Some Radiological Dating with Stoichiometry

[phost]

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

Homework Statement

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 × 10⁹ 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?)

Homework Equations

k = In 2/t_1/2

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.

 

[daveb]

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?

 

[phost]

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.

 

[Borek]

Potassium and argon are only a small part of the sample, so their masses don't have to add to 500 mg.

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?

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.

 

[phost]

Potassium and argon are only a small part of the sample, so their masses don't have to add to 500 mg.

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?

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.

I hadn't considered that. That was really helpful, thanks a lot :D