[PaigeNoel]

The Problem:

One of the isotopes of potassium, ⁴⁰K, occurs at 120 ppm in natural samples. This 0.012% naturally decays with a 1.28X10^9 year half life into argon-40 (4⁴⁰Ar).
If a rock stays crystallized the argon formed will be trapped in the rock matrix. After bringing the same to the lab the argon and potassium can be measured experimentally by isotope ratio mass spectrometry.

Assuming no argon-40 is present in rock when it is formed, if the following plot [see https://drive.google.com/file/d/0B-BclHOW2R0QemZKOFM0WXJJOFc0WS0zay1saTVzYXZ4Qmdr/view?usp=sharing and https://drive.google.com/file/d/0B-BclHOW2R0QaEEwc1NpSDJQZURkOWJQSFZXcHNDOU93SHFz/view?usp=sharing] represents the amount of potassium-40 and argon-40 in a sample of sediment, what is the age of the rock?

The Equations Provided:

ln ([A] / [A]_o) = - kt

t_1/2=0.693/k


The Attempt at a Solution:

k = 0.693/t_1/2

k = 0.693/(1.29X10^9)
 
k = 5.41X10^-10

Ar/K = 0.00052/0.001  I found this using peak heights from above spectrum; I'm mainly confused about what I'm supposed to be plugging in to the equation and how to find that information from the spectrum provided.

= 52%

ln([A]/[A]_o)=-kt

ln([52][0.012]_o)=-(5.41X10^-10)t

t = -1.55X10^10 which doesn't make any sense

I know this is basic stuff and I should know how to do it, but I can't seem to figure out what I'm doing wrong.
Using the isotope ratio mas spectrum, I found the heights of both peaks and divided the height of the ⁴⁰Ar peak by the height of the ⁴⁰K peak. This ends up being 52%. I'm relatively certain that I wasn't supposed to plug this number in to the equation; here is where I'm stumped. If most natural samples are 0.012% 40K, should I multiple 48% (100-52=48) by 0.012% to get the percentage of the sample that is still ⁴⁰K? Even then, the number ends up being too large. But I thought that I'm supposed to be using concentrations in the ln equation. In any case, the percentage of ⁴⁰K should have gone down because some should have decayed, right?  I'm lost and there's going to be a problem like this on our exam tomorrow. I appreciate any help. Thank you!

[Burner]

I only know the basics of radioactive decay, but I spotted something that you may want to double-check on:

k = 0.693/t_1/2

k = 0.693/(1.29X10^9)
 
k = 5.41X10^-10

What is your unit of k? In this case the unit is year^-1. You have to be aware of this.

ln([A]/[A]_o)=-kt

ln([52][0.012]_o)=-(5.41X10^-10)t

A₀ is the initial activity of the material so it should be larger than A. Check if you have substituted something wrong. Also, the 0.012% refers to ⁴⁰K but the 52% seems to be referring to ⁴⁰Ar(maybe I am wrong), you can also double-check it.

[mjc123]

The 0.012% is a red herring, you don't need this number.
If the Ar comes from K, and all the Ar is retained, then if the amount of Ar is 52% of the amount of K remaining, what is the remaining K as a fraction of the original amount?

[Enthalpy]

The problem is flawed.

⁴⁰K decays mainly into ⁴⁰Ca, and only 10.72% of the time to ⁴⁰Ar.
1.25 billion years refers to all decay modes together.
https://en.wikipedia.org/wiki/Potassium-40
and Handbook of Chemistry and Physics, table of the isotopes.

Sorry for bringing mess.

Now, you have to decide whether it's a good idea or not to know more that your professor. My experience tells it's a very bad idea, and you get a better mark if doing as if you didn't notice the flaw.