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Carbon Dating

Practice Carbon Dating
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Carbon Dating

Students will develop an understanding of how carbon dating works and also learn to solve for the age of an artifact using the carbon dating technique.

Key Equations

M = M_0 (\frac{1}{2})^{\frac{t}{t_H}}

The amount of mass M of the substance surviving after an original \mathit{M}_0 nuclei decay for time t with a half-life of \mathit{t}_H .

Radioactive carbon dating is a technique that allows scientists to determine the era in which a sample of biological material died. A small portion of the carbon we ingest every day is actually the radioactive isotope ^{14}\mathrm{C} rather than the usual ^{12}\mathrm{C} . Since we ingest carbon every day until we die (we do this by eating plants; the plants do it through photosynthesis), the amount of ^{14}\mathrm{C} in us should begin to decrease from the moment we die. By analyzing the ratio of the number of ^{14}\mathrm{C} to ^{12}\mathrm{C} atoms in dead carbon-based life forms (including cloth made from plants!), we can determine how long ago the life form died.

Example 1

Question : The half-life of ^{239} Pu is 24,119 years. You have 31.25 micrograms left, and the sample you are studying started with 2000 micrograms. How long has this rock been decaying?

Answer : We will use the equation for time and simply plug in the known values. t=t_H\frac{\ln{\frac{N}{N_0}}}{\ln{\frac{1}{2}}}=24119\mathrm{y}\frac{\ln{\frac{31.25\mu\mathrm{g}}{2000\mu\mathrm{g}}}}{\ln{\frac{1}{2}}}=144,700\mathrm{years}

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  1. The half-life of ^{239}\mathrm{Pu} is 24,119 years. You have 31.25 micrograms left, and the sample you are studying started with 2000 micrograms. How long has this rock been decaying?
  2. A certain fossilized plant is 23,000 years old. Anthropologist Hwi Kim determines that when the plant died, it contained 0.250 \;\mathrm{g} of radioactive ^{14}\mathrm{C} (t_H = 5730 \;\mathrm{years}) . How much should be left now?
  3. Jaya unearths a guinea pig skeleton from the backyard. She runs a few tests and determines that 99.7946 % of the original ^{14}\mathrm{C} is still present in the guinea pig’s bones. The half-life of ^{14}\mathrm{C} is 5730 years. When did the guinea pig die?
  4. You use the carbon dating technique to determine the age of an old skeleton you found in the woods. From the total mass of the skeleton and the knowledge of its molecular makeup you determine that the amount of ^{14}\mathrm{C} it began with was 0.021 grams. After some hard work, you measure the current amount of ^{14}\mathrm{C} in the skeleton to be 0.000054 grams. How old is this skeleton? Are you famous?
  5. Micol had in her lab two samples of radioactive isotopes: ^{151}\mathrm{Pm} with a half-life of 1.183 days and ^{134}\;\mathrm{Ce} with a half-life of 3.15 days. She initially had 100 \;\mathrm{mg} of the former and 50 \;\mathrm{mg} of the latter.
    1. Do a graph of quantity remaining (vertical axis) vs. time for both isotopes on the same graph.
    2. Using the graph determine at what time the quantities remaining of both isotopes are exactly equal and what that quantity is.
    3. Micol can detect no quantities less than 3.00 \;\mathrm{mg} . Again, using the graph , determine how long she will wait until each of the original isotopes will become undetectable.
    4. The Pm goes through \beta- decay and the Ce decays by means of electron capture. What are the two immediate products of the radioactivity?
    5. It turns out both of these products are themselves radioactive; the Pm product goes through \beta- decay before it becomes stable and the Ce product goes through \beta+ decay before it reaches a stable isotope. When all is said and done, what will Micol have left in her lab?

Answers to Selected Problems

  1. t = 144,700 years
  2. 0.0155 \;\mathrm{g}
  3. 17 years
  4. 49,000 years
  5. .

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