 1016
kg) x 0.013 = 6.6 x 1014 kg argon in the atmosphere
Thus, the amount of argon in the atmosphere is about 1.4% of the amount
of argon-40 produced in the Earth's crust over the past 4.5 billion
years. Most of the argon produced in the crust remains locked in the
rock. Only about 1.4% has escaped and entered the atmosphere.
The calculations have a precision of two significant digits. Your answers might be slightly different due to rounding.
Why is the decay of potassium-40 to argon-40 important?
The decay of potassium-40 to argon-40 explains why there is so much
argon in the atmosphere, compared with the other noble gases. Almost all
atmospheric argon (99.6%) is argon-40, whereas the argon in the
Sun and stars, produced by stellar nucleosynthesis
is mostly argon-36. This suggests that primordial argon is in the form
of argon-36 and essentially all of the argon in the atmosphere was
produced by the decay of potassium-40 to argon-40.
The amount of argon still trapped in the Earth's crust is the amount
produced minus the amount that has escaped into the atmosphere:
(4.6 x 1016
) - (6.6 x 1014
) = 4.5 x 1016 kg of argon-40 in the Earth's crust
The crustal abundance of argon is the mass of trapped argon-40 divided by the mass of Earth's crust:
(4.5 x 1016
kg)/(2.4 x 1022
kg) = 2 x 10-6
= 2 parts per million argon in the Earth's crust
Potassium-argon rock dating
dating is a technique used to determine the age of rocks. Because argon
is a noble gas, it does not form compounds and remains a gas, trapped
in the solid rock. When the rock is melted by volcanic processes, the
argon becomes mobile and typically separates from the molten rock. When
the rock solidifies, it starts off with no argon. As the solid rock
ages, the potassium in the rock decays, producing argon that is trapped
in the rock.
By measuring the amount of potassium and argon in a rock sample, the age
of the rock since it solidified can be determined. For example, a rock
taken from a fresh lava flow will have no argon, whereas a rock that is
1.2 billion years old (one potassium-40 half-life) will have an amount
of argon equal to 11% of the amount of potassium-40 remaining in the
rock. One half-life ago, there was twice as much potassium-40 in the
rock, an amount that has decayed away into calcium-40 (89%) and argon-40
Here is a typical rock dating problem:
A rock sample is found to contain 1.00 gram of potassium and 6.0 micrograms of argon. How old is the rock?
At the present time, all natural potassium is 0.012% potassium-40. Thus,
the quantity of potassium-40 currently in the rock sample is:
1.00 gram x 0.00012 = 0.00012 grams = 120 micrograms of potassium-40 at present
The 6.0 micrograms of argon came from the decay of potassium-40.
Potassium-40 decays to calcium-40 89% of the time and to argon-40 11% of
the time. Therefore, the quantity of potassium-40 that decayed to
produced the argon is:
(X micrograms potassium-40) x 0.11 = (6.0 micrograms argon-40)
X = 6.0/0.11 = 55 micrograms potassium-40 decayed to produce argon-40
55 micrograms of potassium-40 decayed away and 120 micrograms remain in
the rock. Therefore, when the rock solidified from a molten form, it contained 120 + 55 = 175
micrograms potassium-40 originally
Now we have the starting amount, the ending amount, and the half-life of
potassium-40, so we can use the decay formula
to find the elapsed time
N(t) = N0(1/2)t/(half-life)
Before we plug in the numbers, let's solve the equation for t
: First divide both sides by N0
N(t)/N0 = (1/2)t/(half-life)
Now take the base-10 logarithm of both sides:
] = log
] log (1/2)
t = (half-life) log
] / log (1/2)
= (1.2 billion years) log
[120/175] / log
(0.5) = 2.3 billion years
Your answer might be slightly different due to rounding.