Anthropologists can estimate the age of a bone or other sample of...

Question

Anthropologists can estimate the age of a bone or other sample of
organic matter by its carbon-14 content. The carbon-14 in a living
organism is constant until the organism dies, after which carbon-
14 decays with first-order kinetics and a half-life of 5730 years.
Suppose a bone from an ancient human contains 19.5% of the
C-14 found in living organisms. How old is the bone?

Accepted Answer

Hey everyone in this example, we're told that an archaeologist wants to determine the age of a cell. Using carbon dating and carbon 14 will decay with first order kinetics. Were told that it has a half life of 5730 years. And we have a fossilized shell with 26.3% carbon found in the living shell. We need to determine the age of the shell. So because we are told that in the prompt, we have first order kinetics being followed, We're going to recall our first order integrated great Law. So we should recall that. That is going to be calculated by taking the natural log of our concentration of our atom at a given time, setting that equal to negative one times a rate constant K. Times the time in seconds, added to the natural log of our concentration of our atom initially. And so what we should do is calculate art rate constant K. So we should recall that the half life is equal to the natural log of two, divided by our rate constant K. And so we would reorganize this to sulfur K. So that the rate constant Is equal to the natural log of two divided by our half life. And so we would say that the rate constant K. Is equal to the natural log of two, divided by our half life given in the prompt as 5730 years. And this would give us a rate constant value equal to 1.2097 times 10 to the negative fourth power inverse years. So now that we know our value for the rate constant, we can determine our age of our shell. And we would say that that is going to equal our integrated rate law where we have the natural log of our percentage of carbon 14 in our fossilized shell after the half life. Which according to the prompt is 26.3%. But we're going to write that in as a decimal so that we have 0.2 63. And this is set equal to negative one times our rate constant, which we just calculated above as 1. times 10 to the negative fourth power inverse years. Which we will now multiply by time T. Which is what we're solving for for our age added to the natural log of 100%. We would assume of our carbon 14 being in the shell. So we would just write that in as one. And so what we would get here is that our time is equal to a value or the age is equal to a value of 1.10 times 10 to the fourth power years. So this would be the age of our shell found from carbon dating. So I hope that everything I explained was clear. If you have any questions, please leave them down below. Otherwise I will see everyone in the next practice video

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