Cosmic rays impinging on our atmosphere generate radioactive \({ }^{14} \mathrm{C}\) from \({ }^{14} \mathrm{~N}\) nuclei. \(^{78}\) These \({ }^{14} \mathrm{C}\) atoms soon team up with oxygen to form \(\mathrm{CO}_{2}\), so that plants absorbing \(\mathrm{CO}_{2}\) from the air will have about one in a trillion of their carbon atoms in this form. Animals eating these plants \(^{79}\) will also have this fraction of carbon in their bodies, until they die and stop cycling carbon into their bodies. At this point, the fraction of carbon atoms in the form of \({ }^{14} \mathrm{C}\) in the body declines, with a half life of 5,715 years. If you dig up a human skull, and discover that only one-eighth of the usual one-trillionth of carbon atoms are \({ }^{14} \mathrm{C}\), how old do you deem the skull to be?

Step by step solution

01

Identify initial and final fractions

The initial fraction of carbon atoms as \({ }^{14} \mathrm{C}\) when the organism was alive is given to be one in a trillion (1/1,000,000,000,000). In the skull, the final fraction of carbon atoms as \({ }^{14} \mathrm{C}\) is one-eighth of the initial fraction i.e., (1/8)*(1/1,000,000,000,000).

02

Calculate the ratio between initial and final fractions

We need to find the ratio between the final and initial fractions of \({ }^{14} \mathrm{C}\). This ratio is equal to the fraction of carbon atoms left, after a certain number of half-lives. Ratio = (Final fraction) / (Initial fraction) = [(1/8)*(1/1,000,000,000,000)] / (1/1,000,000,000,000)

03

Ratio simplification

Now, let's simplify the ratio: Ratio = (1/8)*(1/1,000,000,000,000) / (1/1,000,000,000,000) = (1/8)

04

Calculate the number of half-lives

We will use the following formula for radioactive decay, which relates the remaining fraction of \({ }^{14} \mathrm{C}\) and the number of half-lives (n): Remaining fraction = (1/2)^n Here, the remaining fraction is the ratio calculated in step 3. Let's plug in the values and solve for n: (1/8) = (1/2)^n Take the logarithm of both sides: log(1/8) = n * log(1/2) Therefore, n = log(1/8) / log(1/2)

05

Calculate the age of the skull

Now that we have the number of half-lives (n) that have passed, we can calculate the age of the skull using the following formula: Age = n * Half-life Plug in the values: Age = (log(1/8) / log(1/2)) * 5,715 years After solving the equation, we get the age of the skull: Age ≈ 17,145 years So, the age of the skull is approximately 17,145 years old.

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