Just as carbon dating is used to measure the age of organic material, Argon-40 can be used to measure the age of rocks. A volcanic eruption melts a large chunk of rock, and all gasses are expelled. After cooling, Argon-40 accumulates from the ongoing decay of potassium-40 in the rock (t_1/2 = 1.25E9 years). When a piece of rock is analyzed, it is found to contain 1.38 mmol of potassium-40 and 1.14 mmol of Argon-40. How long did the rock cool?

3.77 mg of K-40 decayed into Ar-40.

Data:

1) K-40, Ca-40, Ar-40: all three have the same atomic mass

2) 90% of the potassium-40 will decay into calcium-40

3) 10% of the potassium-40 will decay into argon-40.

4) K-40 inside the rock = 0.81 mg

5) Ar-40 trapped = 0.377 mg

Soltuion:

1) 0.377 mg of Ar-40 is the 10% of the mass of the K-40 that decayed

=> x * 10% = 0.377 mg => x * 0.1 = 0.377mg

=> x = 0.377 mg / 0.1 = 3.77 mg

That means that 3.77 mg of K-40 decayed into Ar-40. And this is the answer to the question.

Additionaly, you can analyze the content of all K-40 and Ca-40, to understand better the case.

2) The mass of the K-40 that decayed into Ca-40 is 9 times (ratio 9:1) the amount that decayed into Ar-40 =>

mass of K-40 that decayed into Ca-40 = 9 * 0.377 = 3.393 mg

3) Total amount of K-40 that decayed = amount that decayed into Ar-40 + amount that decayed into Ca-40 = 0.377mg + 3.393mg = 3.77 mg

4) Original amount of K-40 = amount of K-40 that decayed + amount of K-40 present in the rock = 3.77mg + 0.81 mg = 4.58 mg

5) amount of K-40 that decayed into Ar-40 as percent

% = [3.77 mg / 4.58mg] * 100 = 82.31 %.