It's kind of baffling me when I'm encountering this question in this sub-chapter. It's just unusual. So I really need your help :D 1. The problem statement, all variables and given/known data If a rock sample was found to contain 1.16 × 10^{-7} mol of argon-40, how much potassium-40 (t_{1/2} = 1.3 × 10^{9} yr) would also have to be present for the rock to be 1.3 × 109 years old? See assumption in Problem 14.84. And the problem 14.84 question is ... A 500 mg sample of rock was found to have 2.45 × 10^{-6} mol of potassium-40 (t_{1/2} = 1.3 × 109 yr) and 2.45 × 10^{-6} mol of argon-40. How old was the rock? (Hint: What assumption is made about the origin of the argon- 40?) 2. Relevant equations k = In 2/t_{1/2} 3. The attempt at a solution I just find out that the both K and Ar in periodic table have a closely enough molecular mass, which is 40 g/mol (39,1 for K and 39,95 for Ar). But it just weird when the molecular mass is multiplied with each moles of Ar and K to find mass, because it doesn't add up for 500 mg. Also I don't have any idea what does the t_{1/2} works for. Of course we could find the rate constant from the equation before for it.
What happens if you assume all the argon-40 originates from the potassium-40 beta minus decay? How much potassium had to be present in the rock for that much to decay?
Well, if I assume that, it is plausible because potassium-40 will lose 1 electron and becoming argon-40 with beta minus decay. But I still don't understand it, was the assumption says the rock originally a pure potassium-40 back then and becoming argon-40 in the whole time? Or is it in the first time the potassium-40 had 4.9x10^{-6} mole and in the meantime because of the beta minus decay it changed and split into two parts, the half of the potassium-40 with 2.45x10^{-6} mole and the other half with the same mole but the argon-40? My apologize if I don't get the idea.
Potassium and argon are only a small part of the sample, so their masses don't have to add to 500 mg. That's what would happen exactly after half time. We assume rock was melted before and degassed, so all argon it contains now comes from the potassium-40 decay. And total number of atoms of both elements is constant.