# Isotopes question

I am legitimately stuck. "A sample of biotite contains 788 ppm of $\ce{^40K}$ and 630 ppm of $\ce{^40Ar}$. What % of the original parent isotope remains?"

an explanation of how this is solved would help tremendously. thanks.

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Welcome @Kyle. Your question had some formatting issues I fixed that would have confused anyone trying to answer this question, the main one being your inconsistent use of capitalization for the element symbols. Also, if you could tell us where you are stuck on this problem and what you know about radioisotope decay, you are more likely to get a more helpful answer. – Ben Norris Dec 12 '12 at 0:14
It would be faster if you don't assume that people know what biotite and the parent isotope are. Also the relevant radioactive decay rates... – sencer Dec 12 '12 at 0:17
Thanks @BenNorris I couldn't figure that out for the life of me. – jonsca Dec 12 '12 at 4:09
@jonsca - it did take me a few minutes to figure out what 40k and 40AR were, but the title did mention isotopes – Ben Norris Dec 12 '12 at 21:25

Since potassium-40 has 21 neutrons and argon-40 has 22 neutrons, decay must involve a proton from potassium combining with an electron to form a neutron, or the other way around. Since the masses of proton and electron add up to less than that of a neutron, the neutron must be decaying to proton and electron, giving off the extra mass as energy. Thus argon is the parent isotope (if you define parent isotope as the one that gives off energy when decaying to the daughter).

Assuming the problem is talking about mole % and not mass %, you can simply assume that all of the potassium came from argon originally. So, calculate the total and find the percent of argon that remains:

$$\frac{630 \ \text{ppm}}{630 \ \text{ppm} + 788 \ \text{ppm}}$$

Notice that the ppm's cancel, and you get a decimal that can be converted to percent.

If you were asked to assume that the higher element number is the parent isotope, the same procedure applies, but put potassium on top instead of argon.

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Even if the original question was "by mass" and not by moles, this calculation works - you would just get mass 5 and not mole %. – Ben Norris Dec 13 '12 at 11:44
There is an error in the above analysis. Argon-40 is stable and does not decay. K-40 is not stable and one of its decay paths leads to Argon-40. So we need to have the 788 ppm in the numerator, not the 630 ppm. – Paul J. Gans Dec 15 '12 at 2:54