Calculation of Potassium-40 Decay Into Argon-40 in the Earth's Crust

Problem

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?
   

Ignore the possible gain or loss of material over time due to mixing between the crust and the mantle.

Solution

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,

  N₀ = 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.

Note: 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

Potassium-argon 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 (11%).

Here is a typical rock dating problem:

Question:

A rock sample is found to contain 1.00 gram of potassium and 6.0 micrograms of argon. How old is the rock?

Answer:

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 t:

 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[ N(t)/N₀ ] =  log [ (1/2)^{t/(half-life)} ]
                       =  [ t/(half-life) ] log (1/2)


 t = (half-life) log[ N(t)/N₀ ] / log (1/2)

   = (1.2 billion years) log[120/175] / log(0.5) = 2.3 billion years

Note: Your answer might be slightly different due to rounding.
 

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