Does Carbon-14 Dating Agree w/ Mt. Vesuvius Eruption?

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Homework Help Overview

The discussion revolves around the application of carbon-14 dating to determine the age of bones uncovered from the eruption of Mount Vesuvius in 79 AD. Participants explore the relationship between the percentage of carbon-14 remaining in the bones and the historical timeline of the eruption.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants examine the calculations related to carbon-14 decay and its implications for dating the bones. There are attempts to verify the original poster's calculations and to suggest alternative methods for determining the age based on the half-life of carbon-14.

Discussion Status

Some participants provide guidance on the correct approach to calculating the decay of carbon-14, while others confirm the accuracy of the calculations presented. Multiple interpretations of the problem are being explored, particularly regarding the assumptions about the timing of the measurements and the implications for historical accuracy.

Contextual Notes

The original poster's calculations assume a specific year for the measurement of carbon-14, which is not explicitly stated in the problem. This assumption is questioned by other participants, leading to further exploration of the implications of different timeframes on the dating results.

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Homework Statement



Nitrogen in the upper atmostphere is convereted by radiation to carbon 14
the half-life of carbon, \tau = 5730 years
carbon 14 makes up a known proportion of living plants and animals, after they die, the proportion of carbon 14 decays.

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History records that an eruption on Mount Vesuvius buried the city Pompeli in 79 AD
excavations uncovered bones and these contain 79.17 % of the original carbon 14:

the question:
Does the radiocarbon dating agree with the historical record?------------------------

The Attempt at a Solution



I'm going to assume that the percentage of undecayed mass, 79.17% was measured in 2010
as the question doesn't say when

2010 - 79 = 1931, the number of years passed

\frac{1931}{\tau} = \frac{1931}{5730} = 0.3369 % \tau

now that means it should be 100% - 33.69% of the original mass
= 0.663%,

So I'm going to say that the radiocarbon dating doesn't agree with the historical record, and that the bones uncovered had been of people that had died before the eruption on Vesuvius,

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Can someone confirm that I've done this correctly?
thanks
 
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No, sorry but you didn't do this correctly at all.

The way to approach half-life is:

x(t) = Ce^{-kt}

where C = x(0) and k is some constant you solve for. Well we know that

x(5730) = \frac{C}{2} = Ce^{-5730k}

So we can easily find k. Now that you've got your full equation, plug in t = 1931 to check if x(t) is about 0.7917*C.
 
x(t) = Ce^{-kt}

x(5730) = Ce^{-k5730} = \frac{C}{2}

ln(Ce^{-k5730}) = ln(\frac{C}{2})

ln(C) + (-k5730) = ln(C) - ln(2)

-k*5730 = -ln(2)

k = \frac{ln(2)}{5730} = 0.000120968094

putting into the equation, e^{-kt}
I get, e^(-0.000120968094 * 1931) =0.79168% != 0.7917% but close enough I reckon

so the radiocarbon dating does match up with the historical record
 
yep, that's perfect. any other questions?
 
Raskolnikov said:
yep, that's perfect. any other questions?

Nope, Thank you. =]
 
Since you are talking about a "half life" of 5730 years, that can be done more simply with x(t)= C(1/2)^{t/5730}

Solve x(t)= C(1/2)^{t/5730}= .7917 C so you need to solve (1/2)^{t/5730}= .7917 for t. If that is the "current year" (whenever the excavations were done) then excavations agree with the rule.