Potassium-40 can decay into either calcium-40 or argon-40. All three of these atoms have essentially the same weight. Ninety percent of the potassium-40 will decay into calcium-40, and only ten percent will decay into argon-40. When argon-40 is produced by the radioactive decay of potassium-40 inside a rock, the argon-40 produced by the decay is a gas and is trapped inside the rock. The amount argon-40 trapped in a rock can be measured by grinding up the rock and capturing the liberated argon-40 gas.
Suppose the amount of potassium-40 inside a rock is measured to be 0.81 milligrams, and the amount of argon-40 gas trapped in the rock is measured to be 0.377 milligrams.
1. How much of the potassium-40 that was originally present inside the rock has undergone radioactive decay to produce argon-40?
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
