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Activity of 1 g of carbon if 1 in 10^12 atoms are carbon-14

  1. Feb 6, 2017 #1
    1. The problem statement, all variables and given/known data
    Carbon-14 decays by β emission and has a half-life of 5570 years.
    1. What is the decay constant of carbon-14?
    2. What is the activity of 1 g of carbon if 1 in 1012 atoms are carbon-14?
    3. After what time will the activity per gram have fallen to 3 Bq?

    2. Relevant equations
    λt½ = ln(2)
    A = A0e-λt
    A = λN

    3. The attempt at a solution
    Question 1

    λt½ = ln(2)
    λ= ln(2)/5570
    = 1.24 * 10-4 year-1

    Question 2
    Mr of carbon-14 is 14 g mol-1
    1 mol = 6.022*1023 atoms
    Atoms in 1 g = (6.022*1023)/14
    = 4.30*1022 atoms

    Don't know what to do from here. Any help would be appreciated.

    Thanks
     
  2. jcsd
  3. Feb 6, 2017 #2

    gneill

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    Staff: Mentor

    Hi WhiteWolf98, Welcome to Physics Forums!

    Remember that not all the atoms are C14. What's the main isotope and how big a contribution does C14 make to the given quantity of the carbon?
     
  4. Feb 6, 2017 #3
    Main isotope of carbon is carbon-12. Do I just divide the number of atoms by 1*10-12?
     
  5. Feb 6, 2017 #4

    gneill

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    Staff: Mentor

    Sure. But first calculate the number of atoms using the overwhelmingly more abundant C12 mass.

    If you ever come across a case where the ratios of isotope abundance are closer to unity you'll have to use a more subtle mathematical approach, but in this case this approximation is all but perfect.
     
  6. Feb 6, 2017 #5
    I don't understand what carbon-12 has to do with this question; the question is asking about carbon-14. Wouldn't introducing more isotopes make it more complicated than it already is?
     
  7. Feb 6, 2017 #6

    gneill

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    Staff: Mentor

    The question asks about 1 gram of carbon, and says that only 1 in 1012 atoms of it are C14. So what is the rest of it made up of? Will C14 make up the whole of the 1 gram mass?
     
  8. Feb 6, 2017 #7
    It seems it does. Well, I have no idea anymore; am I supposed to be working out the activity of carbon-12 or carbon-14? There lies also the existence of carbon-13, so if you consider that too, then this question is unanswerable. I will have to talk with the mastermind who created the question!
     
  9. Feb 6, 2017 #8

    gneill

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    Staff: Mentor

    1 in 1012 is an insignificant contribution to the overall mass. C13 makes a much larger contribution than C14 percentage wise, but it is still small compared to C12 which is the overwhelmingly largest contributor to the makeup of a given sample (unless it's specially prepared to enhance some particular isotope).

    For this question you can assume that the only measurable contribution to the mass of the 1 gram is C12. Work out the number of atoms of C12 that would represent. You won't introduce any significant error by assuming that 1 x 10-12 of them are actually the C14 that you're interested in.
     
  10. Feb 6, 2017 #9
    In other words, the number of carbon-14 atoms is negligible? And all the information given about carbon-14 serves as a red herring?
     
  11. Feb 6, 2017 #10

    gneill

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    It is negligible in that it makes up an insignificant portion of the mass of 1 gram of carbon. No scale could detect its contribution. So when measuring out the 1 gram sample, you can consider it to be pure C12. In order to determine the number of C14 atoms that came along for the ride, you look at the abundance ratio. One in 1012 of the atoms in the sample will be C14 rather than C12.
     
  12. Feb 6, 2017 #11
    I appreciate the help, thanks.
     
  13. Feb 6, 2017 #12

    gneill

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    Staff: Mentor

    You're very welcome.
     
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